98 FUNDAMENTAL SOLUTIONS
Alternatively we may exploit the orthonormalisation of our eigen-
functions to retrieve the same result.
We have therefore
(κ
2
-κ'
2
) ^2π
and hence using (f) we can write
u*(xA)
= f
7^2^,2,
exp
|>'(£-x)]
άκ' (1)
which by simple contour integration gives (taking the real part for our
actual solution)
u*(x. ξ) = — sin (κχ-ξ) = — sin(tcr) (m)
where r is the (positive) distance between x and ξ.
The above example demonstrates the procedure used to obtain the
fundamental solution in Cartesian coordinates. The main steps of the
analysis are given below:
1 Transform the equation a sufficient number of times in order to
obtain an algebraic equation in terms of the transformed
function w*(jc, ξ).
2 Solve this equation in terms of ü*.
3 Invert the transformations to obtain the fundamental solution
u*
(3c,
ξ).
In practice it may not be necessary to perform the full number of
transformations as the resulting transformed equation at some point
may have
a
known fundamental solution. The inverse transforms may
then be performed on this function to obtain the fundamental
solution for the original equation.
As an example of the non-trivial use of this technique, the
fundamental solution for a concentrated load P applied at the point
ξ in an anisotropic elastic space will now be deduced starting from
Navier's equations in the form
3,4
^pqrs ™r, sq "
(4.49)