Disturbance produced by a point source
A source of seismic waves produces on a spherical cavity of radius enclosing the source a step displacement of the form
Starting with equation (2.12a) below, show that the displacement at distance is given by
Is the motion oscillatory? What is the final (permanent) displacement?
When a source, such as an explosion, creates very high stresses, the wave equation does not apply near the the source because the medium does not obey Hooke’s law in this region. For a symmetrical point source, this situation can be handled mathematically by enclosing the source with a spherical surface centered at the source and specifying the displacement at all points on the spherical surface at . If the source generates a wave such that the displacement at each point on the surface of radius is
the displacement is given by
where [see Sheriff and Geldart, 1995, Section 2.4.5, equations (2.76) and (2.77)]. The step function, step (t) , is defined in Sheriff and Geldart, 1995, Section 15.2.5.
Equation (2.12a) gives when . If we let , in the limit when , the displacement of the spherical surface becomes
which is the given type of source. Setting in equation (2.12a) we find that
If the motion is oscillatory, must change sign at least once, that is, the value of the expression in the square brackets must pass through zero. But , so and the exponential term is always positive, therefore oscillation is not possible.
At , which is the permanent displacement.
|Boundary conditions in terms of potential functions
|Far- and near-field effects for a point source
|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
- 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