1063-7710/02/4803- $22.00 © 2002 MAIK “Nauka / Interperiodica” 0347 Acoustical Physics, Vol. 48, No. 3, 2002, pp. 347–352. Translated from Akusticheskiœ Zhurnal, Vol. 48, No. 3, 2002, pp. 400–405. Original Russian Text Copyright © 2002 by Mironov, Pislyakov. GENERALIZED WEBSTER EQUATION The velocity of sound propagation in a narrow (in comparison with the wavelength) tube depends on the acoustic admittance of the tube wall [1]. The elastic-type wall admittance reduces the velocity of propagation. In what follows, we investigate longitudinal sound waves in a narrow waveguide of varying cross section with vary- ing acoustic admittance of the walls. The waveguide pro- vides a smooth decrease in the velocity of wave propaga- tion, which, however, occurs rapidly enough for the propagation velocity to vanish within a finite distance. The possibility of creating specially tapered edges of plates and bars to eliminate the reflection of bending waves was considered earlier [2, 3]. Two conditions determine the choice of the tapering rate. On the one hand, the cross section should vary smoothly to cause no reflection. On the other hand, the cross section should vary rapidly enough for the velocity of a bending wave to vanish within the tapered segment of a finite length. In the case of a narrow, axially symmetric waveguide of varying cross section and with varying wall admittance, the wave equation is derived as fol- lows. We direct the x axis along the symmetry axis and denote by S = S(x) the area of the waveguide cross sec- tion, by r = r(x) the waveguide radius, and by Y = Y(x) the admittance of the walls (Fig. 1). The Euler equation remains intact: (1) ∂ p ∂ x ----- – ρ 0 ∂ v ∂ t ------ , = where v is the projection of the velocity on the x axis, p is the pressure, and ρ 0 is the density of the medium. The equation of continuity can be obtained from the conser- vation of matter in a thin layer of thickness dx (see Fig. 1) where v ⊥ is the projection of the velocity of liquid near the walls on the perpendicular to the symmetry axis and ρ is the acoustic component of the density. Substituting v ⊥ = Yp, we obtain (2) d ρ 0 S v ( ) 2 π r v ⊥ ρ 0 ( ) dx ∂ρ ∂ t ------ Sdx + + 0, = 1 ρ 0 ---- ∂ρ ∂ t ------ – v S ln ( ) x ' ∂ v ∂ x ------ 2 Yp r ---------. + + = One-Dimensional Acoustic Waves in Retarding Structures with Propagation Velocity Tending to Zero M. A. Mironov* and V. V. Pislyakov** * Andreev Acoustics Institute, Russian Academy of Sciences, ul. Shvernika 4, Moscow, 117036 Russia e-mail: mironov@akin.ru ** Moscow Institute of Physics and Technology, Institutskiœ per. 9, Dolgoprudnyœ, Moscow oblast, 141700 Russia Received July 17, 2001 Abstract—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. © 2002 MAIK “Nauka/Interperiodica”. Y = Y(x) r(x) dx x Fig. 1. Derivation of the generalized Webster equation. The waveguide with a varying cross section and varying wall admittance.