The thin‐shape breakdown (TSB) of the Helmholtz integral equation

Abstract

Many numerical implementations of the Helmholtz integral equation exist today that can predict routinely the field scattered by a volume‐holding body. One such object is the ellipsoidal ‘‘core’’ of a typical airborne or submerged vehicle stripped of its thin appendages, i.e., stripped of its finlike control surfaces, etc. The reason for these omissions has often been an inherent limitation of the cited modeling tools rather than a rational dismissal of the potential effect of these neglected appendages on the complete body’s expected scattering cross section. The limitation of existing techniques is this: The standard form of Gauss’ theorem on which they are based, that leads to the Helmholtz integral, becomes meaningless when the thickness of the shape addressed tapers down to zero even over only part of the structure. This paper exposes analytically the fundamental cause of this thin‐shape breakdown (TSB), and in the process derives an alternate boundary element formulation for its cure.