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Previous Page: Introduction
Next Page: Reciprocity and self-adjointness of operators
Previous: Introduction
Up: Vector Wave Reciprocity
Next: How can it be used?
Previous Page: Introduction
Next Page: Reciprocity and self-adjointness of operators
Given a set of dynamic equations, which describe the propagation of a wave in some medium, conservation principles lead to a reciprocal relation. Consider the set of dynamic equations
in a volume V, where are components of a displacement vector,
are components of the body force vector and
are components of the stress tensor. The raised parentheses
indicate various positions of the source.
These equations describe the force balance
of a medium, without specifying the particular way in which
stresses might be related to displacements. It is not necessary
to assume any particular constitutive relation at this point.
The force equations
describe the distribution of boundary forces on the enclosing
surface
,
when the body force
and
are applied while
define the distribution of displacement vectors
on the surface
when the same body forces are applied.
See Figure 1 for an illustration showing the state of
the medium in the two cases.
Figure 1:
[ IMAGE ]
The above equations are a set of equations that describe the physics
of motion and boundary conditions of a wave propagation problem.
The equations of motion 1 and 2
augmented by a constitutive
relationship can be generally written in terms of a linear
differential operator
with appropriate boundary conditions 3 and 4.
In further analysis I assume that initial acceleration and displacements are
zero before some time and represent a causal wave propagation problem.
Forming an inner product of the above equations with and
, respectively, leads in the time domain
to a general integral relation. This relation is often referred to as Betti's
reciprocal theorem [(Aki and Richards,1980)].
It relates the work done in each of the experiments.
It is noteworthy that the dependency of the stress field on the displacement field does not enter explicitly into this equation. In fact it is valid for a large variety of media (inhomogeneous, discontinuous, elastic, anisotropic, etc.). The above integral relation can now be used to derive special properties of the Green's function. The definition of a Green's function, is the solution of the impulse repsonse problem
A particular case of this Green's function would be the elastic case with causal initial conditions
in the Volume , with initial conditions
If satisfies homogeneous boundary conditions on S,
, a relation
between receiver and source positions is possible.
Let and
,
be impulsive forces in the
and
direction,
then the displacements can be expressed as
and
.
Substituting those expressions in equation 6 results in
a reciprocal relationship between the Green's tensor components:
Choosing the reference time to be leaves then the final
reciprocal relation:
It is the spatial part of the reciprocity principle that I will use later.
![[ IMAGE ]](_7428_dummy90.gif){kind=link}