Laplace Domain Methods for the Construction of Transparent Boundary Conditions for Time-Harmonic Problems

Abstract

We consider scattering problems governed by the scalar Helmholtz equation

$$ \Delta u + {k^2}u = 0 $$

((1))

More general equations involving radially symmetric potentials have been considered in [7, 6, 5]. A proper formulation of such problems on infinite domains must involve a radiation condition at infinity. The standard radiation conditions in ℝ^d is Sommerfeld’s radiation

$$ {r^{{(d - 1)/2}}}\left( {\frac{{\partial u}}{{\partial r}} - iku} \right) \to 0,\quad r = \left| x \right| \to \infty $$

((2))

which holds uniformly for all direction $\frac{x}{{\left| x \right|}}$. However, this condition is not valid in general for infinite obstacles and inhomogeneities with infinite support. Special radiation conditions have been devised for particular problems, e.g., for scattering by infinite obstacles, or scattering problems in wave guide structures.