# Boundary conditions in terms of potential functions

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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 2.11

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

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

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{aligned}\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{aligned}}}

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{aligned}\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{aligned}}}

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