The spectral characteristics of an acoustic pressure field scattered by a nondeformable solid of revolution

Abstract

The general solution of the problem of scattering of sound impulses by an elastic solid of revolution of nearly spherical shape was given in [4] and [5] by use of the method of perturbation of the shape of the boundary [2,3,5]. It is known that any deviation of the geometry of the surface of the scatterer from the canonical geometry introduces significant variations in the spectral characteristics of the echo signal: a shift of resonance frequencies, widening of the spectrum, formation of a fine structure [1,[5][6][7][8], and others.In this paper we use the example of immovable acoustically soft and acoustically rigid scatterers in the shape of a solid of revolution differing only slightly in shape from a sphere to carry out a numericalanalytic analysis of the Fourier spectrum of the echo-signals, and we study the convergence of the method of perturbations. We assume that the problem is posed in axisymmetric form. This means that an incident acoustic wave of pressure Pine emanates from a transformer whose acoustical axis coincides with the axis of revolution of the scatterer. We take a function that conformally maps the exterior of a circle of radius a onto the exterior of a meridian contour of the object in the form [3]where (z, R, ~), (r, 0, c2), and (p, C, ~) are respectively the cylindrical, spherical, and curvilinear orthogonal coordinates measured from the center of the scatterer. Thus, expanding the Fourier spectrum of the echo-signal in a series of powers of e, we obtain [4,5] = J, z(k) l (k)hll (kr)P,(cos 0),j=0 Z=0where the coefficients of the expansion of xlJ)(k) for an acoustically soft immovable object (the s-case) and an acoustically rigid immovable object (the r-case) assume the following respective forms: