Numerical search for a breakdown of the Waterman algorithm for approximating the T matrix of a rigid obstacle

Abstract

The convergence of the Waterman algorithm for approximating the T matrix [J. Acoust. Soc. Am. 45, 1417–1429 (1969)] has not been established for scatterers of general shape. One can prove that the convergence criteria of Kristensson, Ramm, and Ström [J. Math. Phys. 24, 2619–2631 (1983)] are never fulfilled if the obstacle is nonspherical and the spherical wave functions have their usual normalization; the proof of this fact shall appear elsewhere. One of the coordinate sequences of this algorithm is the family of regular spherical waves, which fails to be complete in L2 (Γ), Γ denoting the obstacle boundary, when k2 is an interior Dirichlet eigenvalue for the negative Laplacian. These circumstances lead one to conjecture that the method will fail at such frequencies (as it does for the sphere). The results of computations near various of these frequencies for spheroidal scatterers are reported. For the sphere, the widths of the ka intervals of instability are examined, while for the nonspherical case, the object is to determine whether a breakdown occurs at all.