  This paper presents a numerical compression strategy for the boundary
integral equation of acoustic scattering in two dimensions. These equations
have oscillatory kernels that we represent in a basis of wave atoms, and
compress by thresholding the small coefficients to zero. This phenomenon was
perhaps first observed in 1993 by Bradie, Coifman, and Grossman, in the context
of local Fourier bases \cite{BCG}. Their results have since then been extended
in various ways. The purpose of this paper is to bridge a theoretical gap and
prove that a well-chosen fixed expansion, the nonstandard wave atom form,
provides a compression of the acoustic single and double layer potentials with
wave number $k$ as $O(k)$-by-$O(k)$ matrices with $O(k^{1+1/\infty})$
nonnegligible entries, with a constant that depends on the relative $\ell_2$
accuracy $\eps$ in an acceptable way. The argument assumes smooth, separated,
and not necessarily convex scatterers in two dimensions. The essential features
of wave atoms that enable to write this result as a theorem is a sharp
time-frequency localization that wavelet packets do not obey, and a parabolic
scaling wavelength $\sim$ (essential diameter)${}^2$. Numerical experiments
support the estimate and show that this wave atom representation may be of
interest for applications where the same scattering problem needs to be solved
for many boundary conditions, for example, the computation of radar cross
sections.
