 
Summary: A Representation of Acoustic Waves
in Unbounded Domains
BRADLEY K. ALPERT
National Institute of Standards and Technology
AND
YU CHEN
Courant Institute
Abstract
Compact, timeharmonic, acoustic sources produce waves that decay too slowly
to be squareintegrable on a line away from the sources. We introduce an inner
product, arising directly from Green's second theorem, to form a Hilbert space
of these waves and present examples of its computation.1 c 2005 Wiley Peri
odicals, Inc.
1 Introduction
Consider the scattering problem governed by the Helmholtz equation
(1.1) + k2
(1 + q) = 0
in two dimensions for a scatterer q L2
(D) supported in a compact domain D in
the halfplane above the xaxis. A function u : R2
