User:Ageary/Theory of seismic waves

**Problems in Exploration Seismology and their Solutions**
Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Chapter 2 Theory of seismic waves

2.1 The basic elastic constants

2.1a Show that when the only nonzero applied stress is ${\sigma }_{ii}$ , Hooke’s law requires that the normal strains ${\varepsilon }_{yy}={\varepsilon }_{zz}$ , and that Poisson’s ratio ${\sigma }$ , defined as ${\sigma }=-{\varepsilon }_{yy}/{\varepsilon }_{xx}=-{\varepsilon }_{zz}/{\varepsilon }_{xx}$ , satisfies the equation

{\begin{aligned}{\sigma ={\frac {\lambda }{{2}\left({\lambda }+{\mu }\right)}}}.\end{aligned}}

(2.1a)

Background

Stress is force/unit area and is denoted by ${\mathrm {\sigma } }_{xy}$ , etc., where a force in the $$ x $$ -direction acts upon a surface perpendicular to the $$ y $$ -axis. The stresses ${\mathrm {\sigma } }_{xx}$ and ${\mathrm {\sigma } }_{xy}$ are, respectively, a normal stress and a shearing stress.

Stresses produce strains (changes in size and/or shape). If the stresses cause a point $P\left(x,\;y,\;z\right)$ to have displacements $\left(u,\;v,\;w\right)$ along the coordinate axes, the basic strains are given by derivatives of these displacements as follows:

{\begin{aligned}{Normal}\ {strains}:\ \,\ {\mathrm {\varepsilon } }_{xx}={\frac {{\mathrm {\partial } }u}{{\mathrm {\partial } }x}},{\quad }{\mathrm {\varepsilon } }_{yy}={\frac {{\mathrm {\partial } }v}{{\mathrm {\partial } }y}},{\quad }{\mathrm {\varepsilon } }_{zz}={\frac {{\mathrm {\partial } }w}{{\mathrm {\partial } }z}}.\end{aligned}}

(2.1b)

{\begin{aligned}{Shearing}\ {strains}:\ {\mathrm {\varepsilon } }_{xy}={\mathrm {\varepsilon } }_{yx}={\frac {{\mathrm {\partial } }v}{{\mathrm {\partial } }x}}+{\frac {{\mathrm {\partial } }u}{{\mathrm {\partial } }y}},{\quad }{\hbox{and so on for }}{\mathrm {\varepsilon } }_{yz}{\hbox{ and }}{\mathrm {\varepsilon } }_{zx}.\end{aligned}}

(2.1c)

The vector displacement ${\mathrm {\zeta } }$ is

{\begin{aligned}{\mathrm {\zeta } }=ui+vj+wk,\end{aligned}}

(2.1d)

where ${\textbf {i}},{\textbf {j}},{\textbf {k}}$ are unit vectors in the ${\textit {x}}-,{\textit {y}}-,{\textit {z}}$ -directions (see Sheriff and Geldart, 1995, problem 15.3). The dilatation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta is the change in volume per unit volume, i.e.,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \Delta \approx \left(1+{\mathrm{\varepsilon}} _{xx} \right)\left(1+{\mathrm{\varepsilon}} _{yy} \right)\left(1+{\mathrm{\varepsilon}} _{zz} \right)-1\approx {\mathrm{\varepsilon}} _{xx} +{\mathrm{\varepsilon}} _{yy} +{\mathrm{\varepsilon}} _{zz} ={\nabla} \cdot {\mathrm{\zeta}}. \end{align}

(2.1e)

A pressure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathcal P} produces a decrease in volume, the proportionality constant being the bulk modulus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} k=-\mathcal{P}/\Delta. \end{align}

(2.1f)

Sometimes the compressibility, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 1/k , is used instead of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k .

In addition to creating strains, stresses cause rotation of the medium, the vector rotation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\theta} being equal to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\theta}={\mathrm{\theta}}_{x} i+{\mathrm{\theta}} _{y} j+{\mathrm{\theta}} _{z} k=\frac{1}{2} {\nabla} \times {\mathrm{\zeta}}, \end{align}

(2.1g)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\theta}} _{x} =\left(1/2\right)\left({\mathrm{\partial}} w/{\mathrm{\partial}} y-{\mathrm{\partial}} v/{\mathrm{\partial}} z\right) , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\theta}} _{y} =\left(1/2\right)\left({\mathrm{\partial}} u/{\mathrm{\partial}} z-{\mathrm{\partial}} w/{\mathrm{\partial}} x\right) , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\theta}} _{z} =\left(1/2\right)\left({\mathrm{\partial}} v/{\mathrm{\partial}} x-{\mathrm{\partial}} u/{\mathrm{\partial}} y\right).

For small strains and an isotropic medium (where properties are the same regard-less of the direction of measurement), Hooke’s law relates the stresses to the strains as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\sigma}} _{ii} ={\mathrm{\lambda}} {\mathrm{\Delta}} +2{\mu} {\mathrm{\varepsilon}} _{ii},{\qquad} i=x,\; y,\; z; \end{align}

(2.1h)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\sigma}} _{ij} ={\mathrm{\mu}} {\mathrm{\varepsilon}} _{ij},{\qquad}{\qquad}{\quad} i\ne j. \end{align}

(2.1i)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} are Lamé’s constants (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} is usually called the modulus of rigidity or the shear modulus).

Solution

Subtracting equation (2.1h) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i=y from the same equation for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i=z gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\varepsilon}} _{yy} ={\mathrm{\varepsilon}} _{zz} .

Dividing equation (2.1h) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i=y by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\varepsilon}} _{xx} gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 0={\mathrm{\lambda}} \left(1+2\frac{{\mathrm{\varepsilon}}_{yy}}{{\mathrm{\varepsilon}}_{xx}}\right)+2{\mathrm{\mu}} \frac{{\mathrm{\varepsilon}} _{yy} }{{\mathrm{\varepsilon}} _{xx} } ={\mathrm{\lambda}} -2\left({\mathrm{\lambda}} +{\mathrm{\mu}} \right)\sigma, \end{align}

so

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\sigma}} ={\mathrm{\lambda}} /2\left({\mathrm{\lambda}} +{\mathrm{\mu}} \right). \end{align}

2.1b Show that Young’s modulus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E , defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}_{xx} /{\mathrm{\varepsilon}} _{xx}} , is given by the equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {E=\frac{{\mu} \left({3}{\lambda} + {2}{\mu} \right)}{\left({\lambda} +{\mu} \right)}.} \end{align}

(2.1j)

Solution

Adding the three equations (2.1h) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i=x , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z , and recalling that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}}_{yy} =0={\mathrm{\sigma}} _{zz} , we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\sigma}} _{xx} =\left[3{\mathrm{\lambda}} {\mathrm{\Delta}} +2{\mathrm{\mu}} \left({\mathrm{\varepsilon}} _{xx} +{\mathrm{\varepsilon}} _{yy} +{\mathrm{\varepsilon}} _{zz} \right)\right]=\left(3{\mathrm{\lambda}} +2{\mathrm{\mu}} \right)\Delta. \end{align}

Dividing both sides by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\varepsilon}} _{xx} gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E={\mathrm{\sigma}} _{xx} /{\mathrm{\varepsilon}} _{xx} =\left(3{\mathrm{\lambda}} +2{\mathrm{\mu}} \right)\left(1+2{\mathrm{\varepsilon}} _{yy} /{\mathrm{\varepsilon}} _{xx} \right)=\left(3{\mathrm{\lambda}} +2{\mathrm{\mu}} \right)\left(1-2{\mathrm{\sigma}} \right). \end{align}

Using equation (2.1a) we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E=\frac{{\mathrm{\mu}} \left(3{\mathrm{\lambda}} +2{\mathrm{\mu}} \right)}{\left({\mathrm{\lambda}} +{\mathrm{\mu}} \right)}. \end{align}

2.1c A pressure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathbf{\mathcal{P}} is equivalent to stresses Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\sigma}_{xx} ={\sigma} _{yy} ={\sigma} _{zz} =-{\mathcal P} . Derive the following result for the bulk modulus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} k{\mathbf{=}\left({\lambda}\mathbf{ +\frac{{2}}{{3}}} {\mu} \right)}. \end{align}

(2.1k)

Solution

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathcal P}=-{\mathrm{\sigma}} _{xx} =-{\mathrm{\sigma}} _{yy} =-{\mathrm{\sigma}} _{zz} , we add equation (2.1h) for each of the three values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i=x,y,z , obtaining Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): -3{\mathcal P}=\left[3{\mathrm{\lambda}} {\mathrm{\Delta}} +2{\mathrm{\mu}} \left({\mathrm{\varepsilon}} _{xx} +{\mathrm{\varepsilon}} _{yy} +{\mathrm{\varepsilon}} _{zz} \right)\right]=\left(3{\mathrm{\lambda}} +2{\mathrm{\mu}} \right)\Delta , so from equation (2.1f),

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} k=-{\mathcal P}/{\mathrm{\Delta}} =\left({\mathrm{\lambda}} +\frac{2}{3} {\mathrm{\mu}} \right). \end{align}

2.2 Interrelationships among elastic constants

The entries in Table 2.2a express, for isotropic media, the quantities at the heads of the columns in terms of the pairs of elastic constants or velocities at the left ends of the rows. The first three entries in the ninth row are equations (2.1j), (2.1a), and (2.1k) and the next two entries in the same row are formulas for the P- and S- wave velocities Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\alpha} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\beta} (see problem 2.5). Starting from these five relations, derive the other relations in the table.

Background

For isotropic media, any two of the elastic constants can be considered as independent and the others can be expressed in terms of these two. The P- and S-wave velocities, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\alpha}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\beta}} , given by the equations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\rho}}{\mathrm{\alpha}} ^{2} =\left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right),{\quad} {\mathrm{\rho}}\beta ^{2} =\mu \end{align}

(2.2a)

[see Sheriff and Geldart, 1995, equations (2.28) and (2.29)] can also be expressed in terms of any two elastic constants (plus the density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\rho}} ).

Solution

Denoting the equations by row and column (as for matrix elements) and using a comma instead of a period, we use equations (9,1) to (9,3) to derive the equations that do not involve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\alpha}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\beta}} , then we use equations (9,6) and (9,7) (see equation (2.2a)) to derive the rest. From equation (9,1),

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {{\mathrm{\lambda}} \left(E-3{\mathrm{\mu}} \right)=2{\mathrm{\mu}} ^{2} -{\mathrm{\mu}} E,}\\ {{\mathrm{\lambda}} ={\mathrm{\mu}} \left(E-2{\mathrm{\mu}} \right)/\left(3{\mathrm{\mu}} -E\right).} \end{align}

(3,5)

From equation (9,2),

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {{\mathrm{\lambda}} \left(2{\mathrm{\sigma}} -1\right)+2{\mathrm{\mu}} {\mathrm{\sigma}} =0,}\\ {{\mathrm{\lambda}} =2{\mathrm{\mu}} {\mathrm{\sigma}} /\left(1-2{\mathrm{\sigma}} \right).} \end{align}

(5,5)

Solving equation (9,2) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} , we have $2{\mathrm {\mu } }{\mathrm {\sigma } }={\mathrm {\lambda } }\left(1-2{\mathrm {\sigma } }\right)$ ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\mu}} ={\mathrm{\lambda}} \left(1-2{\mathrm{\sigma}} \right)/2{\mathrm{\sigma}}. \end{align}

(6,4)

From equation (9,3),

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\lambda}} =k-\frac{2}{3} {\mathrm{\mu}}. \end{align}

(7,5)

Equating Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} from equations (9,2) and (9,3) [or from equations (6,4) and (7,5)] gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\mu}} ={\mathrm{\lambda}} \left(1-2{\mathrm{\sigma}} \right)/2{\mathrm{\sigma}} =\frac{3}{2} \left(k-{\mathrm{\lambda}} \right), \end{align}

(8,4)

thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} k={\mathrm{\lambda}} \left(1-2{\mathrm{\sigma}} \right)/3{\mathrm{\sigma}} +{\mathrm{\lambda}} ={\mathrm{\lambda}} \left(1+{\mathrm{\sigma}} \right)/3\sigma, \end{align}

(6,3)

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\lambda}} =\frac{3{\mathrm{\sigma}} k}{\left(1+{\mathrm{\sigma}} \right)}. \end{align}

(4,5)

Solving equation (4,5) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}} gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\sigma}} =\frac{{\mathrm{\lambda}} }{3k-{\mathrm{\lambda}} }. \end{align}

(8,2)

Substituting equation (4,5) into equation (8,4), we get,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\mu}} =\frac{3}{2} \left(k-\frac{3{\mathrm{\sigma}} k}{\left(1+{\mathrm{\sigma}} \right)} \right)=\frac{3k\left(1-2{\mathrm{\sigma}} \right)}{2\left(1+{\mathrm{\sigma}} \right)}. \end{align}

(4,4)

We use equation (7,5) to eliminate ${\mathrm {\lambda } }$ from equation (6,3):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} k=\left(k-\frac{2}{3} {\mathrm{\mu}} \right)\left(\frac{1+{\mathrm{\sigma}} }{3{\mathrm{\sigma}} } \right)=\frac{k\left(1+{\mathrm{\sigma}} \right)}{3{\mathrm{\sigma}} } -\frac{2{\mathrm{\mu}} \left(1+{\mathrm{\sigma}} \right)}{9{\mathrm{\sigma}} }, \end{align}

that is,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} k\left(\frac{1+{\mathrm{\sigma}} }{3{\mathrm{\sigma}} } -1\right)=\frac{2{\mathrm{\mu}} \left(1+{\mathrm{\sigma}} \right)}{9{\mathrm{\sigma}} }, \end{align}

so

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} k=\frac{2{\mathrm{\mu}} \left(1+{\mathrm{\sigma}} \right)}{3\left(1-2{\mathrm{\sigma}} \right)}. \end{align}

(5,3)

Solving for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}} , we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\sigma}} =\frac{\left(3k-2{\mathrm{\mu}} \right)}{2\left(3k+{\mathrm{\mu}} \right)}. \end{align}

(7,2)

We use equation (8,4) to express Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} in equation (9,1) in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(k,\; {\mathrm{\lambda}} \right) . Thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E=\frac{\frac{3}{2}\left(k-{\mathrm{\lambda}} \right)\left[3{\mathrm{\lambda}} +3\left(k-{\mathrm{\lambda}} \right)\right]}{{\mathrm{\lambda}} +\frac{3}{2} \left(k-{\mathrm{\lambda}} \right)} =\frac{9k\left(k-{\mathrm{\lambda}} \right)}{\left(3k-{\mathrm{\lambda}} \right)}. \end{align}

(8,1)

Solving equation (8,1) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} gives

{\begin{aligned}{\mathrm {\lambda } }=3k\left(3k-E\right)/\left(9k-E\right).\end{aligned}}

(2,5)

We now eliminate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} from equation (9,1) using equation (7,5)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {E = \frac{{\mu \left( {3k - 2\mu + 2\mu } \right)}}{{\left( {k - 2\mu /3 + \mu } \right)}}}\\ { = 9k\mu /\left( {3k + \mu } \right).} \end{align}

(7,1)

Solving equation (7,1) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} k ={\mathrm{\mu}} E/3\left(3{\mathrm{\mu}} -E\right), \end{align}

(3,3)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\mu}} =3kE/\left(9k-E\right). \end{align}

(2,4)

Next we use equation (6,4) to eliminate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} from equation (9,1):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E={\mathrm{\mu}} \left(\frac{6{\mathrm{\mu}} {\mathrm{\sigma}} +2{\mathrm{\mu}} -4{\mathrm{\mu}} {\mathrm{\sigma}} }{2{\mathrm{\mu}} {\mathrm{\sigma}} +{\mathrm{\mu}} -2{\mathrm{\mu}} {\mathrm{\sigma}} } \right)={\mathrm{\lambda}} \left(1-2{\mathrm{\sigma}} \right)\left(1+{\mathrm{\sigma}} \right)/{\mathrm{\sigma}}. \end{align}

(6,1)

Using equations (9,1) and (5,5) we get

{\begin{aligned}{E={\mathrm {\mu } }\left(6{\mathrm {\mu } }{\mathrm {\sigma } }\right)/\left(2{\mathrm {\mu } }+{\mathrm {\mu } }-2{\mathrm {\mu } }{\mathrm {\sigma } }\right)}\\{{\quad }=2{\mathrm {\mu } }^{2}\left(1+{\mathrm {\sigma } }\right)/{\mathrm {\mu } }=2{\mathrm {\mu } }\left(1+{\mathrm {\sigma } }\right).}\end{aligned}}

(5,1)

Solving equation (5,1) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\sigma}} =E/2{\mathrm{\mu}} -1=\left(E-2{\mathrm{\mu}} \right)/2\mu, \end{align}

(3,2)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\mu}} =E/2\left(1+{\mathrm{\sigma}} \right). \end{align}

(1,4)

Using equations (4,4) and (4,5) to replace Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left({\mathrm{\lambda}}, \; {\mathrm{\mu}} \right) in equation (9,1) by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left({\mathrm{\sigma}}, \; k\right) gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E=\frac{3k\left(1-2{\mathrm{\sigma}} \right)}{2\left(1+{\mathrm{\sigma}} \right)} \times \frac{\left(9{\mathrm{\sigma}} k+3k-6{\mathrm{\sigma}} k\right)}{\left[3{\mathrm{\sigma}} k+\left(3k-6{\mathrm{\sigma}} k\right)/2\right]} =3k\left(1-2{\mathrm{\sigma}} \right). \end{align}

(4,1)

Solving equation (4,1) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}} , we get

{\begin{aligned}k=E/3\left(1-2{\mathrm {\sigma } }\right),\end{aligned}}

(1,3)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\sigma}} =\left(3k-E\right)/6k. \end{align}

(2,2)

The last equation (of this group) can be obtained by substituting equation (1,3) into equation (4,5):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\lambda}} =\frac{3{\mathrm{\sigma}} k}{\left(1+{\mathrm{\sigma}} \right)} =\frac{3{\mathrm{\sigma}} E}{3\left(1-2{\mathrm{\sigma}} \right)\left(1+{\mathrm{\sigma}} \right)} =\frac{{\mathrm{\sigma}} E}{\left(1+{\mathrm{\sigma}} \right)\left(1-2{\mathrm{\sigma}} \right)}. \end{align}

(1,5)

Equations (10,1) to (10,3) express Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k in terms of the P- and S-wave velocities, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\alpha}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\beta}} (see equation 2.2a). To introduce Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\alpha}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\beta}} , we write equations (10,1) to (10,3) in terms of $\left({\mathrm {\lambda } }+2{\mathrm {\mu } }\right)$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} . Thus,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {E ={\mathrm{\mu}} \left(\frac{3{\mathrm{\lambda}} +2{\mathrm{\mu}} }{{\mathrm{\lambda}} +{\mathrm{\mu}} } \right)={\mathrm{\mu}} \left[\frac{3\left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right)-4{\mathrm{\mu}} }{\left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right)-{\mathrm{\mu}} } \right] }\\ { {\qquad}{\quad}=\frac{{\mathrm{\rho}}\beta ^{2} \left(3{\mathrm{\rho}}{\mathrm{\alpha}} ^{2} -4{\mathrm{\rho}}\beta ^{2} \right)}{{\mathrm{\rho}}\left({\mathrm{\alpha}} ^{2} -\beta ^{2} \right)} =\frac{{\mathrm{\rho}}\beta ^{2} \left(3{\mathrm{\alpha}} ^{2} -4\beta ^{2} \right)}{{\mathrm{\alpha}} ^{2} -\beta ^{2} }}, \end{align}

(10,1)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\sigma}} =\frac{{\mathrm{\lambda}} }{2\left({\mathrm{\lambda}} +{\mathrm{\mu}} \right)} =\frac{\left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right)-2{\mathrm{\mu}} }{2\left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right)-2{\mathrm{\mu}} } =\frac{{\mathrm{\alpha}} ^{2} -2\beta ^{2} }{2\left({\mathrm{\alpha}} ^{2} -\beta ^{2} \right)}, \end{align}

(10,2)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} k={\mathrm{\lambda}} +\frac{2{\mathrm{\mu}} }{3} =\left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right)-\frac{4}{3} {\mathrm{\mu}} ={\mathrm{\rho}}\left({\mathrm{\alpha}} ^{2} -\frac{4}{3} \beta ^{2} \right). \end{align}

(10,3)

Equation (10,4) is the second equation in equation (2.2a). To get equation (10,5), we write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\lambda}} =\left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right)-2{\mathrm{\mu}} ={\mathrm{\rho}}\left({\mathrm{\alpha}} ^{2} -2\beta ^{2} \right). \end{align}

(10,5)

To verify column 6, we start with equation (9,6) and express Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} in terms of the required pair of constants. Thus, using equations (1,4) and (1,5) we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {{\mathrm{\rho}}{\mathrm{\alpha}} ^{2} =\left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right)=\frac{E{\mathrm{\sigma}} }{\left(1+{\mathrm{\sigma}} \right)\left(1-2{\mathrm{\sigma}} \right)} +\frac{2E}{2\left(1+{\mathrm{\sigma}} \right)}}\\ {{\qquad}=\left(\frac{E}{1+{\mathrm{\sigma}} } \right)\left(\frac{{\mathrm{\sigma}} }{1-2{\mathrm{\sigma}} } +1\right)=\frac{E\left(1-{\mathrm{\sigma}} \right)}{\left(1+{\mathrm{\sigma}} \right)\left(1-2{\mathrm{\sigma}} \right)}}. \end{align}

(1,6)

Following the same procedure, using equations (2,4) and (2,5), we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {{\mathrm{\rho}}{\mathrm{\alpha}} ^{2} =3k\left(\frac{3k-E}{9k-E} \right)+\left(\frac{6kE}{9k-E} \right) }\\ {{\qquad}=\left(\frac{3k}{9k-E} \right)\left(3k-E+2E\right)=\frac{3k\left(3k+E\right)}{9k-E}.} \end{align}

(2,6)

In the same way, we get the following results:

{\begin{aligned}{{\mathrm {\rho } }{\mathrm {\alpha } }^{2}={\mathrm {\mu } }\left({\frac {E-2{\mathrm {\mu } }}{3{\mathrm {\mu } }-E}}\right)+2{\mathrm {\mu } }=\left({\frac {\mathrm {\mu } }{3{\mathrm {\mu } }-E}}\right)\left(E-2{\mathrm {\mu } }+6{\mathrm {\mu } }-2E\right)}\\{{\qquad }=\left({\frac {\mathrm {\mu } }{3{\mathrm {\mu } }-E}}\right)\left(4{\mathrm {\mu } }-E\right),}\end{aligned}}

(3,6)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {{\mathrm{\rho}}{\mathrm{\alpha}} ^{2} =\left(\frac{3k{\mathrm{\sigma}} }{1+{\mathrm{\sigma}} } \right)+3k\left(\frac{1-2{\mathrm{\sigma}} }{1+{\mathrm{\sigma}} } \right)=\left(\frac{3k}{1+{\mathrm{\sigma}} } \right)\left({\mathrm{\sigma}} +1-2{\mathrm{\sigma}} \right)} \\ {{\qquad} =\frac{3k\left(1-{\mathrm{\sigma}} \right)}{\left(1+{\mathrm{\sigma}} \right)},} \end{align}

(4,6)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\rho}}{\mathrm{\alpha}} ^{2} =\left(\frac{2{\mathrm{\mu}} {\mathrm{\sigma}} }{1-2{\mathrm{\sigma}} } \right)+2{\mathrm{\mu}} =2{\mathrm{\mu}} \left(\frac{{\mathrm{\sigma}} }{1-2{\mathrm{\sigma}} } +1\right)=2{\mathrm{\mu}} \left(\frac{1-{\mathrm{\sigma}} }{1-2{\mathrm{\sigma}} } \right), \end{align}

(5,6)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\rho}}{\mathrm{\alpha}} ^{2} ={\mathrm{\lambda}} +\frac{{\mathrm{\lambda}} \left(1-2{\mathrm{\sigma}} \right)}{{\mathrm{\sigma}} } =\frac{{\mathrm{\lambda}} }{{\mathrm{\sigma}} } \left(1-{\mathrm{\sigma}} \right), \end{align}

(6,6)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\rho}}{\mathrm{\alpha}} ^{2} =\left(k-\frac{2}{3} {\mathrm{\mu}} \right)+2{\mathrm{\mu}} =k+\frac{4}{3} {\mathrm{\mu}}, \end{align}

(7,6)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\rho}}{\mathrm{\alpha}} ^{2} ={\mathrm{\lambda}} +3\left(k-{\mathrm{\lambda}} \right)=3k-2\lambda. \end{align}

(8,6)

Column 7 is merely the square root of column 4 after dividing by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\rho}} . Column 8 is obtained by dividing column 7 by column 6.

2.3 Magnitude of disturbance from a seismic source

2.3a Firing an air gun in water creates a pressure transient a small distance away from the air gun with peak pressure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathcal{P} of 5 atmospheres (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 5.0\times 10^{5} Pa). If the compressibility of water is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 4.5\times 10^{-10} /Pa, what is the peak energy density?

Background

Air guns (see problem 7.7) suddenly inject a bubble of high‐pressure air into the water to generate a seismic wave.

Stresses acting upon a medium cause energy to be stored as strain energy, because the stresses are present while the medium is being displaced. Strain energy density (energy/unit volume) $$ E $$ is equal to [see Sheriff and Geldart, 1995, equation (2.22)]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {E=\frac{1}{2} \sum\limits_{i} \sum\limits_{j} {\mathrm{\sigma}} _{ij} {\mathrm{\varepsilon}} _{ij} =\frac{1}{2} {\mathrm{\lambda}} {\mathrm{\Delta}} ^{2} +{\mathrm{\mu}} \left({\mathrm{\varepsilon}} _{xx}^{2} +{\mathrm{\varepsilon}} _{yy}^{2} +{\mathrm{\varepsilon}} _{zz}^{2} \right)} \\ {{\qquad}{\qquad}{\qquad}{\qquad} +\frac{1}{2} {\mathrm{\mu}} \left({\mathrm{\varepsilon}} _{xy}^{2} +{\mathrm{\varepsilon}} _{yz}^{2} +{\mathrm{\varepsilon}} _{zx}^{2} \right).} \end{align}

(2.3a)

Solution

From problem 2.1c, we see that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k=1/\left(4.5\times 10^{-10} \right)=2.2\times 10^{9} Pa. Also, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} =0 for water, so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E=\frac{1}{2} {\mathrm{\lambda}} {\mathrm{\Delta}} ^{2} . From equation (7,5) of Table 2.2a we find that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} =k when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} =0 . Also, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k=-{\mathcal P}/{\mathrm{\Delta}} (see equation (2.1f)), so

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \Delta =\left|{\cal P}/k\right|=5.0\times 10^{5} /2.2\times 10^{9} =2.3\times 10^{-4}. \end{align}

Using equation (2.3a) we find that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E=\frac{1}{2} \left(2.2\times 10^{9} \times 2.3^{2} \times 10^{-8} \right)=58\,\,\,\mathrm{J/m}^{3}. \end{align}

[The dimensions of $$ E $$ are the same as those of stress, since strains are dimensionless. Thus, stress units are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm N/m}^{2} = {\rm Nm/m}^{3} = {\rm J/m}^{3} .]

2.3b If the same wave is generated in rock with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} ={\mathrm{\mu}} =3.0\times 10^{10} Pa, what is the peak energy density? Assume a symmetrical Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P -wave with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\varepsilon}} _{xx} ={\mathrm{\varepsilon}} _{yy} ={\mathrm{\varepsilon}} _{zz}, {\mathrm{\varepsilon}} _{ij} =0 for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i\ne j .

Solution

We have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\Delta}} ={\mathrm{\varepsilon}} _{xx} +{\mathrm{\varepsilon}} _{yy} +{\mathrm{\varepsilon}} _{zz} =3{\mathrm{\varepsilon}} _{xx} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\varepsilon}} _{ij} =0, i\ne j , so equation (2.3a) becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E=\frac{1}{2} {\mathrm{\lambda}} {\mathrm{\Delta}} ^{2} +3{\mathrm{\mu}} {\mathrm{\varepsilon}} _{xx}^{2} ={\mathrm{\lambda}} {\mathrm{\Delta}} ^{2} \left(\frac{1}{2} +\frac{1}{3} \right)=\frac{5}{6} {\mathrm{\lambda}} ({\mathcal P}/k)^{2}. \end{align}

Equation (9,3) in Table 2.2a gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k={\mathrm{\lambda}} +\frac{2}{3} {\mathrm{\mu}} =\frac{5}{3} {\mathrm{\lambda}} so that

${\begin{aligned}E={\frac {5}{6}}{\mathrm {\lambda } }{\mathcal {P}}^{2}\left/\right.\left({\frac {5}{3}}{\mathrm {\lambda } }\right)^{2}=0.3\times (5.0\times 10^{5})^{2}/3\times 10^{10}=2.5\ \mathrm {J/m} ^{3}.\end{aligned}}$

2.4 Magnitudes of elastic constants

To illustrate the relationships and magnitudes of the elastic constants, complete Table 2.4a.

[Note that these values apply to specific specimens; the elastic constants for rocks range considerably, especially as porosity and pressure change.]

Solution

We use the row-column notation to designate equations from Table 2.2a.

Water: Since water is a fluid we know that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} =0=\beta . Equation (4,1) shows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E=0 also. From equation (4,5) we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} =k=2.1\times 10^{9} Pa.

**Table 2.4a. Magnitudes of elastic constants and velocities.**
Constant	Water	Stiff mud	Shale	Sandstone	Limestone	Granite
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E\,\,\,({\times}10^9 \hbox{Pa})				16	54	50
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k\,\,\,({\times}10^9 \hbox{Pa})	2.1
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}}\,\,\,({\times}10^9 \hbox{Pa})
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}}\,\,\,({\times}10^9 \hbox{Pa})
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}}	0.50	0.43	0.38	0.34	0.25	0.20
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\rho}}\,\,\,(\hbox{g/cm}^{3})	1.0	1.5	1.8	1.9	2.5	2.7
${\alpha }\,\,\,({\hbox{km/s}})$	1.5	1.6	3.2
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\beta}\,\,\,(\hbox{km/s})

Stiff mud: Because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}}<0.5 , stiff mud is equivalent to a solid, hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} \ne 0 . From Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\rho}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\alpha}} we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right) using equation (9,6), while equation (9,2) expresses Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}} in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} , thus enabling us to find both ${\mathrm {\lambda } }$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} .

We have:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {{\rho} {{\alpha} ^2} = \left( {{\lambda} + 2{\mu} } \right) = \left( {1.5\;{\rm{g/c}}{{\rm{m}}^3}} \right) {\times} {{(1.6 {\times} {{10}^5}\;{\rm{cm/s}})}^2}}\\ {{\qquad} = 3.8 {\times} {{10}^{10}}\;{\rm{dynes/c}}{{\rm{m}}^2} = 3.8 {\times} {{10}^9}\;{\rm{Pa}},}\\ {{\quad}{\sigma} = 0.43 = {\lambda} /2\left( {{\lambda} + {\mu} } \right),\;{\rm{i}}.{\rm{e}}.,\;0.86{\mu} = 0.14{\lambda} .} \end{align}

Solving the two equations, we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} =2.9{\times} 10^{9} Pa, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} =0.47{\times} 10^{9} Pa. Using equation (6,1),

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {E={\mathrm{\lambda}} \frac{\left(1+{\mathrm{\sigma}} \right)\left(1-2{\mathrm{\sigma}} \right)}{{\mathrm{\sigma}} } =2.9{\times} 10^{9} {\times} 1.43{\times} 0.14/0.43}\\ {{\qquad}{\qquad}{\qquad}{\qquad}{\quad}=1.4{\times} 10^{9}\ \mathrm{Pa}.} \end{align}

Equation (6,3) gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} k={\mathrm{\lambda}} \frac{\left(1+{\mathrm{\sigma}} \right)}{3{\mathrm{\sigma}} } =2.9{\times} 10^{9} {\times} 1.43/1.29=3.2{\times} 10^{9}\ \mathrm{Pa}. \end{align}

Finally, to get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\beta}} we note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} , and $$ k $$ are in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): N/m^{2} , and so we must express Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\rho}} in appropriate units of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): kg/m^{3} , i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\rho}}=1.5{\times} 10^{3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): kg/m^{3} . We now have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \beta =({\mathrm{\mu}} /{\mathrm{\rho}})^{1/2} =\left(0.47{\times} 10^{9} /1.5{\times} 10^{3} \right)=0.56\ \mathrm{\hbox{km/s}}.

Shale. As with stiff mud, we have been given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\rho}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}} , so that again ${\mathrm {\rho } }$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\alpha}} give us Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}} gives us Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} /2\left({\mathrm{\lambda}} +{\mathrm{\mu}} \right) so that we can solve these equations for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} , then find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E using equations (9,3) and (9,1). Thus,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \left(\lambda +2\mu \right)=\rho\alpha^{2} =\left(1.8\ \mathrm{g/cm}^{3} \right)\times (3.2\times 10^{5}\ \mathrm{cm/s})^{2} =18.4\times 10^{9}\ \mathrm{Pa}, \end{align}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \sigma =0.38=\lambda /2\left(\lambda +\mu \right),\quad \mathrm{i.e.}, 0.76\mu =0.24\lambda. \end{align}

Solving the two equations gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} =11{\times} 10^{9} Pa, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} =3.6{\times} 10^{9} Pa.

From equation (9,3), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k={\mathrm{\lambda}} +\frac{2}{3} {\mathrm{\mu}} =14{\times} 10^{9} Pa, and equation (9,1) gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E={\mathrm{\mu}} \frac{\left(3{\mathrm{\lambda}} +2{\mathrm{\mu}} \right)}{\left({\mathrm{\lambda}} +{\mathrm{\mu}} \right)} =9.9{\times} 10^{9}\ \mathrm{Pa}. \end{align}

Finally,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \beta =({\mathrm{\mu}} /{\mathrm{\rho}})^{1/2} =(3.6{\times} 10^{9} /1.8{\times} 10^{3} )^{1/2} =1.4\ \mathrm{\hbox{km/s}}. \end{align}

Sandstone: We are given the elastic constants Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E and ${\mathrm {\sigma } }$ (plus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\rho}} ), so we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\beta}} using equations (1,3) to (1,7) in Table 2.2a. Thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {k=16{\times} 10^{9} /3{\times} 0.32=17{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\mu}} =16{\times} 10^{9} /2{\times} 1.34=6.0{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\lambda}} =16{\times} 10^{9} {\times} 0.34/1.34{\times} 0.32=13{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\alpha}} =[16{\times} 10^{9} {\times} 0.66/1.34{\times} 0.32{\times} 1.9{\times} 10^{3} ]^{1/2} =3.6\ \mathrm{\hbox{km/s}},}\\ {{\mathrm{\beta}} =[16{\times} 10^{9} /2{\times} 1.34{\times} 1.9{\times} 10^{3} ]^{1/2} =1.8\ \mathrm{\hbox{km/s}}.} \end{align}

[We could also have obtained Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\alpha}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\beta}} by using equations (9,6) and (9,7).]

Limestone: We solve in the same way as with sandstone since we are given the same constants:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {k =54{\times} 10^{9} /3{\times} 0.50=36{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\mu}} =54{\times} 10^{9} /2{\times} 1.25=22{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\lambda}} =54{\times} 10^{9} {\times} 0.25/1.25{\times} 0.50=22{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\alpha}} =[54{\times} 10^{9} {\times} 0.75/1.25{\times} 0.50{\times} 2.5{\times} 10^{3} ]^{1/2} =5.1\ \mathrm{\hbox{km/s}},}\\ {{\mathrm{\beta}} =[54{\times} 10^{9} /2{\times} 1.25{\times} 2.5{\times} 10^{3} ]^{1/2} =2.9\ \mathrm{\hbox{km/s}}.} \end{align}

Granite: Again the solution is the same as for sandstone.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {k=50{\times} 10^{9} /3{\times} 0.60=28{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\mu}} =50{\times} 10^{9} /2{\times} 1.2=21{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\lambda}} =50{\times} 10^{9} {\times} 0.20/1.2{\times} 0.60=14{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\alpha}} =[50{\times} 10^{9} {\times} 0.80/1.2{\times} 0.60{\times} 2.7{\times} 10^{3} ]^{1/2} =4.5\ \mathrm{\hbox{km/s}},}\\ {{\mathrm{\beta}} =[50{\times} 10^{9} /2{\times} 1.2{\times} 2.7{\times} 10^{3} ]^{1/2} =2.8\ \mathrm{\hbox{km/s}}.} \end{align}

Table 2.4b summarizes the results.

**Table 2.4b Magnitudes of elastic constants.**
Constant	Water	Stiffmud	Shale	Sandstone	Limestone	Granite
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E\,\,\,({\times} 10^{9} \hbox{Pa})	0	1.4	9.9	16	54	50
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): K\,\,\,({\times} 10^{9} \hbox{Pa})	2.1	3.2	13	17	36	28
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}}\,\,\,({\times} 10^{9} \hbox{Pa})	0	0.47	3.6	6.0	22	21
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}}\,\,\,({\times} 10^{9} \hbox{Pa})	2.1	2.9	11	13	22	14
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}}	0.50	0.43	0.38	0.34	0.25	0.20
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}}\,\,\,(\hbox{g/cm}^{3})	1.0	1.5	1.8	1.9	2.5	2.7
${\mathrm {\alpha } }\,\,\,({\hbox{km/s}})$	1.5	1.6	3.2	3.6	5.1	4.5
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\beta}}\,\,\,(\hbox{km/s})	0	0.56	1.4	1.8	2.9	2.8

2.5 General solutions of the wave equation

2.5a Verify that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi =f\left(x-Vt\right) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi =g\left(x+Vt\right) are solutions of the wave equation (2.5b).

Background

When unbalanced stresses act upon a medium, the strains are propagated throughout the medium according to the general wave equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\nabla} ^{2} \psi =\frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} x^{2} } +\frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} y^{2} } +\frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} z^{2} } =\frac{1}{V^{2} } \frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} t^{2} }, \end{align}

(2.5a)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi being a disturbance such as a compression or rotation. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi is propagated with velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V (see Sheriff and Geldart, 1995, Section 2.2). The disturbance is the result of unbalanced normal stresses, shearing stresses, or a combination of both. When normal stresses create the wave, the result is a volume change and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi is the dilitation [see equation (2.1e)], and we get the P-wave equation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V becoming the P-wave velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\alpha}} . Shearing stresses create rotation in the medium and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi is one of the components of the rotation given by equation (2.lg) ; the result is an S-wave traveling with velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\beta}} . Various expressions for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\alpha}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\beta}} in isotropic media are given in Table 2.2a.

In one dimension the wave equation (2.5a) reduces to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} x^{2} } =\frac{1}{V^{2} } \frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} t^{2}}. \end{align}

(2.5b)

Solution

We use subscripts to denote partial derivatives and primes to denote derivatives with respect to the argument of the function. Then, writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \zeta =\left(x-Vt\right) , we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {{{\mathrm{\psi}} _x} = \frac{{\partial {\mathrm{\psi}} }}{{\partial x}} = \left( {\frac{{df}}{{d{\mathrm{\zeta}} }}} \right)\left( {\frac{{\partial {\mathrm{\zeta}} }}{{\partial x}}} \right) = \frac{{df}}{{d{\mathrm{\zeta}} }} = f',}\\ {{{\mathrm{\psi}} _{xx}} = \left( {\frac{{df'}}{{d{\mathrm{\zeta}} }}} \right)\left( {\frac{{\partial {\mathrm{\zeta}} }}{{\partial x}}} \right) = f'',}\\ {{{\mathrm{\psi}} _t} = \left( {\frac{{df}}{{d{\mathrm{\zeta}} }}} \right)\left( {\frac{{\partial {\mathrm{\zeta}} }}{{\partial t}}} \right) = - Vf',}\\ {{{\mathrm{\psi}} _{tt}} = - V\left( {\frac{{df'}}{{d{\mathrm{\zeta}} }}} \right)\left( {\frac{{\partial {\mathrm{\zeta}} }}{{\partial t}}} \right) = {V^2}f''} \end{align}

Substituting in equation (2.5b), we get the identity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f''=f'' so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f\left(x-Vt\right) is a solution. We get the same result when $\psi =g\left(x+Vt\right)$ . A sum of solutions is also a solution, so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f\left(x-Vt\right)+g\left(x+Vt\right) is a solution.

2.5b Verify that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\psi =f\left(\ell x+my+nz-Vt\right)+g\left(\ell x+my+nz+Vt\right)} is a solution of equation (2.5a), where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\left(\ell, {\rm \; }m,\; n\right)} are direction cosines.

Solution

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(\ell x\; +my+nz-Vt\right)=\zeta , \left(\ell x\; +my+nz+Vt\right)=\xi . We now must show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi =f\left(\zeta \right)+g\left(\xi \right) is a solution of equation (2.5a). Proceeding as before, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {{\mathrm{\psi}} _{x} =\left(\frac{df}{d\zeta } \right)\left(\frac{{\mathrm{\partial}} \zeta }{{\mathrm{\partial}} x} \right)+\left(\frac{dg}{d\xi } \right)\left(\frac{{\mathrm{\partial}} \xi }{{\mathrm{\partial}} x} \right)=\ell (f'+g'),}\\ {{\mathrm{\psi}} _{xx} =\ell ^{2} (f''+g'').} \end{align}

In the same way we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \psi _{yy} =m^{2} (f''+g''),\;{\quad}\psi _{zz} =n^{2} (f''+g''). \end{align}

But Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(\ell ^{2} +m^{2} +n^{2} \right)=1 (see Sheriff and Geldart, 1995, problem 15.9a), so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(\psi _{xx} +\psi _{yy} +\psi _{zz} \right)= (f''+g'') .

Following the same procedure we find that $\left(1/V^{2}\right)\psi _{tt}=(f''+g'')$ thus verifying that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi =f\left(\ell x+my+nz-Vt\right)+g\left(\ell x+my+nz+Vt\right) is a solution of equation (2.5a).

2.5c Show that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \psi \left(r,\; t\right)=\left(1/r\right)f\left(r-Vt\right)+\left(1/r\right)g\left(r+Vt\right) \end{align}

is a solution of the wave equation in spherical coordinates (see problem 2.6b) when the wave motion is independent of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\theta} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \phi :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \frac{1}{V^{2} } \frac{{\mathrm{\partial}} ^{2} {\mathrm{\psi}} }{{\mathrm{\partial}} t^{2} } =\frac{1}{r^{2} } \left[\frac{{\mathrm{\partial}} }{{\mathrm{\partial}} r} \left(r^{2} \frac{{\mathrm{\partial}} {\mathrm{\psi}} }{{\mathrm{\partial}} r} \right)\right]. \end{align}

(2.5c)

Solution

The wave equation in spherical coordinates is given in problem 2.6b. When we drop the derivatives with respect to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\theta}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\phi}} , the equation reduces to equation (2.5c). Writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\zeta}} = (r-Vt) , we proceed as in part (a). Starting with the right-hand side, we ignore Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): g\left(r+Vt\right) for the time being and obtain

${\begin{aligned}{\mathrm {\psi } }_{r}=-\left(1/r^{2}\right)f\left({\mathrm {\zeta } }\right)+\left(1/r\right)f'\left({\mathrm {\zeta } }\right),\\r^{2}{\mathrm {\psi } }_{r}=-f\left({\mathrm {\zeta } }\right)+rf'\left({\mathrm {\zeta } }\right),\\{\frac {\mathrm {\partial } }{{\mathrm {\partial } }r}}\left(r^{2}{\mathrm {\psi } }_{r}\right)=-f'\left({\mathrm {\zeta } }\right)+f'\left({\mathrm {\zeta } }\right)+rf''\left({\mathrm {\zeta } }\right)=rf''\left({\mathrm {\zeta } }\right),\\\left(1/r^{2}\right){\frac {\mathrm {\partial } }{{\mathrm {\partial } }r}}\left(r^{2}{\mathrm {\psi } }_{r}\right)=\left(1/r\right)f''\left({\mathrm {\zeta } }\right),\\\psi _{t}=-\left(V/r\right)f';{\quad }{\mathrm {\phi } }_{tt}=\left(V^{2}/r\right)f''.\end{aligned}}$

Substitution in equation (2.5c) shows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(1/r\right)f\left({\mathrm{\zeta}} \right) is a solution. In the same way we can show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(1/r\right)g\left({\mathrm{\xi}} \right) is also a solution, hence the sum is a solution.

2.6 Wave equation in cylindrical and spherical coordinates

2.6a Show that the wave equation (2.5a) can be written in cylindrical coordinates (see Figure 2.6a as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \frac{\partial ^{2} \psi }{\partial r^{2} } +\frac{1}{r} \frac{\partial \psi }{\partial r} +\frac{1}{r^{2} } \frac{\partial ^{2} \psi }{\partial \theta^{2} } +\frac{\partial ^{2} \psi }{\partial z^{2} } =\frac{1}{V^{2} } \frac{\partial ^{2} \psi }{\partial t^{2} } . \end{align}

(2.6a)

Solution

We shall solve by direct substitution. We have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=r\cos{\mathrm{\theta}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y=r\sin{\mathrm{\theta}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z=z , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r^{2} = x^{2} + y^{2} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {{\mathrm{\theta}}} =\tan^{-1} \left(y/x\right) . The following solution is lengthy, so we use subscripts to denote partial derivatives and write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} a=\sin {\mathrm{\theta}} = y/r,{{\mathrm{\quad}}} b= \cos{\mathrm{\theta}} =x/r. \end{align}

We shall require the derivatives:

${\begin{aligned}{\mathrm {\partial } }r/{\mathrm {\partial } }x=r_{x}=x/r=\cos {\mathrm {\theta } }=b,\\{\mathrm {\partial } }r/{\mathrm {\partial } }y=r_{y}=y/r=\sin {\mathrm {\theta } }=a;\\{\mathrm {\partial } }\theta /{\mathrm {\partial } }x=\theta _{x}=\left[{\frac {\mathrm {\partial } }{{\mathrm {\partial } }x}}\left(y/x\right)\right]/[1+\left(y/x\right)^{2}]=-{\frac {y}{x^{2}}}\left({\frac {1}{1+({\frac {y}{x}})^{2}}}\right)\\=-\left(\sin \theta \right)/r=-a/r,\\{\mathrm {\partial } }\theta /{\mathrm {\partial } }y=\theta _{y}=\left[{\frac {\mathrm {\partial } }{{\mathrm {\partial } }y}}\left(y/x\right)\right]/[1+\left(y/x\right)^{2}]={\frac {1}{x}}\left({\frac {1}{1+({\frac {y}{x}})^{2}}}\right)\\=\left(\cos \theta \right)/r=b/r.\end{aligned}}$

To replace Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\psi}} _{xx} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\psi}} _{yy} with derivatives with respect to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\theta} , we write:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \psi _{x} &=\psi _{r} r_{x} +\psi _{\theta} \theta _{x} =\psi _{r} b-\psi _{\theta} a/r,\\ \psi _{y} &=\psi _{r} r_{y} +\psi _{\theta} \theta _{y} =\psi _{r} a+\psi _{\theta} b/r. \end{align}

Then,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\psi}} _{xx} =\frac{{\mathrm{\partial}} }{{\mathrm{\partial}} r} \left({\mathrm{\psi}} _{r} b-{\mathrm{\psi}} _{\mathrm{{\mathrm{\theta}}}} a/r\right)r_{x} +\frac{{\mathrm{\partial}} }{{\mathrm{\partial}} {\mathrm{\theta}} } \left({\mathrm{\psi}} _{r} b-{\mathrm{\psi}} _{\mathrm{\theta}} a/r\right)\theta _{x}\\ =\left({\mathrm{\psi}} _{rr} b-{\mathrm{\psi}} _{r{\mathrm{\theta}}} a/r+{\mathrm{\psi}} _{\mathrm{{\mathrm{\theta}}}} a/r^{2} \right)b\\ {\quad} +\left({\mathrm{\psi}} _{r{\mathrm{\theta}}} b-{\mathrm{\psi}} _{r} a-{\mathrm{\psi}} _{\theta\theta} a/r-{\mathrm{\psi}} _{\mathrm{\theta}} b/r\right)\left(-a/r\right)\\ =[{\mathrm{\psi}} _{rr} b^{2} -{\mathrm{\psi}} _{r{\mathrm{\theta}}} \left(2ab/r\right)+{\mathrm{\psi}} _{\mathrm{\theta}} \left(2ab/r^{2} \right)+{\mathrm{\psi}} _{r} \left(a^{2} /r\right)+{\mathrm{\psi}} _{\theta\theta} \left(a/r)^{2} \right],\\ {\mathrm{\psi}} _{yy} =\frac{{\mathrm{\partial}} }{{\mathrm{\partial}} r} \left({\mathrm{\psi}} _{r} a+{\mathrm{\psi}} _{\mathrm{\theta}} b/r\right)r_{y} +\frac{{\mathrm{\partial}} }{{\mathrm{\partial}} {\mathrm{\theta}} } \left({\mathrm{\psi}} _{r} a+{\mathrm{\psi}} _{\mathrm{\theta}} b/r\right)\theta _{y} \\ =\left({\mathrm{\psi}} _{rr} a+{\mathrm{\psi}} _{r{\mathrm{\theta}}} b/r-{\mathrm{\psi}} _{\mathrm{{\mathrm{\theta}}}}, b/r^{2} \right)a\\ {\quad} +\left({\mathrm{\psi}} _{r{\mathrm{\theta}}} a+\psi _{r} b+\psi _{\theta\theta} b/r-\psi _{\mathrm{\theta}} a/r\right)\left(b/r\right)\\ =\left({\mathrm{\psi}} _{rr} a^{2} +\psi _{r{\mathrm{\theta}}} 2ab/r-{\mathrm{\psi}} _{\mathrm{\theta}} 2ab/r^{2} \right)+\left({\mathrm{\psi}} _{r} b^{2} /r+{\mathrm{\psi}} _{\theta\theta} b^{2} r^{2} \right). \end{align}

Thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\nabla} ^{2} {\mathrm{\psi}} ={\mathrm{\psi}} _{rr} \left(a^{2} +b^{2} \right)+{\mathrm{\psi}} _{r} \left(a^{2} +b^{2} \right)/r+{\mathrm{\psi}} _{\theta\theta} \left(a^{2} +b^{2} \right)/r^{2} +{\mathrm{\psi}} _{zz}, \end{align}

so

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \frac{{\mathrm{\partial}} ^{2} {\mathrm{\psi}} }{{\mathrm{\partial}} r^{2} } +\frac{1}{r} \frac{{\mathrm{\partial}} {\mathrm{\psi}} }{{\mathrm{\partial}} r} +\frac{1}{r^{2} } \frac{{\mathrm{\partial}} ^{2} {\mathrm{\psi}} }{{\mathrm{\partial}} {\mathrm{\theta}} ^{2} } +\frac{{\mathrm{\partial}} ^{2} {\mathrm{\psi}} }{{\mathrm{\partial}} z^{2} } =\frac{1}{V^{2} } \frac{{\mathrm{\partial}} ^{2} {\mathrm{\psi}} }{{\mathrm{\partial}} t^{2} }.

2.6b Transform the wave equation into spherical coordinates (see Figure 2.6b), showing that it becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \frac{1}{r^{2} } \left[\frac{\partial }{\partial r} \left(r^{2} \frac{\partial \psi }{\partial r} \right)+\frac{1}{\sin \theta} \frac{\partial }{\partial \theta} \left(\sin \theta\frac{\partial \psi }{\partial \theta} \right)+\frac{1}{{\sin}^{2} \theta} \frac{\partial ^{2} \psi }{\partial \phi ^{2} } \right]=\frac{1}{V^{2} } \frac{\partial ^{2} \psi }{\partial t^{2} }. \end{align}

(2.6b)

Solution

Spherical coordinates $\left(r,\;\theta ,\;\phi \right)$ and rectangular coordinates are related as follows (see Figure 2.6b):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} x=r\sin \theta \cos \phi, \\ y=r\sin \theta \sin \phi, \\ z=r\cos \theta, \\ r=(x^{2} +y^{2} +z^{2} )^{1/2}, \\ \theta ={\rm cos}^{-1} \left(z/r\right), \\ \phi =\tan ^{-1} \left(y/x\right). \end{align}

We continue to use subscripts to denote derivatives and letters to denote sines and cosines:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \begin{array}{l} a=\sin \theta, \\ m=\sin \phi, \\ a_{\mathrm{\theta}} =b, \\ m_{\phi} = n, \end{array}{\quad} \begin{array}{l} b=\cos\theta, \\ n=\cos\phi, \\ b_{\mathrm{\theta}} = -a, \\ n_{\phi} = -m. \end{array} \end{align}

The derivatives of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\theta} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \phi now become:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} r_{x} &=x/r=an,\\ r_{y} &=y/r=am,\\ r_{z} &=z/r=b;\\ \theta _{x} &=\frac{\partial }{\partial x} [{\rm cos}^{-1} \left(z/r\right)]=-\frac{\partial }{\partial r} \left(z/r\right)[1-(z/r)^{2} ]^{-1/2} =\frac{Z}{r^{2} } r_{x} \left(1/a\right)=bn/r,\\ \theta _{y} &=-\frac{\partial }{\partial y} \left(z/r\right)[1-(z/r)^{2} ]^{-1/2} =-\frac{Z}{r^{2} } r_{y} \left(1/a\right)=bm/r,\\ \theta _{z} &=-\frac{\partial }{\partial z} \left(z/r\right)[1-(z/r)^{2} ]^{-1/2} =-\left(1/r-z/r^{2} r_{z} \right)[1-(z/r)^{2} ]^{-1/2} \\ &=\left(-1/r\right)\left(1-zb/r\right)\left(1/a\right)=\left(-1/r\right)\left(1-b^{2} \right)\left(1/a\right)=-a/r,\\ \phi _{x} &=\frac{\partial }{\partial x} [\tan ^{-1} \left(y/x\right)]=\frac{\partial }{\partial x} \left(y/x\right)[1+(y/x)^{2} ]^{-1} =-\left(y/x^{2} \right)[1+(y/x)^{2} ]^{-1} \\ &=-\left(y/x\right)\left(1/x\right)[1+(y/x)^{2} ]^{-1} =-\left(m/n\right)\left(1/ran\right)(1/n^{2} )^{-1} =-m/ar,\\ \phi_{y} &=\frac{\partial }{\partial y} \left(y/x\right)[1+(y/x)^{2} ]^{-1} =\left(1/x\right)[1+(y/x)^{2} ]^{-1} =\left(1/ran\right)(1/n^{2} )^{-1} =n/ar,\\ \phi_z&= 0\ {\text{(because }\phi \text{ is not a function of }z)}. \end{align}

Summarizing these results, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \begin{array}{l} r_{x} =an, \\ \theta _{x} =bn/r,\\ \phi_{x} =-m/ar, \end{array}\quad \begin{array}{l} r_{y} =am, \\ \theta _{y} =bm/r,\\ \phi_{y} =n/ar, \end{array}\quad \begin{array}{l} r_{z} =b, \\ \theta_{z} =-a/r, \\ \phi_z=0. \end{array} \end{align}

We now calculate the derivatives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \phi _{xx} , etc.:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \psi _{x} &=\psi _{r} r_{x} +\psi _{\theta} \theta _{x} +\psi _{\phi} \phi_{x}\\ &=\psi _{r} an+\psi _{\theta} bn/r-\psi _{\phi} m/ar;\\ \psi _{xx} &=\frac{\partial }{\partial r} \left(\psi _{r} an+\psi _{\theta} bn/r-\psi _{\phi} m/ar\right)\left(an\right)\\ &\quad +\frac{\partial }{\partial \theta } \left(\psi _{r} an+\psi _{\theta} bn/r-\psi _{\phi} m/ar\right)\left(bn/r\right)\\ &\quad+\frac{\partial }{\partial \theta } \left(\psi _{r} an+\psi _{\theta} bn/r-\psi _{\phi} m/ar\right)\left(-m/ar\right), \\ &=\left(\psi _{rr} an+\psi _{r\theta} bn/r-\psi _{\theta} bn/r^{2} -\psi _{\gamma \phi} m/ar+\psi _{\phi} m/ar^{2} \right)an\\ &\quad +(\psi _{r\theta} an+\psi _{r} bn+\psi _{\theta\theta} bn/r-\psi _{\theta} an/r-\psi _{\theta\phi} m/ar\\ &\quad+\psi _{\phi} mb/a^{2} r)\left(bn/r\right)+(\psi _{\gamma \phi} an-\psi _{r} am+\psi _{\theta\phi} bn/r\\ &\quad -\psi _{\theta} bm/r-\psi _{\phi\phi} m/ar-\psi _{\phi} n/ar)\left(-m/ar\right)\\ &=\psi _{rr} a^{2} n^{2} +\psi _{r\theta} \left(2abn^{2} /r\right)-\psi _{\gamma \phi} \left(2mn/r\right)+\psi _{r} \left(\frac{b^{2} n^{2} +m^{2} }{r} \right)\\ &\quad +\psi _{\theta\theta} \; \left(\frac{bn}{r} \right)^{2} +\psi _{\theta\phi} \left(\frac{b}{a} \right)\left(\frac{2mn}{r} \right)+\psi _{\theta} \; \left(\frac{bm^{2} }{ar^{2} } -\frac{2abn}{r^{2} } \right)\\ &\quad +\psi _{\phi\phi} \left(\frac{m}{ar}\right)^{2} +\psi _{\phi} \left(\frac{2mn}{a^{2} r^{2} } \right) ;\\ \psi _{y} &=\psi _{r} r_{y} +\psi _{\theta} \theta _{y} +\psi _{\phi} \phi _{y} \\ &=\psi _{r} am+\psi _{\theta} bm/r+\psi _{\phi} n/ar;\\ \psi _{yy} &=\frac{\partial }{\partial r} \left(\psi _{r} am+\psi _{\theta} bm/r+\psi _{\phi} n/ar\right)\left(am\right)\\ &\quad +\frac{\partial }{\partial \theta } \left(\psi _{r} am+\psi _{\theta} bm/r+\psi _{\phi} n/ar\right)\left(bm/r\right)\\ &\quad +\frac{\partial }{\partial \phi } \left(\psi _{r} am+\psi _{\theta} bm/r+\psi _{\phi} n/ar\right)\left(n/ar\right)\\ &=\left(\psi _{rr} am+\psi _{r\theta} bm/r-\psi _{\theta} bm/r^{2} +\psi _{\gamma \phi} n/ar-\psi _{\phi} n/ar^{2} \right)am\\ &\quad +(\psi _{r\theta} am+\psi _{r} bm+\psi _{\theta\theta} bm/r-\psi _{\theta} am/r+\psi _{\theta\phi} n/ar\\ &\quad -\psi _{\phi} bn/a^{2} r)\left(bm/r\right)\\ &\quad+(\psi _{\gamma\phi} am+\psi _{r} an+\psi _{\theta\phi} bm/r+\psi _{\theta} bn/r+\psi _{\phi\phi} n/ar\\ &\quad-\psi _{\phi} m/ar)\left(n/ar\right)\\ &=\psi _{rr} a^{2} m^{2} +\psi _{r\theta} \; \left(\frac{2abm}{r} \right)+\psi _{\gamma\phi} \left(\frac{2mn}{r} \right)+\psi _{r} \left(\frac{1-a^{2} m^{2} }{r} \right)\\ &\quad +\psi _{\theta\theta} \; \left(\frac{bm}{r}\right)^{2} +\psi _{\theta\phi} \left(\frac{b}{a} \right)\left(\frac{2mn}{r^{2} } \right)+\psi _{\theta} \; \left(-\frac{2abm^{2} }{r} +\frac{bn^{2} }{ar^{2} } \right)\\ &\quad+\psi _{\phi\phi} \left(\frac{n}{ar}\right)^{2_{-\psi } } \phi\left(\frac{mn}{r^{2} } \right)\left(1+\frac{b^{2} }{a^{2} } +\frac{1}{a^{2} } \right) ;\\ \psi _{z} &=\psi _{r} r_{z} +\psi _{\theta} \theta _{z} +\psi _{\phi} \phi_{z} =\psi _{r} b-\psi _{\theta} a/r;\\ \psi _{zz} &=\left(\psi _{rr} b-\psi _{r\theta} a/r+\psi _{\theta} a/r^{2} \right)b\\ &\quad+\left(\psi _{r\theta} b-\psi _{r} a-\psi _{\theta\theta} a/r-\psi _{\theta} b/r\right)\left(-a/r\right)\\ &=[\psi _{rr} b^{2} -\psi _{r\theta} \left(2ab/r\right)+\psi _{r} \left(a^{2} /r\right)+\psi _{\theta\theta} \left(a/r)^{2} +\psi _{\theta} \left(2ab/r^{2} \right)\right]. \end{align}

Adding the three derivatives, we get

${\begin{aligned}\nabla ^{2}\psi &=\psi _{rr}\left(a^{2}m^{2}+a^{2}n^{2}+b^{2}\right)+\psi _{r\theta }\;\left[{\frac {2ab\left(m^{2}+n^{2}\right)}{r}}-{\frac {2ab}{r}}\right]\\&\quad +\psi _{r}\left[{\frac {\left(b^{2}n^{2}+m^{2}\right)+\left(1-a^{2}m^{2}\right)+a^{2}}{r}}\right]\\&\quad +\psi _{\theta \theta }\;\left({\frac {b^{2}n^{2}+b^{2}m^{2}+a^{2}}{r^{2}}}\right)\\&\quad +\psi _{\theta }\;\left[{\frac {\left(bm^{2}/a-2abm^{2}\right)+\left(-2abm^{2}+bn^{2}/a+2ab\right)}{r}}\right]\\&\quad +\psi _{\phi \phi }\left[(-m/ar)^{2}+\left(n/ar\right)^{2}\right]\\&=\psi _{rr}+\left({\frac {2}{r}}\right)\psi _{r}+\left({\frac {1}{r^{2}}}\right)\psi _{\theta \theta }\;+\left({\frac {\cot \theta }{r^{2}}}\right)\psi _{\theta }\;+\left({\frac {1}{a^{2}r^{2}}}\right)\psi _{\phi \phi }.\end{aligned}}$

Substituting the values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n , we get for the wave equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} r^{2} } +\frac{2}{r} \frac{{\mathrm{\partial}} \psi }{{\mathrm{\partial}} r} +\frac{1}{r^{2} } \frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} \theta ^{2} } +\left(\frac{{\rm \; cot\; }\theta }{r^{2} } \right)\frac{{\mathrm{\partial}} \psi }{{\mathrm{\partial}} \theta } +\left(\frac{1}{r^{2} {\rm sin}^{2} \theta } \right)\frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} \phi ^{2} } =\frac{1}{V^{2} } \frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} t^{2} }. \end{align}

This is often written in the more compact form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \frac{1}{r^{2} } \left[\frac{{\mathrm{\partial}} }{{\mathrm{\partial}} r} \left(r^{2} \frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} r} \right)+\frac{1}{\sin \theta } \frac{{\mathrm{\partial}} }{{\mathrm{\partial}} \theta } \left(\sin \theta \frac{{\mathrm{\partial}} \psi }{{\mathrm{\partial}} \theta } \right)+\frac{1}{{\rm sin}^{2} \theta } \frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} \phi ^{2} } \right]=\frac{1}{V^{2} } \frac{{\mathrm{\partial}} ^{2} \psi }{{\mathrm{\partial}} t^{2}}. \end{align}

2.7 Sum of waves of different frequencies; Group velocity

2.7a A pulse composed of two frequencies, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \omega _{0} \pm {\mathrm{\Delta}} \omega , can be represented by factors involving the sum and difference of the two frequencies. If the two components have the same amplitudes, we can write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} A \cos \left(\kappa_{1} x-\omega _{1} t\right),{\quad} A \cos \left(\kappa_{2} x- \omega_{2} t\right), \end{align}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \omega _{1} =\omega _{0} +\Delta \omega, \omega_2= \omega_0 -\Delta \omega, k_{0} =2\pi /\lambda _{0} =\omega _{0} /V ,

${\begin{aligned}\kappa _{1}&\approx \kappa _{0}+\Delta \kappa \approx \left(\omega _{0}+\Delta \omega \right)/V,\ \mathrm {and} \\\kappa _{2}&\approx \kappa _{0}-\Delta \kappa \approx \left(\omega _{0}-\Delta \omega \right)/V.\end{aligned}}$

Show that the composite wave is given approximately by the expression

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} B\cos (\kappa_{0} x-\omega _{0} t), \end{align}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B=2A\cos \left\{{\mathrm{\Delta}} \kappa\left[x-\left({\mathrm{\Delta}} \omega /{\mathrm{\Delta}} \kappa\right)t\right]\right\} .

Background

When different frequency components in a pulse have different phase velocities Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V (the velocity with which a given frequency travels), the pulse changes shape as it moves along. The group velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): U is the velocity with which the envelope of the pulse travels.

The envelope of a pulse comprises two mirror-image curves that are tangent to the waveform at the peaks and troughs, and therefore define the general shape of the pulse.

Solution

Adding the two components and using the identity

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \cos \theta +\cos \phi =2\cos \left[\frac{1}{2} \left(\theta +\phi \right)\right]\cos \left[\frac{1}{2} \left(\theta -\phi \right)\right], \end{align}

we get for the composite wave

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 2A\cos \left\{{\mathrm{\Delta}} \kappa\left[x-\left({\mathrm{\Delta}} \omega /{\mathrm{\Delta}} \kappa\right)t\right]\right\}\cos \left(\kappa_{0} x-\omega _{0} t\right)=B\cos \left(\kappa_{0} x-\omega _{0} t\right). \end{align}

2.7b Why do we regard Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B as the amplitude? Show that the envelope of the pulse is the graph of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B plus its reflection in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x -axis.

Solution

The solution in (a) can be written $\psi =B\psi _{0}$ . We regard Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B as the amplitude for two reasons: (1) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B repeats every time that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x increases by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2\pi /{\mathrm{\Delta}} \kappa or each time that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t increase by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2\pi /{\mathrm{\Delta}} \omega . But Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\Delta}} \kappa\ll \kappa_{0} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\Delta}} \omega \ll \omega _{0} \} , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B must repeat more slowly than $\psi _{0}$ . (2) Each time that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi _{0} attains its limiting values, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \pm 1 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi has the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \pm B and therefore never exceeds Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left|B\right| ; thus the curves of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): +\left|B\right| and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): -\left|B\right| pass through the maxima and minima of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi _{0} and therefore constitute the envelope.

2.7c Show that the envelope moves with the group velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): U where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} U=\frac{{\mathrm{\Delta}} \omega }{{\mathrm{\Delta}} \kappa} \approx \frac{\mathrm{d}\omega }{\mathrm{d}\kappa} \approx V-{\mathrm{\lambda}} \frac{\mathrm{d}V}{\mathrm{d}{\mathrm{\lambda}} } =V+\omega \frac{\mathrm{d}V}{\mathrm{d}\omega } \end{align}

(2.7a)

(see Figure 2.7a).

Solution

If we consider the quantity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B as a wave superimposed on the primary wavelet Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \cos \left(\kappa_{0} x-\omega _{0} t\right) , comparison with the basic wave type Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(x-Vt) of problem 2.5a shows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left[x-\left({\mathrm{\Delta}} \omega /{\mathrm{\Delta}} \kappa\right)t\right] takes the place of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (x-Vt) , i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left({\mathrm{\Delta}} \omega /{\mathrm{\Delta}} \kappa\right) is the velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): U with which the envelope travels. In the limit, $$ U $$ is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} U=\mathrm{d}\omega /\mathrm{d}{\kappa} =\mathrm{d}f/\mathrm{d}\left(1/{\mathrm{\lambda}} \right)=\left(\mathrm{d}f/\mathrm{d}{\mathrm{\lambda}} \right)\left[\mathrm{d}{\mathrm{\lambda}} /\mathrm{d}\left(1/{\mathrm{\lambda}} \right)\right], \end{align}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f is the frequency. The result is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} U=-{\mathrm{\lambda}} ^{2} \frac{\mathrm{d}f}{\mathrm{d}\lambda}. \end{align}

We introduce the phase velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V by noting that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V={\mathrm{\lambda}} f , so

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mathrm{d}V={\mathrm{\lambda}} \mathrm{d}f+f\mathrm{d}\lambda, \; {\mathrm{\lambda}} \frac{\mathrm{d}f}{\mathrm{d}{\mathrm{\lambda}} } =\frac{\mathrm{d}V}{\mathrm{d}{\mathrm{\lambda}} } -f. \end{align}

Thus,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} U=-{\mathrm{\lambda}} \left({\mathrm{\lambda}} \frac{\mathrm{d}V}{\mathrm{d}{\mathrm{\lambda}} } \right)={\mathrm{\lambda}} f-{\mathrm{\lambda}} \frac{\mathrm{d}V}{\mathrm{d}{\mathrm{\lambda}} } =V-{\mathrm{\lambda}} \frac{\mathrm{d}V}{\mathrm{d}\lambda}. \end{align}

To replace the derivative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{d}V/\mathrm{d}{\mathrm{\lambda}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{d}V/\mathrm{d}f , we find the relation between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}}/\mathrm{d}{\mathrm{\lambda}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f/\mathrm{d}f ; to do this we let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V be constant so that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V={\mathrm{\lambda}} f,\; \mathrm{d}V=0={\mathrm{\lambda}} \mathrm{d}f+f\mathrm{d}\lambda, \end{align}

hence,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\lambda}} /\mathrm{d}{\mathrm{\lambda}} =-f/\mathrm{d}f,\ \mathrm{and}\ U=V+f\frac{\mathrm{d}V}{\mathrm{d}f} =V+\omega \frac{\mathrm{d}V}{\mathrm{d}\omega }. \end{align}

2.8 Magnitudes of seismic wave parameters

The magnitudes of period $$ T $$ , wavelength Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\lambda} , wavenumber Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\kappa} , frequency Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f , and angular frequency Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\omega} are important in practical situations. Calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\lambda} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\kappa} for 15 and 60 Hz for the following velocity situations:

weathering, 100 and 500 m/s (minimum and average values);
water, 1500 m/s;
sands and shales, 2000 (poorly consolidated) and 3300 m/s;
limestone, 4300 (porous) and 5500 m/s;
salt, 4600 m/s;
anhydrite, 6100 m/s.

Solution

The period Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T equals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 1/\it{f} , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \it{T}=0.067 s for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \it{f}=15 Hz and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \it{T}=0.017 s for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \it{f}=60 Hz. Also, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\lambda} =V/f , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\kappa}=2\mathrm{\pi} /\mathrm{\lambda} =2\mathrm{\pi} f/V . Using these equations we get the values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\lambda} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\kappa} in Table 2.8a.

Table 2.8a. Magnitudes of **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T** , **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\lambda}** , and $\mathrm {\kappa }$ .
		For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \it{f} = 15 Hz			For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \it{f} = 60 Hz
	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V (km/s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\lambda} (m)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\kappa}(\mathrm{m}^{-1})	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\lambda} (m)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\kappa}(\mathrm{m}^{-1})
Weathering (min.)	0.1	0.067	7	0.7	0.017	2	4
Weathering (avg.)	0.5	0.067	30	0.2	0.017	8	0.8
Water	1.5	0.067	100	0.063	0.017	25	0.25
Poorly consolidated sandstone-shales at 0.75 km	2.0	0.067	130	0.047	0.017	33	0.19
Tertiary clastics at 3.00 km	3.3	0.067	220	0.029	0.017	55	0.11
Porous limestone	4.3	0.067	290	0.022	0.017	72	0.088
Dense limestone	5.5	0.067	370	0.017	0.017	92	0.069
Salt	4.6	0.067	310	0.020	0.017	77	0.082
Anhydrite	6.1	0.067	410	0.015	0.017	100	0.062

2.9 Potential functions used to solve wave equations

2.9a Show that equation (2.9a) relating the potential functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\phi} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\chi} to the vector displacement Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\zeta} requires that ${\Delta }$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\theta_{z}} [see equations (2.1e) and (2.1g)] be solutions of the P- and S-wave equations, that is, of equation (2.5a) with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\psi} replaced by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\Delta} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\theta_{z}} , respectively.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathbf{\zeta =\nabla \left(\mathrm{\phi} +\frac{\partial \chi }{\partial z} \right)-\nabla ^{2} \mathrm{\chi} k}},

(2.9a)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\phi} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\chi} being solutions of the P- and S-wave equations, respectively.

Background

The dilatation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\Delta} and component of rotation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\theta}_{x} are defined in equations (2.1e,g).

While solutions of the wave equation (see problem 2.5) furnish values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\Delta} or a component of rotation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\theta}_{i} , we often need to know the displacements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(u,\; v,\; w\right) (defined in problem 2.1) which are not easily found given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\Delta} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\theta}_{i} . This difficulty can be avoided by using potential functions that are solutions of the wave equations and from which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(u,\; v,\; w\right) , hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\theta}_{i} also, can be found by differentiation.

Note that derivatives of a solution of a differential equation are also solutions.

The vector operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \nabla (called “del”) and its properties are discussed in Sheriff and Geldart, 1995, Section 15.1.2c.

Solution

From equation (2.1e) and the definition of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \nabla , we get for the dilatation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \Delta &={\nabla} {\cdot} {\zeta} =\nabla ^{2} \left(\mathrm{\phi} +\frac{\partial \mathrm{\chi} }{\partial z} \right)-\left[{\nabla} {\cdot} {k}\right]\left(\nabla ^{2} \mathrm{\chi} \right) \\ &=\nabla ^{2} \mathrm{\phi} +\nabla ^{2} \left(\frac{\partial \mathrm{\chi} }{\partial z} \right)-\left(\frac{\partial }{\partial z} \right)\left(\nabla ^{2} \mathrm{\chi} \right)\\ &=\nabla ^{2} \mathrm{\phi},\end{align}

(2.9b)

since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \nabla ^{2} \left(\frac{\partial \mathrm{\chi} }{\partial z} \right)=\frac{\partial }{\partial z} \left(\nabla ^{2} \mathrm{\chi} \right) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \nabla \cdot k=\partial /\partial z . Because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\phi} is a solution of the P-wave equation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\Delta} must also be a solution.

We have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\theta}&=\frac{1}{2} \nabla \times {\zeta} =\frac{1}{2} {\nabla} \times {\nabla} \left(\mathrm{\phi} +\frac{\partial \mathrm{\chi} }{\partial z} \right)-\frac{1}{2} {\nabla} \times \left[\left(\nabla ^{2} \mathrm{\chi} \right)k\right]\\ &=0-\frac{1}{2} {\nabla} \times \left[\left(\nabla ^{2} \mathrm{\chi} \right)k\right]=\frac{1}{2} \nabla ^{2} \left(\chi _{x} -\chi _{y} \right)\left(\chi _{y} i-\chi _{x} j\right) \end{align}

[see Sheriff and Geldart, 1995, equations (15.13) and (15.9)]. Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\chi} is a solution of the S-wave equation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\theta} is also a solution.

2.9b In two dimensions, the potential function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \zeta can be defined as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {{\zeta = {\mathbf{\nabla \phi}} +{\nabla} \times {\chi}, \quad {\chi} =-\left|{\chi} \right|j}}. \end{align}

(2.9c)

Show how to obtain the displacements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathbf{\left(u,\; v,\; w\right)}} , the dilatation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\Delta} , and rotation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\theta} from this equation (see Sheriff and Geldart, 1995, Section 15.1.2c and problem 15.5c).

Solution

From equation (2.1d) we see that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u is the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x -component of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\zeta} , that is, of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\nabla}\mathrm{\phi} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\nabla} \times {\chi} , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u=\frac{\partial \mathrm{\phi}}{\partial x} +x -component of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\nabla} \times {\chi} . From Sheriff and Geldart, 1995, equation (15.13) we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\nabla} \times {\chi} =\left|\begin{array}{ccc} i & j & k \\ {\frac{\partial }{\partial x} } & {\frac{\partial }{\partial y} } & {\frac{\partial }{\partial z} } \\ {0} & {-\mathrm{\chi} } & {0} \end{array}\right|=\frac{\partial \mathrm{\chi} }{\partial z} i-\frac{\partial \mathrm{\chi} }{\partial x} k=\mathrm{\chi}_{z} i-\mathrm{\chi}_{x} k. \end{align}

Thus,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} u=\mathrm{\phi}_{x} +\mathrm{\chi}_{z} \end{align}

(2.9d)

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} w= \mathrm{\phi}_z -\mathrm{\chi}_{x}. \end{align}

(2.9e)

To get the dilatation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta , we use equation (2.1e) and Sheriff and Geldart, 1995, problem 15.5c and obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \Delta ={\nabla} {\cdot} {\zeta} = \nabla ^{2} \mathrm{\phi}. \end{align}

(2.9f)

The rotation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\theta} can be obtained by taking the curl of equation (2.9c) but an easier method is to substitute equations (2.9d) and (2.9e) in equation (2.1g). This gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\theta} = \frac{1}{2}\left|\begin{array}{ccc} i & j & k \\ \frac{\partial}{\partial x} & 0 & \frac{\partial}{\partial z} \\ u & 0 & w \end{array}\right| = \frac{1}{2}\left(\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}\right)j \end{align}

Thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\theta} has only a $$ y $$ -component given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mathrm{\theta}_{y} =\frac{1}{2} \left[\frac{\partial }{\partial z} \left(\mathrm{\phi}_{x} +\mathrm{\chi}_{z} \right)-\frac{\partial }{\partial x} \left(\mathrm{\phi}_z -\mathrm{\chi}_{x} \right)\right]=\frac{1}{2} \nabla ^{2} \mathrm{\chi}. \end{align}

2.10 Boundary conditions at different types of interfaces

2.10a Justify on physical grounds the boundary conditions for solid-fluid media in contact.

Background

The basic principles underlying the boundary conditions are (i) all stresses must be continuous (that is, no changes in values) at the interface; this must be true to avoid unbalanced forces which would produce accelerations; and (ii) strains must be continuous at the interface; this must be true for normal strains to avoid interpenetration of one medium into the other, or creation of a vacuum between them, or sliding of solid on solid for unequal tangential strains.

When a wave is incident on a boundary, one to four boundary conditions must be satisfied, depending on the types of media. The angles of reflection and refraction are fixed by the laws of reflection and refraction [see equation (3.1a)], so the only parameters that can be adjusted to satisfy these conditions are the relative amplitudes of reflected and/or refracted P- and S-waves generated by the incident wave. An S-wave generated by an incident P-wave, or a P-wave generated by an incident S-wave, is called a converted wave.

Solution

Normal stress and displacement (strains) are continuous, stress to avoid normal acceleration and displacement to avoid interpenetration or a vacuum. Tangential stress is zero everywhere in the fluid and so the tangential stress in the solid must be zero at the interface. There is no restriction on tangential displacement.

2.10b Justify on physical grounds the boundary conditions for solid-vacuum media in contact.

Solution

Stresses are zero in a vacuum, so normal and tangential stresses in the solid vanish at the interface.

2.10c Justify on physical grounds the boundary conditions for fluid-fluid media in contact.

Solution

Normal stresses and displacements are continuous at the interface for the same reasons as in (a). Tangential stresses cannot exist in a fluid, and tangential displacements are zero.

2.10d Justify on physical grounds the boundary conditions for fluid-vacuum media in contact.

Solution

Stresses are zero in a vacuum, so the normal stress in the fluid vanishes at the interface.

2.11 Boundary conditions in terms of potential functions

Using the definitions of stress and strain in problem 2.1 and the potential functions in equation (2.9b), show that the boundary conditions at the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {xy} -plane separating two semi-infinite solids require that, for a wave traveling in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {xz} -plane, the following functions must be continuous:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \left({\phi}_{z} -{\chi}_{x} \right),\qquad \left({\phi}_{x} +{\chi}_{z} \right), \end{align}

(2.11a)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathbf{\lambda \nabla ^{2} \phi}} +{\mathbf{2\mu}}\left({\phi}_{zz} -{\chi}_{xz} \right),\qquad {\mu} \left({\mathbf{2\phi}}_{xz} +{\chi}_{zz} -{\chi}_{xx} \right), \end{align}

(2.11b)

where subscripts denote partial derivatives. These terms are, respectively, the normal and tangential stressses and the normal and tangential displacements.

Background

As stated in problem 2.10, all stresses and displacements must be continuous at an interface between two different media.

Solution

From equations (2.9d) and (2.9e) we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} u=\mathrm{\phi}_{x} +\mathrm{\chi}_{z}, \; w=\mathrm{\phi} -\mathrm{\chi}_{x}. \end{align}

The normal displacement is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): w while the tangential displacement is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u , so these two functions must be continuous.

The normal stress is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\sigma}_{zz} and equations (2.1b), (2.1h), (2.9e), and (2.9f) show that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mathrm{\sigma}_{zz} =\mathrm{\lambda}\Delta +2\mathrm{\mu} \mathrm{\varepsilon}_{zz} =\mathrm{\lambda} \nabla ^{2} \mathrm{\phi} +2\mathrm{\mu} w_{z} =\mathrm{\lambda} \nabla ^{2} \mathrm{\phi} +2\mathrm{\mu} \left(\mathrm{\phi}_{z} -\mathrm{\chi}_{xz} \right), \end{align}

the last step being obtained by differentiating the above expression for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): w .

The tangential stress is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\sigma}_{xz} and equations (2.1c), (2.1i), (2.9d), and (2.9e) give

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mathrm{\sigma}_{xz} =\mathrm{\mu \varepsilon}_{xz} =\mathrm{\mu} \left(u_{z} +w_{x} \right)=\mathrm{\mu} \left(2\mathrm{\phi}_{xz} +\mathrm{\chi}_{zz} -\mathrm{\chi}_{xx} \right). \end{align}

Since the normal and tangential stresses must be continuous, these two functions must also be continuous.

2.12 Disturbance produced by a point source

A source of seismic waves produces on a spherical cavity of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathbf{\mathit{r_{0}}}} enclosing the source a step displacement of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mathrm{step}_0 (t)&=k,\qquad t\ge 0,\\ &=0,\qquad t\le 0. \end{align}

Starting with equation (2.12a) below, show that the displacement at distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {r} is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} u=\frac{r_{0}^{2} k}{r} \left[\left(\frac{1}{r_0}-\frac{1}{r}\right)e^{-V\zeta /r_{0}} + \frac{1}{r} \right]. \end{align} Is the motion oscillatory? What is the final (permanent) displacement?

Background

When a source, such as an explosion, creates very high stresses, the wave equation does not apply near the the source because the medium does not obey Hooke’s law in this region. For a symmetrical point source, this situation can be handled mathematically by enclosing the source with a spherical surface centered at the source and specifying the displacement at all points on the spherical surface at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t=0 . If the source generates a wave such that the displacement at each point on the surface of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r_{0} is

${\begin{aligned}u_{0}\;\left(r_{0},\;t\right)&=ke^{-at},\quad k>0,\;t\geq 0,\;a>0,\\&=0,\qquad \quad t\leq 0,\end{aligned}}$

the displacement Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u\left(r,\; t\right) is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} u(r, t)=\frac{r_{0} k}{r\left(V/r_{0} -a\right)} \left(\frac{V}{r_{0} } e^{-V\zeta /r_{0} } -ae^{-a\zeta } -\frac{V}{r} e^{-V\zeta /r_{0} } +\frac{V}{r} e^{-a\zeta } \right), \end{align}

(2.12a)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \zeta =t-\left(r-r_{0} \right)/V [see Sheriff and Geldart, 1995, Section 2.4.5, equations (2.76) and (2.77)]. The step function, step (t) , is defined in Sheriff and Geldart, 1995, Section 15.2.5.

Solution

Equation (2.12a) gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u\left(r,\; t\right) when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u_{0} \left(r_{0}, \; t\right)=ke^{-at} . If we let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a\to 0 , in the limit when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a=0 , the displacement of the spherical surface becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mathrm{step}_0 (t)&=k,\quad t\ge 0, \\ &=0,\quad t\le 0, \end{align}

which is the given type of source. Setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a=0 in equation (2.12a) we find that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} u\left(r,\; t\right)=\left(r_{0}^{2} k/r\right)\left[\left(1/r_{0} -1/r\right)e^{-V\zeta /r_{0} } +1/r\right]. \end{align}

If the motion is oscillatory, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u\left(r,\; t\right) must change sign at least once, that is, the value of the expression in the square brackets must pass through zero. But Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r>r_{0} , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(r_{0}^{2} k/r\right)(1/r_{0} -1/r)>0 and the exponential term is always positive, therefore oscillation is not possible.

At Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t=+\infty, \zeta =+\infty, u\left(r,\; \infty \right)=k(r_{0} /r)^{2} , which is the permanent displacement.

2.13 Far- and near-field effects for a point source

(a) Show that for harmonic waves of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \phi =\left(A/r\right)\cos [\omega (r/V-t)], \end{align}

(2.13a)

the displacement Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u\left(r,\; t\right) is

{\begin{aligned}u\left(r,\;t\right)=-\left({\frac {A}{r^{2}}}\right)\cos \left[\omega \left(r/V-t\right)\right]-\left({\frac {A}{r}}\right)\left({\frac {\omega }{V}}\right)\sin[\omega (r/V-t)].\end{aligned}}

(2.13b)

Background

If we set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\chi} =0 in equation (2.9a), we obtain the result Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\zeta} =\nabla \mathrm{\phi} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\phi} is a solution of the P-wave equation. Furthermore if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\phi} is independent of latitude and longitude, the wave equation reduces to equation (2.5c), and the solution of problem 2.5c shows that equation (2.13a) is a P-wave solution of equation (2.5c).

Solution

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\phi} in equation (2.13a) is a solution of equation (2.5c), it represents a spherically symmetrical P-wave and therefore the only displacement is along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {r} . If we take the x-axis along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {r} , equation (2.9d) shows that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} u\left(r,\; t\right)=\frac{\partial \phi }{\partial r} =-\left(\frac{A}{r^{2} } \right)\cos \left[\omega \left(r/V-t\right)\right]-\left(\frac{A}{r} \right)\left(\frac{\omega }{V} \right)\sin [\omega (r/V-t)]. \end{align}

2.13b Show that the two terms in equation (2.13b), which decay at different rates, are of equal importance at distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathbf{r=\lambda /2\pi}} .

Solution

The two terms are of equal importance when the two amplitudes are equal, that is, when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A/r^{2} =\left(A\omega /rV\right) or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r=V/\mathrm{\omega} =\mathrm{\lambda} /2\pi .

2.14 Rayleigh-wave relationships

2.14a Show that, when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathbf{\sigma =0.25}} , the Rayleigh‐wave potentials, equation (2.14c), become

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathbf{\phi =Ae^{-0.847\kappa z} e^{\mathrm{j}\kappa\left(x-V_{R} t\right)}}}, \end{align}

and

${\begin{aligned}{\mathbf {\chi =1.466jAe^{-0.394{\kappa z}}e^{j\kappa \left(x-V_{R}t\right)}} },\end{aligned}}$

and that the displacements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {u} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {w} at depth Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {z} are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {u}&={\mathbf{\kappa A\left(-e^{-0.847\kappa z} +0.578e^{-0.394{\kappa z} } \right)\sin \kappa\left(x-V_{R} t\right)}}, \end{align}

(2.14a)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {w}&={\mathbf{\kappa A\left(-0.847e^{-0.847\kappa z} +1.466e^{-0.394{\kappa z} } \right)\cos \kappa\left(x-V_{R} t\right)}}. \end{align}

(2.14b)

Background

When a medium is divided by the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): xy -plane into two semi-infinite media having different properties, surface waves are propagated parallel to the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): xy -plane, the amplitude decreasing with increasing distance from the plane. When one medium is a solid and the other a vacuum, the surface wave is known as a Rayleigh wave. The near-equivalent at the surface of the real earth, a pseudo-Rayleigh wave, is called ground roll.

We take the potential functions of equation (2.9b) in the form

{\begin{aligned}\mathrm {\phi } =Ae^{-m\mathrm {\kappa } z}e^{j\mathrm {\kappa } \left(x-V_{R}t\right)}\;,\;\chi =Be^{-n\mathrm {\kappa } z}e^{j\mathrm {\kappa } \left(x-V_{R}t\right)},\end{aligned}}

(2.14c)

where the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z -axis is positive downward, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n are real positive constants (so that the amplitudes decrease as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z increases) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{R} is the Rayleigh-wave velocity. We can take either the real or the imaginary parts of the functions as a solution, the only difference being the phase. When we substitute these functions in the P- and S-wave equations [equation (2.5a) with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\psi} replaced with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\theta}_{y} , respectively), we find that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n must satisfy the equations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} m^{2} =\left[1-\left(V_{R} /\mathrm{\alpha} \right)^{2} \right]\;, \; n^{2} =\left[1-\left(V_{R} /\mathrm{\beta} \right)^{2} \right]. \end{align}

(2.14d)

Because both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n must be real, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{R} <\mathrm{\beta} <\mathrm{\alpha} .

When we apply the boundary conditions of problem 2.10b, we find that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} B=\left(\frac{2\mathrm{j}m}{1+n^{2} } \right)A, \end{align}

and that the velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{R} is a root of the equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} x^{3} -8x^{2} +\left[24-16(\mathrm{\beta} /\mathrm{\alpha} )^{2} \right]x+16\left[\left(\mathrm{\beta} /\mathrm{\alpha} \right)^{2} -1\right]=0, \end{align}

(2.14e)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=(V_{R} /\mathrm{\beta} )^{2}, (\mathrm{\beta} /\mathrm{\alpha} )^{2} =\left(1-2\mathrm{\sigma} \right)/2\left(1-\mathrm{\sigma} \right) from equation (1,8) in Table 2.2a.

The left-hand side of equation (2.14e) is negative when $$ x=0 $$ , positive for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=+1 , so a root must exist in the range Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 0<x<+1 . When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\sigma} =0.25 , the root is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{R} =0.919\mathrm{\beta} .

The angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\theta} in Figure 2.14a is given by the equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mathrm{\theta} = \tan^{-1} \left(-w/u\right) \end{align}

(2.14f)

(the minus sign is necessary because we have taken Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): w positive downward).

