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Relation between formulations in Laplace's equation and Helmholtz's equation

We can consider Laplace's equation to be the special case Helmholtz's equation as k tends to 0. Therefore one expects that the FMM formulation for Laplace's equation is obtained as the limit of $k \rightarrow 0$ in the FMM formulation for Helmholtz's equation. However, the formulation obtained above does not allow such operation in the form as it is. It is because the behaviours of $j_n(k\vert\overrightarrow{O\mbox{\boldmath$\space x $ }}\vert)$ and $h_n(k\vert\overrightarrow{O\mbox{\boldmath$\space x $ }}\vert)$ near k=0 are O(k-n-1) and O(kn) and, hence,

\begin{eqnarray*}\lim_{k\rightarrow0}{\cal I}_n^m(k,\overrightarrow{O\mbox{\bold...
...\cal O}_n^m(k,\overrightarrow{O\mbox{\boldmath$ x $ }}) = \infty
\end{eqnarray*}


In order to obtain a formulation allowing an obvious transition between these governing equations we need some modifications to cancel these behaviours.

Noting the power series of $j_n(k\vert\overrightarrow{O\mbox{\boldmath$\space y $ }}\vert)$ and $h_n(k\vert\overrightarrow{O\mbox{\boldmath$\space x $ }}\vert)$, we obtain the following formulation

 
$\displaystyle \frac{e^{ik R}}{4 \pi R}
=\frac{1}{4 \pi}\sum_{n=0}^{\infty}\sum_...
...boldmath$ y $ }})\hat{{\cal O}}_n^m(k,\overrightarrow{O\mbox{\boldmath$ x $ }})$     (3.134)

where $\hat{{\cal I}}_n^m(k,\overrightarrow{O\mbox{\boldmath$\space y $ }})$ and $\hat{{\cal O}}_n^m(k,\overrightarrow{O\mbox{\boldmath$\space x $ }})$ are defined as

\begin{eqnarray*}\hat{{\cal I}}_n^m(k,\overrightarrow{O\mbox{\boldmath$ y $ }})=...
... x $ }}\vert) S_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }}).
\end{eqnarray*}


Since

\begin{eqnarray*}\lim_{k\rightarrow0} \hat{{\cal I}}_n^m(k,\overrightarrow{O\mbo...
...math$ x $ }})=S_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }}),
\end{eqnarray*}


the expression in (3.135) is seen to approach (3.120) as $k \rightarrow 0$.
next up previous contents
Next: Three-dimensional scattering of elastic Up: Three-dimensional scattering of scalar Previous: Many penny-shaped cracks
Ken-ichi Yoshida
2001-07-28