General solutions of the wave equation

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Problem 2.5a

Verify that and 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


(2.5a)


being a disturbance such as a compression or rotation. is propagated with velocity (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 is the dilitation [see equation (2.1e)], and we get the P-wave equation, becoming the P-wave velocity . Shearing stresses create rotation in the medium and is one of the components of the rotation given by equation (2.lg) ; the result is an S-wave traveling with velocity . Various expressions for and in isotropic media are given in Table 2.2a.

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


(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 , we have


Substituting in equation (2.5b), we get the identity so is a solution. We get the same result when . A sum of solutions is also a solution, so is a solution.

Problem 2.5b

Verify that is a solution of equation (2.5a), where are direction cosines.

Solution

Let . We now must show that is a solution of equation (2.5a). Proceeding as before, we have


In the same way we get


But (see Sheriff and Geldart, 1995, problem 15.9a), so .

Following the same procedure we find that thus verifying that is a solution of equation (2.5a).

Problem 2.5c

Show that


is a solution of the wave equation in spherical coordinates (see problem 2.6b) when the wave motion is independent of and :


(2.5c)

Solution

The wave equation in spherical coordinates is given in problem 2.6b. When we drop the derivatives with respect to and , the equation reduces to equation (2.5c). Writing , we proceed as in part (a). Starting with the right-hand side, we ignore for the time being and obtain


Substitution in equation (2.5c) shows that is a solution. In the same way we can show that is also a solution, hence the sum is a solution.

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