Traveling wave and SEM representations for transient scattering by a circular cylinder

Abstract

When observed over long observation times, scattered returns from targets illuminated by broadband acoustic signals reveal distinctive features in the early and late time regimes. The features can be explained in terms of the different wave phenomena, progressive and oscillatory, that are dominant during these respective intervals. The former represent wave front arrivals, and the latter full body resonances. A hybrid theory, recently developed by Heyman and Felsen [IEEE Trans. Antennas Propag. AP-31, 426–437 (1983)], has formalized the connection between them and has provided new interpretations that have not emerged from more conventional treatments. These interpretations can clarify issues which have arisen within the resonance expansion procedure, in particular, the adequacy of the expansion at early times. In this paper, the canonical problem of scattering by a circular cylinder serves as a rigorous test for general observations made elsewhere [E. Heyman and L. B. Felsen, IEEE Trans. Antennas Propag. AP-31, 706–718 (1985)], concerning the traveling wave interpretation of the resonance series, and especially the role of the entire function in the complex frequency plane that is included to repair resonance series deficiencies of the Singularity Expansion Method (SEM). The resonance series of damped global oscillatory wave fields corresponding to the complex resonant frequencies represents, collectively, the multiple effects of diffracted (creeping) waves traveling around the cylinder in the positive and negative directions, but it cannot incorporate the specularly reflected field in the lit region, which has not sampled the obstacle surface as a whole. This contribution remains as a free term that can be identified in the complex frequency plane as an intrinsic (nonremovable) entire function. The analysis also shows how delaying the turn-on time of the resonance series, by retaining some early creeping wave front arrivals intact, can generate, in addition, a nonintrinsic (i.e., removable) entire function. These considerations, having rigorous foundation and appealing physical content, aid in the understanding of wave phenomena associated with broadband scattering from targets observed over early and late time periods.