Solution

When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\sigma} =0.25, V_{R} /\mathrm{\beta} =0.919 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (\mathrm{\beta} /\mathrm{\alpha} )^{2} =\left(1-2\mathrm{\sigma} \right)/2\left(1-\mathrm{\sigma} \right)=1/3 , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(V_{R} /\mathrm{\alpha} \right)=\left(V_{R} /\mathrm{\beta} \right)\times \left(\mathrm{\beta} /\mathrm{\alpha} \right)=0.919/\sqrt{3} =0.531 . Using these values, equation (2.14d) gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m= 0.848 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n=0.393 ; also, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B=2\mathrm{j}m/\left(1+n^{2} \right)A=1.468\mathrm{j}A ; the j indicates that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\phi} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\chi} are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 90^{\circ} out of phase. The potential functions are now

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mathrm{\phi} =Ae^{-0.848 \mathrm{\kappa} z} e^{\mathrm{j}\mathrm{\kappa} \left(x-V_{R} t\right)}, \end{align}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mathrm{\chi} =1.468\mathrm{j}Ae^{-0.393 \mathrm{\kappa} z } e^{\mathrm{j}\mathrm{\kappa} \left(x-V_{R} t\right)}. \end{align}

From equations (2.9d) and (2.9e) we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} u=\mathrm{\phi} _{x} +\mathrm{\chi} _{z} =\mathrm{j}\mathrm{\kappa} \mathrm{\phi} -0.393 \mathrm{\kappa}\mathrm{\chi} \;, \; w=\mathrm{\phi}_z -\mathrm{\chi} _{x} =-0.848 \mathrm{\kappa} \mathrm{\phi} -\mathrm{j}\mathrm{\kappa} \mathrm{\chi}. \end{align}

Taking the real part of the solution (see Sheriff and Geldart, 1995, Section 15.1.5) we obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} u&=\mathrm{\kappa} A\{ e^{-0.848\mathrm{\kappa} z } \left[-\sin \mathrm{\kappa}\left(x-V_{R} t\right)\right] \\ &\quad +0.393\times 1.468e^{-0.393\mathrm{\kappa} z } \sin \mathrm{\kappa}\left(x-V_{R} t\right)\} \\ &=\mathrm{\kappa} A\left(-e^{-0.848 \mathrm{\kappa} z } +0.577e^{-0.393 \mathrm{\kappa} z } \right)\sin \mathrm{\kappa}\left(x-V_{R} t\right), \end{align}

(2.14g)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} w&=\mathrm{\kappa} A\left(-0.848e^{-0.848 \mathrm{\kappa} z } +1.468e^{-0.393\mathrm{\kappa} z } \right)\cos \mathrm{\kappa}\left(x-V_{R} t\right) \end{align}

(2.14h)

2.14b What are the values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {u} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {w} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\theta} (i) when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {z=0} ; (ii) when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {z=1/2\mathrm{\kappa}} ; (iii) when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {z=1/\mathrm{\kappa}} ?

Solution

i) At the surface Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{z}=0 , and equations (2.14g,h) give

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} u&=-0.423\mathrm{\kappa} A\sin \mathrm{\kappa}\left(x-V_{R} t\right), \\ w&=0.620\mathrm{\kappa} A\cos \mathrm{\kappa}\left(x-V_{R} t\right). \end{align}

From equation (2.14f), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \tan \mathrm{\theta} =1.465 \cot \mathrm{\kappa} \left(x-V_{R} t\right)

ii) When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z=1/2 \mathrm{\kappa} , equations (2.14f,g,h) give for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): w , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} u&=\mathrm{\kappa} A\left(-e^{-0.424} +0.578e^{-0.197} \right)\sin \mathrm{\kappa} \left(x-V_{R} t\right)\\ &=-0.180\mathrm{\kappa} A\sin \mathrm{\kappa} \left(x-V_{R} t\right), \\ w&=\mathrm{\kappa} A\left(-0.847e^{-0.424} +1.466e^{-0.197} \right)\cos \mathrm{\kappa} \left(x-V_{R} t\right)\\ &=0.650\mathrm{\kappa} A\cos \mathrm{\kappa} \left(x-V_{R} t\right), \\ \tan\mathrm{\theta} &=\left(0.650/0.180\right)\cot \mathrm{\kappa} \left(x-V_{R} t\right)=3.61\cot \mathrm{\kappa} \left(x-V_{R} t\right). \end{align}

iii) For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z=1/\mathrm{\kappa} , we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} u&=\mathrm{\kappa} A\left(-e^{-0.847} +0.578e^{-0.394} \right)\sin \mathrm{\kappa}\left(x-V_{R} t\right)\\ &=-0.039 \mathrm{\kappa} A\sin \mathrm{\kappa}\left(x-V_{R} t\right), \\ w&= \mathrm{\kappa} A\left(-0.847e^{-0.847} +1.466e^{-0.394} \right)\cos \mathrm{\kappa}\left(x-V_{R} t\right)\\ &=0.625 \mathrm{\kappa} A\cos \mathrm{\kappa}\left(x-V_{R} t\right), \\ \tan \mathrm{\theta} &=\left(0.625/0.039\right)\cot \mathrm{\kappa}\left(x-V_{R} t\right)=16.0 \cot \mathrm{\kappa}\left(x-V_{R} t\right). \end{align}

2.14c Is the motion retrograde for all values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {z} ?

Solution When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z are fixed, the argument of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \cot \mathrm{\kappa} \left(x-V_{R} t\right) decreases as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t increases, and so the cotangent increases, that is, the angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\theta} in Figure 2.14a also increases. Because the wave is progressing in the positive direction of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x -axis, this counter-clockwise rotation is said to be retrograde. For the motion not to be retrograde, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\theta} must change sign, and thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(-w/u\right) must also change sign, that is, either Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): w but not both must change signs. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\kappa} z=0 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u is negative while Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): w is positive. For $$ u $$ to change sign, it must pass through zero; in this case

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} u=0=-e^{-0.847\mathrm{\kappa} z } +0.578e^{-0.394\mathrm{\kappa} z} \end{align}

or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): e^{0.453\mathrm{\kappa} z } =1.73, \mathrm{\kappa} z=1.21. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\kappa} z >1.21 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u is positive. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): w to change sign,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} w=0=-0.847e^{-0.847\mathrm{\kappa} z } +1.466e^{-0.394\mathrm{\kappa} z}, \end{align}

or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): e^{0.453\mathrm{\kappa} z } =0.578, \mathrm{\kappa} z=-1.21 . Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\kappa} z is always positive, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): w can never be zero. Consequently, the motion is retrograde in the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 0<\mathrm{\kappa} z<1.21 and prograde when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\kappa} z>1.21 .

2.14d What are the values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathbf{V_{R}}} , the Rayleigh-wave velocity, when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\sigma =0.4} and when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\sigma =0.2} ? What are the corresponding values of the constants in the expressions for $$ {u} $$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {w} in equation (2.14g,h)?

Solution

From Figure 2.14b we find that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{R} /\mathrm{\beta} \approx 0.95 when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\sigma} =0.4 and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{R} /\mathrm{\beta} \approx 0.92 when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\sigma} =0.2 . To get more accurate values, we solve equation (2.14e) using one of the standard methods of solving cubic equations.

i) For $\mathrm {\sigma } =0.4,(\mathrm {\beta } /\mathrm {\alpha } )^{2}=(1-2\mathrm {\sigma } )/2\left(1-\mathrm {\sigma } \right)=1/6$ . Equation (2.14e) now becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align}x^{3} -8x^{2} +\left(64/3\right)x-\left(40/3\right)=0=x^{3} +px^{2} +qx+r=0,\end{align}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): p=-8 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): q=64/3 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r=-40/3 . Next we eliminate the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x^{2} -term by substituting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=y-p/3=y+8/3 . This gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y^{3} +ay+b=0 , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a=\left(q-p^{2} /3\right)=0, b=\frac{1}{27}(2p^3-9pq+27r)=152/27 To check on the nature of the roots, we calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (\frac{b^2}{4}+\frac{a^3}{27}) ; the value is ${\frac {1}{4}}({\frac {152}{27}})^{2}$ , that is, it is positive, so we have one real root and two complex ones.

We now calculate the quantities

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} C=\left[-\frac{b}{2} +\left(\frac{b^{2} }{4} +\frac{a^{3} }{27} \right)^{1/2} \right]^{1/3} ,\quad D=\left[-\frac{b}{2} -\left(\frac{b^{2} }{4} +\frac{a^{3} }{27} \right)^{1/2} \right]^{1/3}. \end{align}

The three roots of the equation are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(C+D\right),-\frac{1}{2} \left(C+D\right)\pm 3j/\sqrt{2} . The last two roots are complex, so we are left with only the first root, that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y=\left(C+D\right) . Substituting the values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b , we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C=0 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): D=-1.78 . Thus, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y=-1.78=x-8/3 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=0.887=(V_{R} /\mathrm{\beta} )^{2} , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{R} /\mathrm{\beta} =0.942 .

To get the values of the constants in part (a) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\sigma} =0.4 , we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V_{R} /\beta &=0.942,\;\quad n=[1-(V_{R} /\mathrm{\beta} )^{2} ]^{1/2} =0.336,\\ V_{R} /\mathrm{\alpha} &=\left(V_{R} /\mathrm{\beta} \right)\left(\mathrm{\beta} /\mathrm{\alpha} \right)=0.942(1/6)^{1/2} =0.385,\\ m&=[1-(V_{R} /\mathrm{\alpha} )^{2} ]^{1/2} =0.923,\\ B/A&=2\mathrm{j}m/\left(1+n^{2} \right)=1.659\mathrm{j},\\ \mathrm{\phi} &=Ae^{-0.923\mathrm{\kappa} z } e^{\mathrm{j}\mathrm{\kappa} \left(x-V_{R} t\right)}, \\ \mathrm{\chi} &=1.659\mathrm{j}Ae^{-0.336\mathrm{\kappa} z } e^{j\mathrm{\kappa} \left(x-V_{R} t\right)}, \\ u&=\mathrm{\phi} _{x} +\mathrm{\chi} _{z} =j\mathrm{\kappa} A\left(e^{-0.923\mathrm{\kappa} z} -0.557e^{-0.336\mathrm{\kappa} z } \right)e^{\mathrm{j}\mathrm{\kappa} \left(x-V_{R} t\right)}, \\ w&=\mathrm{\phi}_z -\mathrm{\chi} _{x} \\ &=\mathrm{\kappa} A\left(-0.923e^{-0.923\mathrm{\kappa} z } +1.659e^{-0.336\mathrm{\kappa} z } \right)e^{\mathrm{j}\mathrm{\kappa} \left(x-V_{R} t\right)}. \end{align}

ii) For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\sigma} =0.2 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (\mathrm{\beta} /\mathrm{\alpha} )^{2} =\left(1-2\mathrm{\sigma} \right)/2\left(1-\mathrm{\sigma} \right)=3/8 . This gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} x^{3} -8x^{2} +18x-10=0,\quad \mathrm{so}\ p=-8,\; q=18,\; r=-10. \end{align}

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=y+8/3 , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y^{3} +ay+b=0 , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a=q-p^{2} /3=-10/3 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b=r- pq/3+\left(2/27\right)p^{3} =2/27 .

The discriminant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(\frac{b^{2} }{4} +\frac{a^{3} }{27} \right)<0 , so there are three real unequal roots; in this case a trigonometric solution is convenient. We find the value of

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align}\cos \mathrm{\gamma} =-\frac{b}{2} \left(-\frac{27}{a^{3} }\right)^{1/2} =-0.0316,\quad \gamma =91.8^{\circ} \;, \quad \gamma /3=30.6^{\circ}. \end{align}

Next we calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2\sqrt{-a /3}=2.11 . The roots are now

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} y&=2.11\cos 30.6^{\circ} ,\; 2.11 \cos \left(30.6^{\circ} +2\pi /3\right),\; 2.11\cos \left(30.6^{\circ} +4\pi /3\right)\\ &=1.82,\; -1.84,\; 0.022;\\ x&=y+8/3=4.49,\; 0.830,\; 2.69. \end{align}

But Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x<1 , so the only valid root is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align}x=0.830=(V_{R} /\mathrm{\beta} )^{2} \;, \; \left(V_{R} /\mathrm{\beta} \right)=0.911.\end{align}

Hence, for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\sigma} =0.2,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V_{R} /\mathrm{\beta} &=0.911\;,\quad \; n=[1-(V_{R} /\mathrm{\beta} )^{2} ]^{1/2} =0.412,\\ V_{R} /\mathrm{\alpha} &=0.911\left(\mathrm{\beta} /\mathrm{\alpha} \right)=0.911(3/8)^{1/2} =0.558,\\ m&=(1-0.558^{2} )^{1/2} =0.830,\\ B/A &=2\mathrm{j}m/\left(1+n^{2} \right)=1.419\mathrm{j},\\ \phi &=Ae^{-0.830\mathrm{\kappa} z } e^{\mathrm{j}\mathrm{\kappa} \left(x-V_{R} t\right)}, \\ \chi &=1.419\mathrm{j}Ae^{-0.412\mathrm{\kappa} z } e^{\mathrm{j}\mathrm{\kappa} \left(x-V_{R} t\right)}, \\ u&=\phi _{x} +\chi _{z} =\mathrm{j}\mathrm{\kappa} A\left(e^{-0.830\mathrm{\kappa} z} -0.585e^{-0.412\mathrm{\kappa} z } \right)e^{\mathrm{j}\mathrm{\kappa} \left(x-V_{R} t\right)}, \\ w&=\phi_{z} -\chi _{x} =\mathrm{\kappa} A\left(-0.830e^{-0.830\mathrm{\kappa} z } -1.419e^{-0.412\mathrm{\kappa} z } \right)e^{\mathrm{j}\mathrm{\kappa} \left(x-V_{R} t\right)}. \end{align}

2.15 Directional geophone responses to different waves

Assume three geophones so oriented that one records only the vertical component of a seismic wave, another only the horizontal component in the direction of the source, and the third only the horizontal component at right angles to this. Draw the responses of the three geophones for the following cases:

A P-wave traveling directly from the source to the geophones
A P-wave reflected at a deep horizon
An S-wave generated by reflection of a P-wave at an interface
A Rayleigh wave generated by the source
A Love wave generated by the source

Assume a simple waveshape, that there is a small vertical velocity gradient, that the source generates an initial compression for the direct wave, and an initial up- and away-thrust for the horizontal phone that is in line with the source. Compare the relative magnitudes of the components for short- and long-offset distances.

Background

A converted S-wave is generated in a solid medium when an incident P-wave strikes an interface at an angle (see problem 2.10). A Love wave is a type of surface wave (see Sheriff and Geldart, 1995, Section 2.5.4) in which the earth motion is parallel to the surface and perpendicular to the direction of wave travel.

Standard polarity for a minimum-phase waveshape (see problem 9.11) is that a compression produces a negative deflection.

Solution

The components of the waves (Figure 2.15a) are all in-phase except for the Rayleigh wave, where they are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 90^{\circ} out-of-phase.

2.16 Tube-wave relationships

2.16a A tube wave has a velocity of 1050 m/s. The fluid in the borehole has a bulk modulus of $2.15\times 10^{9}$ Pa and a density of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 1.20 g/\mathrm{cm}^{3} . The wall rock has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\sigma} =0.250 and density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2.50 g/\mathrm{cm}^{3} . Calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\mu} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\alpha} for the wall rock.

