Potential functions used to solve wave equations

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Problem 2.9a

Show that equation (2.9a) relating the potential functions and to the vector displacement requires that and [see equations (2.1e) and (2.1g)] be solutions of the P- and S-wave equations, that is, of equation (2.5a) with replaced by and , respectively.


and being solutions of the P- and S-wave equations, respectively.


The dilatation and component of rotation are defined in equations (2.1e,g).

While solutions of the wave equation (see problem 2.5) furnish values of or a component of rotation , we often need to know the displacements (defined in problem 2.1) which are not easily found given or . This difficulty can be avoided by using potential functions that are solutions of the wave equations and from which , hence also, can be found by differentiation.

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

The vector operator (called “del”) and its properties are discussed in Sheriff and Geldart, 1995, Section 15.1.2c.


From equation (2.1e) and the definition of , we get for the dilatation


since and . Because is a solution of the P-wave equation, must also be a solution.

We have

[see Sheriff and Geldart, 1995, equations (15.13) and (15.9)]. Since is a solution of the S-wave equation, is also a solution.

Problem 2.9b

In two dimensions, the potential function can be defined as


Show how to obtain the displacements , the dilatation , and rotation from this equation (see Sheriff and Geldart, 1995, Section 15.1.2c and problem 15.5c).


From equation (2.1d) we see that is the -component of , that is, of and , so -component of . From Sheriff and Geldart, 1995, equation (15.13) we have





To get the dilatation , we use equation (2.1e) and Sheriff and Geldart, 1995, problem 15.5c and obtain


The rotation 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

Thus has only a -component given by

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