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Integral equation

Our problem is to find a solution u of Helmholtz's equation

  \begin{eqnarray*}(\Delta + k^2) \ u(\mbox{\boldmath$ x $ })= 0 \quad \mbox{in } R^3 \setminus \overline{S},
\end{eqnarray*}


subject to the boundary condition
 
$\displaystyle \frac{\partial u^{\pm}(\mbox{\boldmath$ x $ })}{\partial n}= 0
\quad \mbox{on} \ S,$     (3.114)

regularity condition
 
$\displaystyle \mbox{\boldmath$ \phi $ }(\mbox{\boldmath$ x $ }) := \mbox{\boldm...
...{\boldmath$ u $ }^{-}(\mbox{\boldmath$ x $ }) = 0 \quad
\mbox{on} \ \partial S,$     (3.115)

and the radiation condition (Sommerfeld [75]) for $u(\mbox{\boldmath$\space x $ }) -
u_I(\mbox{\boldmath$\space x $ })$, where $\Delta$, k, u, $\phi$ and uI indicate the Laplacian operator, the wave number, the total wave field, the discontinuity of u on S and the incident field.

The solution $u(\mbox{\boldmath$\space x $ })$ to this problem has an integral representation given by

 
$\displaystyle u(\mbox{\boldmath$ x $ }) = u_I(\mbox{\boldmath$ x $ }) + \int_{S...
...dmath$ y $ }) dS_y, \quad \mbox{\boldmath$ x $ }\in R^3 \setminus \overline{S},$     (3.116)

where $G(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ })$ is the fundamental solution of Helmholtz's equation given by
 
$\displaystyle G(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }) = \frac{e^{ikR}}{4 \pi R},\quad R=\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert.$     (3.117)

Using (3.114) and (3.116), one obtains the hypersingular integral equation:

 
$\displaystyle -\frac{\partial u_I}{\partial n_x}(\mbox{\boldmath$ x $ }) =
\pfi...
...tial n_y}
\phi(\mbox{\boldmath$ y $ }) dS_y, \quad \mbox{\boldmath$ x $ }\in S,$     (3.118)

where $\pfint_{}$  stands for the finite part of a divergent integral. The hypersingular integral equation (3.118) can be regularised as (See Appendix L.4)
 
$\displaystyle -\frac{\partial u_I}{\partial n_x}(\mbox{\boldmath$ x $ })
= -n_j...
...box{\boldmath$ x $ }-\mbox{\boldmath$ y $ }) \phi(\mbox{\boldmath$ y $ }) dS_y.$     (3.119)

where $\vpint_{} $  stands for the Cauchy principal value of a singular integral. This regularised integral equation (3.119) is utilised for the direct computation in FM-BIEM.
next up previous contents
Next: FM-BIEM Up: Three-dimensional scattering of scalar Previous: Three-dimensional scattering of scalar
Ken-ichi Yoshida
2001-07-28