Potential functions used to solve wave equations

From SEG Wiki
Jump to navigation Jump to search
ADVERTISEMENT

Problem 2.9a

Show that equation (2.9a) relating the potential functions $ {\phi } $ and $ {\chi } $ to the vector displacement $ {\zeta } $ requires that $ {\Delta } $ and $ {\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 $ {\psi } $ replaced by $ {\Delta } $ and $ {\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.

Problem 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 $ 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 Failed to parse (SVG (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 -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}

Continue reading

Previous section Next section
Magnitudes of seismic wave parameters Boundary conditions at different types of interfaces
Previous chapter Next chapter
Introduction Partitioning at an interface

Table of Contents (book)

Also in this chapter

External links

find literature about
Potential functions used to solve wave equations
Retrieved from "http://wiki.seg.org/w/index.php?title=Potential_functions_used_to_solve_wave_equations&oldid=119891"