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Previous Page: Wave equations and reciprocity
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Previous: Wave equations and reciprocity
Up: Wave equations and reciprocity
Next: Reciprocity in approximations
Previous Page: Wave equations and reciprocity
Next Page: Reciprocity in approximations
Reciprocity and self-adjointess of operators are
closely related to each other.
The adjoint of an operator , defined generally as in 6,
is obtained from the solution of the problem
is then the adjoint operator of the
operator
and
is the adjoint
Green's function of
.
The operator
is said to be self adjoint if
.
To see how self-adjointness relates to reciprocity, use the generalization of the kernel in Green's theorem.
and
are arbitrary solutions to the problem.
is called the ``bilinear concomittant'' and is
some linear combination of functions of
and
.
Assuming homogeneous boundary conditions on the surface S of the volume V, we can integrate Eqn. 12 in time and space and see that the right hand side vanishes leaving us with the relation:
The structure of equation 6 and equations 12
and 13 are very similar, in that a dot product between
the dual fields and
, and
and
are formed.
If the dot product test, equation 13, for a self-adjoint
operator is valid for any arbitrary solutions
and
, then
Betti's theorem is automatically satisfied. A convenient reciprocity
principle for its Green's function can be derived.
In contrast, however, the fact that an operator is reciprocal, does not
imply self adjointness. An example of the latter would be the diffusion
equation.