Rayleigh-wave relationships

From SEG Wiki
Jump to navigation Jump to search
ADVERTISEMENT

Problem 2.14a

Show that, when $ {\mathbf {\sigma =0.25} } $, the Rayleigh‐wave potentials, equation (2.14c), become

$ {\begin{aligned}{\mathbf {\phi =Ae^{-0.847\kappa z}e^{\mathrm {j} \kappa \left(x-V_{R}t\right)}} },\end{aligned}} $

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 $ {u} $ and $ {w} $ at depth $ {z} $ are


$ {\begin{aligned}{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{aligned}} $ (2.14a)


$ {\begin{aligned}{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{aligned}} $ (2.14b)

Background

When a medium is divided by the $ xy $-plane into two semi-infinite media having different properties, surface waves are propagated parallel to the $ 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 $ z $-axis is positive downward, $ 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 $ 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 $ 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 Failed to parse (SVG (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 , 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 , $ (\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)

Problem 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 $ {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

$ {\begin{aligned}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{aligned}} $

Problem 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 $ 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 Failed to parse (SVG (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 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 $ 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 .

Figure 2.14a  Retrograde motion.

Problem 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.

Figure 2.14b  Failed to parse (SVG (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} as a function 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{\sigma} .

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 Failed to parse (SVG (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{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 , $ 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

$ {\begin{aligned}x=0.830=(V_{R}/\mathrm {\beta } )^{2}\;,\;\left(V_{R}/\mathrm {\beta } \right)=0.911.\end{aligned}} $

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}

Continue reading

Previous section Next section
Far- and near-field effects for a point source Directional geophone responses to different waves
Previous chapter Next chapter
Introduction Partitioning at an interface

Table of Contents (book)

Also in this chapter

External links

find literature about
Rayleigh-wave relationships
Retrieved from "http://wiki.seg.org/w/index.php?title=Rayleigh-wave_relationships&oldid=119873"