Boundary conditions at different types of interfaces

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.