Boundary conditions in terms of potential functions
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| Series | Geophysical References Series |
|---|---|
| Title | Problems in Exploration Seismology and their Solutions |
| Author | Lloyd P. Geldart and Robert E. Sheriff |
| Chapter | 2 |
| Pages | 7 - 46 |
| DOI | http://dx.doi.org/10.1190/1.9781560801733 |
| ISBN | ISBN 9781560801153 |
| Store | 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 $ {xy} $-plane separating two semi-infinite solids require that, for a wave traveling in the $ {xz} $-plane, the following functions must be continuous:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \left({\phi}_{z} -{\chi}_{x} \right),\qquad \left({\phi}_{x} +{\chi}_{z} \right), \end{align} ()
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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} ()
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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} u=\mathrm{\phi}_{x} +\mathrm{\chi}_{z}, \; w=\mathrm{\phi} -\mathrm{\chi}_{x}. \end{align}
The normal displacement is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): w while the tangential displacement is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u , so these two functions must be continuous.
The normal stress is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{\sigma}_{zz} and equations (2.1b), (2.1h), (2.9e), and (2.9f) show that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): w .
The tangential stress is $ \mathrm {\sigma } _{xz} $ and equations (2.1c), (2.1i), (2.9d), and (2.9e) give
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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.
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| Boundary conditions at different types of interfaces | Disturbance produced by a point source |
| Previous chapter | Next chapter |
| Introduction | Partitioning at an interface |
Also in this chapter
- The basic elastic constants
- Interrelationships among elastic constants
- Magnitude of disturbance from a seismic source
- Magnitudes of elastic constants
- General solutions of the wave equation
- Wave equation in cylindrical and spherical coordinates
- Sum of waves of different frequencies and group velocity
- Magnitudes of seismic wave parameters
- Potential functions used to solve wave equations
- Boundary conditions at different types of interfaces
- Boundary conditions in terms of potential functions
- Disturbance produced by a point source
- Far- and near-field effects for a point source
- Rayleigh-wave relationships
- Directional geophone responses to different waves
- Tube-wave relationships
- Relation between nepers and decibels
- Attenuation calculations
- Diffraction from a half-plane
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