EAS 557
Introduction to Seismology
Robert L. Nowack
Lecture 11
Elastodynamic Equation in an Unbounded Medium: Point Sources
The solution of the elastodynamic equation when a source term is included is now investigated. The elastodynamic equation can be written
Moving 
to the right side and
writing in vector notation
Now, using the vector identity 
, this can be written
(1)
where 
and 
are the P and S speeds. If we let 
be the input source
function and 
the resulting ground
displacement, then we can write
where T is the differential equation in equation (1), is the ground motion,
and 
is the source function. Using an input-output diagram,
If the source function is a localized delta function in time
and space, then the output ground is written 
which is called the
Green’s function. This is similar to the
impulse response discussed earlier with respect to linear systems.
Let’s first
look at the simple wave equation for the scalar potential 
where 
Then,
where 
. We will use Fourier
transforms to solve this. Let,
and
This is a multi-dimensional, Fourier transform pair.
Note that similar to 1-D Fourier transforms, if
are Fourier transforms, then
Then, the scalar wave equation in the Fourier domain is 
. If 
, then 
. The solution to this
can be written,
Transforming back to the time and space domain, then
where
= the Green’s function
for scalar wave equation
For the case where 
is a constant, then
there are analytic solutions in each of the domains. The solution in different dimensions is given
below, where c is wave speed.
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for
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, the outgoing term is
The geometric spreading term in 1, 2 or 3 dimensions for the wave amplitude is
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2-D |
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In addition, the more complicated 2-D case has an amplitude
term proportional to 
in the 
domain. In fact, Green’s functions for the scalar
wave equation in all even dimensional spaces are more complicated than Green’s
functions for odd dimensional spaces.
The basic idea is that if we know the Green’s function G(x,t), then for an arbitrary forcing function in time
then,
where 
is a time convolution
In 3-D, the Green’s function is
and for a source time function f(t), then
For a general distributed source function in x and time, then
Let’s next look at the elasticdynamic Green’s function
(2)
Let
which is a directed point force of position where 
is acting in the x1 direction and f(t)
is the source time function. The
multi-dimensional Fourier transform pair is
If
then,
In the Fourier domain, equation (1) can be written
(3)
Taking a dot product with k gives
Since 
, then
Now using 
, equation (3) can be written
Then, using 
for the first term on
the right side
then
and
(4)
Fourier transforming equation (4) back to the time and space domain gives
where
and c is either 
or 
.
Recall for the scalar wave equation in 3-D
Here,
If 
, where k is the
component in x1 direction,
then 
, where i is the
component of displacement and 1 is the force direction. This is the Green’s function for the
elastodynamic equation.
For a point force in the x1 direction with time function f(t), then
then,
for the field “P”
wave
for the field “S”
wave
near field terms of
combined P and S energy
where 
and 
are radiation patterns
and particle direction motion terms where 
is the direction
cosine from the source to receiver.
1) Properties of the far field P waves for single force
a) Attenuates as 
.
b) Propagates waveform f(t) at speed .
c) The displacement motion is parallel to the outward normal. uP has longitudinal motion since the particle motion is in the direction of propagation.
d) The directional radiation pattern motion is
proportional to 
where 
is shown in the figure
below.
2) Properties of the far field S waves for single force
a) Attenuates as .
b) Propagates waveform f(t) at speed .
c) The direction of particle motion is perpendicular to the outward normal. Thus, uS has transverse motion
d) The directional radiation pattern is shown below.
The Green’s function for homogeneous elastic medium at one position with time for directed point force looks like
where the energy between 
and 
results from the near
field terms. Although the Green’s
function is for a single directed point force, in the Earth, forces usually
come in balanced combinations.
A superposition of directed point forces can be used to simulate realistic sources for homogeneous, as well as heterogeneous media. Then, for a heterogeneous media, the Green’s functions G may need to be computed numerically.
Examples of combinations of directed point forces include,
1) Explosion
6 directed point forces (3 dipoles)
2) Slip on small crack
A double set of couples
More complicated sources can be constructed by a superposition of Green’s function solutions.