Reflection/refraction at a solid/solid interface and displacement of a free surface

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Problem 3.2a

Derive the Zoeppritz equations for a P-wave incident on a solid/solid interface.

Background

The normal and tangential displacements plus the normal and tangential stresses must be continuous when a P-wave is incident at the angle on an interface between two solid media (see problem 2.10).

Solution

We use the functions in equations (3.1b,c,d,e) to represent the displacements of the waves, the positive direction of displacement for the waves being shown in Figure 3.1a. (We omit the factor because the boundary conditions do not depend upon the time , hence this factor cancels out).

We first derive the equations expressing the continuity of normal and tangential displacements, and . These equations are obtained by resolving the various wave displacements into - and -components. Thus,


(3.2a)


(3.2b)


(3.2c)


(3.2d)

At the interface, and , . The exponentials all reduce to , hence cancel out, and we get for the normal and tangential displacements, respectively,


(3.2e)


(3.2f)

To apply the boundary conditions for the normal and tangential stresses, we differentiate equations (3.2a,b,c,d) with respect to and . Equations (3.1d,e) show that the differentiation with respect to and multiplies each function by and either or . The common factor will cancel in the end, so we simplify the derivation by taking


(3.2g)

From equations (2.1b,c,e,h,i) we get for the normal and tangential stresses:

where , , and , are partial derivatives with respect to and . This allows us to find the normal and tangential stresses in each medium and equate them at . The result for the normal stresses is

Writing (see equations (9,6) and (9,7) in Table 2.2a) and recalling that , , the equation can be changed to the form


(3.2h)

where , ; and are called impedances.

In the same way we get for the tangential stresses the equation

This can be simplified using equation (3.1a) to give


(3.2i)

Equations (3.1e,f,h,i) are known as the Zoeppritz equations. For ease of reference, we have collected them below:

(3.2e)


(3.2f)


(3.2h)


(3.2i)

Problem 3.2b

3.2b Derive the equations below for the tangential and normal displacements, and , of a free surface for an incident P-wave of amplitude :


(3.2e)

where , , and being the angles of incidence of the P- and S-waves, respectively.

Solution

To determine the displacements at a free surface, we start by disregarding equations (3.2e,f) because there are no constraints on displacements at a free surface. After setting , we are left with


(3.2j)


(3.2k)

where , , , and we have dropped unnecessary subscripts. These equations can be written


(3.2l)

where


(3.2m)


(3.2n)

The solution of equations (3.2) is

In equations (3.2a,c) we set , the factor drops out, and we get


(3.2o)


(3.2p)

We now reinsert the values of and in terms of m and , and equations (3.2o,p) become


(3.2q)


(3.2r)

Problem 3.2c

3.2c Show that the displacements of a free surface at normal incidence are

Solution

For normal incidence at the surface, , . Equations (3.2j,k) give , . Substituting in equations (3.2o,p), we get , .

Problem 3.2d

3.2d Show that the displacements of a free surface of a solid, where , km/s, , , are

Solution

For , km/s, , , that is, , so . From the definitions of m and n, we get

Equations (3.2q,r) now give (omitting the factor ).

Problem 3.2e

3.2e Show that the displacements at the surface of the ocean are , .

Solution

In a fluid and equation (3.2j) gives , so equations (3.20,p) show that and .

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