Background

Several types of tube waves exist (Sheriff and Geldart, 1995, Section 2.5.5). The classical type consists of a P-wave traveling in a fluid within a tubular cavity (such as a borehole) in a solid medium, the wall of the tube expanding and contracting as the pressure wave passes. Because the wall material interacts with the fluid, the tube-wave velocity depends upon the properties of both the wall material and the fluid. The formula for the tube-wave velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {V_T} is [see Sheriff and Geldart, 1995, equation (2.97)]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V_{T}^{2} =\frac{1}{p} \left(\frac{1}{k} +\frac{1}{\mu } \right)^{-1}, \end{align}

(2.16a)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho being the density and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k the bulk modulus of the fluid while $\mu$ is the rigidity modulus of the wall material.

Solution

Assuming the wall material to be rock, we write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho_{r} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mu _{r} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \lambda _{r} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sigma _{r} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha _{r} for the rock, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho_{f} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k_{f} for the fluid. We solve equation (2.16a) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mu _{r} but first we note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k and $\mu$ are in units of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): N/m^{2} , so we express Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho_{f} as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 1200\; \mathrm{kg}/m^{3} . Then,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mu _{r} =\left(\frac{1}{p_{f} V_{T}^{2} } -\frac{1}{k_{f} } \right)^{-1} = \left(\frac{1}{1200\times 1050^{2} } -\frac{10^{-9} }{2.15}\right)^{-1} =3.44\times 10^{9}\ \mathrm{Pa}. \end{align}

Using equation (5,5) of Table 2.2a, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \lambda _{r} =\mu _{r} \left[2\sigma _{r} /\left(1-2\sigma _{r} \right)\right]=\mu _{r} =3.44\times 10^{9}\ \mathrm{Pa}. \end{align}

To get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha _{r} , we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \alpha _{r} &=[\left(\lambda _{r} +2\mu _{r} \right)/\rho_{r} ]^{1/2} \\ &=(3\times 3.44\times 10^{9} /2500)^{1/2} =2.03\ \mathrm{km/s}. \end{align}

2.16b. Repeat for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{T} =1200 m/s and 1300 m/s. What do you conclude about the accuracy of this method for determining Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mu ?

Solution

When $V_{T}=1.20km/s$ ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mu _{r} =8.80\times 10^{9}\ \mathrm{Pa} =\lambda _{r}; \alpha _{r} =3.25\ \mathrm{km/s}. \end{align}

When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{T} =1.30 km/s ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mu _{r} =35.7\times 10^{9}\ \mathrm{Pa} =\lambda _{r}; \alpha _{r} =6.55\ \mathrm{km/s}. \end{align}

Summarizing the results for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mu _{r} versus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{T} , we get the following:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_T	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mu_r	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta V_T	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta \mu_r
1.05	3.44 $\times 10^{9}$	—	—
1.20	6.80 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \times 10^{9}	+14Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \%	+156Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \%
1.30	35.7 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \times 10^{9}	+24Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \%	+938Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \%

Since the relative change in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mu _{r} is very much larger than the relative change in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{T} , the method is very sensitive to changes in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{T} , hence the accuracy is very poor.

2.17 Relation between nepers and decibels

The natural logarithm of the ratio of two amplitudes is measured in nepers. Show that one neper = 8.68 dB.

Background

By definition, if $E_{1}$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E_{2} are energies, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \log_{10} \left(E_{2} /E_{1} \right) is the value of the ratio in bels. One bel = 10 decibels (dB), and energy is proportional to (amplitude)Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): ^{2} , so

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mathrm{dB}=10\log_{10} \left(E_{2} /E_{1} \right)=20 \log_{10} \left(A_{2} /A_{1} \right). \end{align}

(2.17a)

Solution

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): N={} value measured in nepers, dB = same value in decibels. Then,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} N= \ln \left(A_{2} /A_{1} \right)=\left(\log_{e} 10\right)\log_{10} \left(A_{2} /A_{1} \right)=2.3026\log_{10} \left(A_{2} /A_{1} \right). \end{align}

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{dB}=20 \log_{10} \left(A_{2} /A_{1} \right) .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} N=\left(20/2.3026\right)\ \mathrm{dB} = 8.686\ \mathrm{dB}. \end{align}

2.18 Attenuation calculations

A refraction seismic wavelet assumed to be essentially harmonic with a frequency of 40 Hz is found to have amplitudes of 5.00 and 4.57 mm on traces 2500 and 3000 m from the source. Assuming a velocity of 3200 m/s, constant subsurface conditions, and ideal recording conditions, what is the ratio of the amplitudes on a given trace of the first and fourth cycles? What percentage of the energy is lost over three cycles? What is the value of the damping factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h ?

Background

As a wave travels through a medium, the energy of the wave is gradually absorbed by the medium. This results in attenuation of the wave, the decrease in amplitude being approximately exponential with both distance and time. For a fixed time $$ t $$ , we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} A_{1} =A_{0} e^{-\eta x}, \end{align}

(2.18a)

where the initial amplitude Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A_{0} has decreased to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A_{1} after the wave travels a distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x;\eta is the absorption coefficient. On the other hand, at a fixed location, the amplitude varies with time according to the equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} A_{1} =A_{0} e^{-ht}, \end{align}

(2.18b)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h being the damping factor. During a period Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T , the wave travels a distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \lambda , hence equations (2.18a) and (2.18b) show that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} hT=\eta \lambda. \end{align}

(2.18c)

A damped harmonic wave can be written

${\begin{aligned}A_{1}=A_{0}e^{-ht}\cos \omega t.\end{aligned}}$

The logarithmic decrement (“log dec”) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta is defined as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \delta = \ln \left(\frac{\mathrm{amplitude}}{\hbox{amplitude one cycle later}}\right). \end{align}

(2.18d)

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T is the period, equations (2.18b) and (2.18d) show that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \delta=hT=h/f=2\pi h/\omega =\eta \lambda. \end{align}

(2.18e)

The quality factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q is another attenuation constant; it is defined by the relation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} Q=2\pi /\left(\Delta E/E\right), \end{align}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta E/E is the fractional wave energy loss/cycle. Because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E is proportional, to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A^{2} , $E=E_{0}e^{-2ht}$ . Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta E={} energy loss in one period Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \Delta E/E=\Delta \left(e^{-2ht} \right)/e^{-2ht} =2h\left(\Delta t\right)=2hT=2\delta. \end{align}

(2.18f)

(Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta E is the loss per cycle, so we have dropped the minus sign and set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta t=T ). Thus, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} Q=\pi /hT=\pi /\delta. \end{align}

(2.18g)

Solution

The wavelength is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \lambda =3200/40=80 m. From equation (2.18a) we get

${\begin{aligned}\eta =\left(1/x\right)\ln \left(A_{0}/A_{1}\right)=\left(1/0.50\right)\ln \left(5.00/4.57\right)=0.180\ \mathrm {km} ^{-1}.\end{aligned}}$

From equation (2.18e),

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \log \mathrm{dec} =\delta=\eta \lambda =0.180\times 0.080=0.0144. \end{align}

Equation (2.18b) shows that the amplitude ratio decreases by the same fraction over each interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T , hence the decrease in the ratio from the first to the fourth cycle is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \ln \left(A_{1} /A_{4} \right)=3hT=38=0.0432, \end{align}

so

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} A_1/A_4 = e^{3\delta} = e^{0.0432} = 1.044, \end{align}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} A_4 = 0.958\ A_1. \end{align}

The fraction of the energy lost per cycle, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta E/E , is equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2\delta from equation (2.18f). For 3 cycles, the fractional energy loss is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 6\delta=0.0864=8.64\% .

From equation (2.18e), $h=f\delta =40\times 0.0144=0.576$ sFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): ^{-1} .

2.19 Diffraction from a half-plane

The general equation for determining the diffraction effect of a plane surface is given by equation (2.19b). Show that the diffraction effect of the half-plane in Figure 2.19a is given by the integral Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \int\nolimits_{t=tr}^{+\infty } \phi \left(t\right)\mathrm{d}t where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \phi \left(t\right)&=\frac{\left(4chy_{0} /\pi V^{3} t\right)}{\left(t^{2} +t_{y}^{2} -t_{r}^{2} \right)(t^{2} -t_{r}^{2} )^{1/2} }, \quad t>t_{r}.\\ &=0,\qquad\qquad\qquad\qquad\qquad\quad\ \ t<t_{r}, \end{align}

(2.19a)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): c is a constant, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V is the velocity (assumed to be constant), while Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h,y_{0}, , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi are defined in Figure 2.19a, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t , $t_{y}$ , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{r} are two-way traveltimes along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi, y_{0} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r .

Background

The use of rays to describe wave propagation simplifies the phenomenon by ignoring diffraction (spreading of energy radiating from a virtual point source). Since a wave is reflected by all parts ofa surface, we can consider each point on the surface as a point source (Huygens's principle, see problem 3.1) and integrate over the surface to get the correct total effect. For a coincident source and receiver, the integral of the effects of point sources over the surface Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S can be transformed into a line integral around the boundary of the surface (see Sheriff and Geldart, 1995, Section 2.8.2). When the origin is over the surface, the integration gives two terms, one term representing the reflection given by ray theory, the other the diffraction. When the origin is not over the surface, the reflection term is zero leaving only the diffraction term.

The diffraction response of a plane area to a unit impulse (see Sheriff and Geldart, 1955, Section 15.2.5) emitted by a source at the origin and recorded at the origin is obtained by integrating the quantity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \phi \left(t\right) around the entire boundary of the area, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \phi \left(t\right) is given by equation (2.19b) [see Sheriff and Geldart, 1995, equation (2.131)]:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \phi \left(t\right)=\left(ch/\pi V^{2} t^{2} \right)\left(\frac{\mathrm{d}\theta }{\mathrm{d}t} \right), \end{align}

(2.19b)

where $\phi \left(t\right)$ is the response of a unit element of the boundary, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t is the two-way traveltime from the origin to the element of the boundry, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta is the angle shown in Figure 2.19a.

Solution

In Figure 2.19a

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} OA&=\xi, \\ \xi ^{2} &=r^{2} +x^{2} =r^{2} +y_{0}^{2} \tan^{2} \theta, \end{align}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y_{0} is normal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): BD and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=O'A . The points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): D are in fact at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \pm \infty ; thus, on integrating, $$ x $$ goes from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): -\infty to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): +\infty while 6 goes from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): -\pi /2 to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): +\pi /2 . Dividing by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(V/2\right) , we get the following relation between the traveltimes:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} (2\xi /V)^{2} =t^{2} =t_{r}^{2} +t_{y}^{2} \tan ^{2} \theta. \end{align}

(2.19c)

Then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \frac{\mathrm{d}t}{\mathrm{d}\theta } =\left(t_{y}^{2} /t\right)\tan\theta \sec^{2} \theta \end{align}

(2.19d)

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \frac{\mathrm{d}\theta }{\mathrm{d}t} =\left(t/t_{y}^{2} \right) \cot\theta \cos^{2} \theta. \end{align}

(2.19e)

From equation (2.19c) we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\tan}^{2} \theta &=\left(t^{2} -t_{r}^{2} \right)/t_{y}^{2}, \\ {\cos}^{2} \theta &=1/\left(1+{\tan}^{2} \theta \right)\\ &=1/\left[1+\left(t^{2} -t_{r}^{2} \right)/t_{y}^{2} \right]\\ &=t_{y}^{2} /\left(l^{2} +l_{y}^{2} -t_{2}^{r} \right), \end{align}

so

${\begin{aligned}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}&=\left({\frac {t}{t_{y}^{2}}}\right){\frac {t_{y}}{(t^{2}-t_{r}^{2})^{1/2}}}{\frac {t_{y}^{2}}{\left(t^{2}+t_{y}^{2}-t_{r}^{2}\right)}}\\&={\frac {t_{y}t}{(t^{2}+t_{y}^{2}-t_{r}^{2})(t^{2}-t_{r}^{2})^{1/2}}}.\end{aligned}}$

Substituting this expression in equation (2.19b), we obtain the following result for the diffraction effect of a unit length of the boundary in Figure 2.19a:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \phi \left(t\right)=\frac{\left(ch/\pi V^{2} t^{2} \right)\left(t_{y} t\right)}{\left(t^{2} + t_{y}^{2} -t_{r}^{2} \right)(t^{2} -t_{r}^{2} )^{1/2} }. \end{align}

Substituting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{y} =2y_{0} /V in the numerator, we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \phi \left(t\right)=\frac{\left(2chyo/\pi V^{3} t\right)}{\left(t^{2} + t_{y}^{2} -t_{r}^{2} \right)(t^{2} -t_{r}^{2} )^{1/2} }. \end{align}

To get the total diffraction effect of the half-plane, we integrate this expression around the four sides of the half-plane. Three of the four sides are at infinity so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t is infinite and the fraction, being proportional to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t^{-2} , vanishes. Therefore the effect reduces to the integral along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): BD . Because of symmetry, we can integrate along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): OD and double the result. Thus, the diffraction function for a half-plane is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \phi \left(t\right)&=\frac{\left(4chyo/\pi V^{3} t\right)}{\left(t^{2} +t_{y}^{2} - t_{r}^{2} \right)(t^{2} -t_{r}^{2} )^{1/2} }, \quad t\ge t_{r}, \\ &=0,\qquad\qquad\qquad\qquad\qquad\quad\ \, t\le t_{r}. \end{align}

References

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User:Ageary/Theory of seismic waves

Contents

Chapter 2 Theory of seismic waves

2.1 The basic elastic constants

2.2 Interrelationships among elastic constants

2.3 Magnitude of disturbance from a seismic source

2.4 Magnitudes of elastic constants

2.5 General solutions of the wave equation

2.6 Wave equation in cylindrical and spherical coordinates

2.7 Sum of waves of different frequencies; Group velocity

2.8 Magnitudes of seismic wave parameters

2.9 Potential functions used to solve wave equations

2.10 Boundary conditions at different types of interfaces

2.11 Boundary conditions in terms of potential functions

2.12 Disturbance produced by a point source

2.13 Far- and near-field effects for a point source

2.14 Rayleigh-wave relationships

2.15 Directional geophone responses to different waves

2.16 Tube-wave relationships

2.17 Relation between nepers and decibels

2.18 Attenuation calculations

2.19 Diffraction from a half-plane

References

See also

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User:Ageary/Theory of seismic waves

Chapter 2 Theory of seismic waves

2.1 The basic elastic constants

2.2 Interrelationships among elastic constants

2.3 Magnitude of disturbance from a seismic source

2.4 Magnitudes of elastic constants

2.5 General solutions of the wave equation

2.6 Wave equation in cylindrical and spherical coordinates

2.7 Sum of waves of different frequencies; Group velocity

2.8 Magnitudes of seismic wave parameters

2.9 Potential functions used to solve wave equations

2.10 Boundary conditions at different types of interfaces

2.11 Boundary conditions in terms of potential functions

2.12 Disturbance produced by a point source

2.13 Far- and near-field effects for a point source

2.14 Rayleigh-wave relationships

2.15 Directional geophone responses to different waves

2.16 Tube-wave relationships

2.17 Relation between nepers and decibels

2.18 Attenuation calculations

2.19 Diffraction from a half-plane

References

See also

External links

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