# Boundary conditions in terms of potential functions

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 2.11

Using the definitions of stress and strain in problem 2.1 and the potential functions in equation (2.9b), show that the boundary conditions at the $\displaystyle {xy}$ -plane separating two semi-infinite solids require that, for a wave traveling in the $\displaystyle {xz}$ -plane, the following functions must be continuous:

 \displaystyle \begin{align} \left({\phi}_{z} -{\chi}_{x} \right),\qquad \left({\phi}_{x} +{\chi}_{z} \right), \end{align} (2.11a)

 \displaystyle \begin{align} {\mathbf{\lambda \nabla ^{2} \phi}} +{\mathbf{2\mu}}\left({\phi}_{zz} -{\chi}_{xz} \right),\qquad {\mu} \left({\mathbf{2\phi}}_{xz} +{\chi}_{zz} -{\chi}_{xx} \right), \end{align} (2.11b)

where subscripts denote partial derivatives. These terms are, respectively, the normal and tangential stressses and the normal and tangential displacements.

### Background

As stated in problem 2.10, all stresses and displacements must be continuous at an interface between two different media.

### Solution

From equations (2.9d) and (2.9e) we have

\displaystyle \begin{align} u=\mathrm{\phi}_{x} +\mathrm{\chi}_{z}, \; w=\mathrm{\phi} -\mathrm{\chi}_{x}. \end{align}

The normal displacement is $\displaystyle w$ while the tangential displacement is $\displaystyle u$ , so these two functions must be continuous.

The normal stress is $\displaystyle \mathrm{\sigma}_{zz}$ and equations (2.1b), (2.1h), (2.9e), and (2.9f) show that

\displaystyle \begin{align} \mathrm{\sigma}_{zz} =\mathrm{\lambda}\Delta +2\mathrm{\mu} \mathrm{\varepsilon}_{zz} =\mathrm{\lambda} \nabla ^{2} \mathrm{\phi} +2\mathrm{\mu} w_{z} =\mathrm{\lambda} \nabla ^{2} \mathrm{\phi} +2\mathrm{\mu} \left(\mathrm{\phi}_{z} -\mathrm{\chi}_{xz} \right), \end{align}

the last step being obtained by differentiating the above expression for $\displaystyle w$ .

The tangential stress is $\displaystyle \mathrm{\sigma}_{xz}$ and equations (2.1c), (2.1i), (2.9d), and (2.9e) give

\displaystyle \begin{align} \mathrm{\sigma}_{xz} =\mathrm{\mu \varepsilon}_{xz} =\mathrm{\mu} \left(u_{z} +w_{x} \right)=\mathrm{\mu} \left(2\mathrm{\phi}_{xz} +\mathrm{\chi}_{zz} -\mathrm{\chi}_{xx} \right). \end{align}

Since the normal and tangential stresses must be continuous, these two functions must also be continuous.