Anomalies of Acoustic Wave Scattering Near the Cut-off Points of Continuous Spectrum (A Review)

Abstract

Several anomalies of wave scattering that occur in acoustic waveguides with cylindrical or corrugated rigid walls at frequencies close to the cut-off points (thresholds) of continuous spectrum are considered. The notion of threshold resonances generated by “almost standing waves”, which cause no energy transfer to infinity, is introduced. For corrugated waveguides, examples are presented to illustrate the opening of spectral gaps (wave stopping zones) and eigenvalues near their edges and common or degenerate thresholds. Weinstein’s and Wood’s anomalies are described, which occur above and below the thresholds and manifest themselves in “almost complete” reflection and transmission of waves, and in disproportionally fast variation of the diffraction pattern, respectively. Examples of complete wave transmission (“invisibility of obstacle”) are discussed along with the procedures of sharpening and smoothing of Wood’s anomalies, specifically, formation of eigenvalues embedded into the continuous spectrum and corresponding trapped waves. The Sommerfeld, Umov–Mandelshtam, and limiting absorption principles are compared along with the specific features of their application at thresholds.