One-dimensional acoustic waves in retarding structures with propagation velocity tending to zero

A retarding structure that allows the effective admittance of a tube wall to increase smoothly along the tube axis is considered. The sound velocity gradually decreases along a finite segment of the tube and finally vanishes at some cross section. The time of the sound propagation along this segment is infinitely long. A wave incident on the input cross section cannot reach the other end of the tube within a finite time, and, hence, it is not reflected from it. The wave is completely absorbed, the absorption being caused by the energy accumulation in the cross section where the velocity of sound vanishes, rather than by the energy transformation to heat, as in common sound absorbers. A differential equation is obtained to describe the sound propagation in a one-dimensional waveguide with a varying cross section and varying acoustic admittance of the walls. The solutions to this equation are analyzed in the WKB approximation. An exact solution is determined for the case of some specific functions describing the variations of the cross section and admittance. Calculated results for the input admittance of the waveguide are presented. A possible similarity to the problem of shear waves in sea sediments is pointed out.