Copyright © 1996 Clifford O. Bloom. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The asymptotic behavior as λ→∞ of the function U(x,λ) that satisfies the reduced wave equation Lλ[U]=∇⋅(E(x)∇U)+λ2N2(x)U=0 on an infinite 3-dimensional region, a Dirichlet condition on
∂V
, and an outgoing radiation condition is investigated. A function
UN(x,λ)
is constructed that is a global approximate solution as
λ→∞ of the problem satisfied by U(x,λ)
. An estimate for
WN(x,λ)=U(x,λ)−UN(x,λ)
on V
is obtained, which implies that
UN(x,λ)
is a uniform asymptotic approximation of
U(x,λ)
as
λ→∞, with an error that tends to zero as rapidly as
λ−N(N=1,2,3,...). This is done by applying a priori estimates of the function
WN(x,λ) in terms of its boundary values, and the L2 norm of rLλ[WN(x,λ)]
on V. It is assumed that
E(x), N(x), ∂V
and the boundary data are smooth, that
E(x)−I and N(x)−1 tend to zero algebraically fast as r→∞, and finally that
E(x) and
N(x) are slowly varying;
∂V may be finite or infinite.
The solution
U(x,λ) can be interpreted as a scalar potential of a high frequency acoustic or electromagnetic field radiating from the boundary of an impenetrable object of general shape. The energy of the field propagates through an inhomogeneous, anisotropic medium; the rays along which it propagates may form caustics. The approximate solution (potential) derived in this paper is defined on and in a neighborhood of any such caustic, and can be used to connect local “geometrical optics” type approximate solutions that hold on caustic free subsets of V.
The result of this paper generalizes previous work of Bloom and Kazarinoff [C. O. BLOOM and N. D. KAZARINOFF, Short Wave Radiation Problems in Inhomogeneous Media: Asymptotic Solutions, SPRINGER VERLAG, NEW YORK, NY, 1976].