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.
The asymptotic behavior as of the function that satisfies the reduced wave equation on an infinite 3-dimensional region, a Dirichlet condition on
, and an outgoing radiation condition is investigated. A function
is constructed that is a global approximate solution as
of the problem satisfied by
. An estimate for
is obtained, which implies that
is a uniform asymptotic approximation of
, with an error that tends to zero as rapidly as
. This is done by applying a priori estimates of the function
in terms of its boundary values, and the norm of
on . It is assumed that
and the boundary data are smooth, that
and tend to zero algebraically fast as , and finally that
are slowly varying;
may be finite or infinite.
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 .
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].