Boundary conditions in terms of potential functions
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 -plane separating two semi-infinite solids require that, for a wave traveling in the -plane, the following functions must be continuous:
where subscripts denote partial derivatives. These terms are, respectively, the normal and tangential stressses and the normal and tangential displacements.
As stated in problem 2.10, all stresses and displacements must be continuous at an interface between two different media.
From equations (2.9d) and (2.9e) we have
The normal displacement is while the tangential displacement is , so these two functions must be continuous.
The normal stress is and equations (2.1b), (2.1h), (2.9e), and (2.9f) show that
the last step being obtained by differentiating the above expression for .
The tangential stress is and equations (2.1c), (2.1i), (2.9d), and (2.9e) give
Since the normal and tangential stresses must be continuous, these two functions must also be continuous.
|Previous section||Next section|
|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