# Boundary conditions at different types of interfaces

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Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 2 7 - 46 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Problem 2.10a

Justify on physical grounds the boundary conditions for solid-fluid media in contact.

### Background

The basic principles underlying the boundary conditions are (i) all stresses must be continuous (that is, no changes in values) at the interface; this must be true to avoid unbalanced forces which would produce accelerations; and (ii) strains must be continuous at the interface; this must be true for normal strains to avoid interpenetration of one medium into the other, or creation of a vacuum between them, or sliding of solid on solid for unequal tangential strains.

When a wave is incident on a boundary, one to four boundary conditions must be satisfied, depending on the types of media. The angles of reflection and refraction are fixed by the laws of reflection and refraction [see equation (3.1a)], so the only parameters that can be adjusted to satisfy these conditions are the relative amplitudes of reflected and/or refracted P- and S-waves generated by the incident wave. An S-wave generated by an incident P-wave, or a P-wave generated by an incident S-wave, is called a converted wave.

### Solution

Normal stress and displacement (strains) are continuous, stress to avoid normal acceleration and displacement to avoid interpenetration or a vacuum. Tangential stress is zero everywhere in the fluid and so the tangential stress in the solid must be zero at the interface. There is no restriction on tangential displacement.

## Problem 2.10b

Justify on physical grounds the boundary conditions for solid-vacuum media in contact.

### Solution

Stresses are zero in a vacuum, so normal and tangential stresses in the solid vanish at the interface.

## Problem 2.10c

Justify on physical grounds the boundary conditions for fluid-fluid media in contact.

### Solution

Normal stresses and displacements are continuous at the interface for the same reasons as in (a). Tangential stresses cannot exist in a fluid, and tangential displacements are zero.

## Problem 2.10d

Justify on physical grounds the boundary conditions for fluid-vacuum media in contact.

### Solution

Stresses are zero in a vacuum, so the normal stress in the fluid vanishes at the interface.