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:
534.26In a harmonic mode of scattering of planar acoustic waves we study the problem of determining the shape of a prolate acoustically rigid solid of revolution. From experimental measurements we assume that the complex amplitude of inverse scattering is known on a discrete set of test wave numbers in directions that are nearly perpendicular to the axis of revolution of the scatterer.We consider a closed prolate acoustically rigid solid of revolution immersed in a homogeneous isotropic acoustic medium. In cylindrical coordinates (r, ~p, z) we describe the surface S of the body by the equation r = dF(z/a), tzl < a, F(:t:I) = 0, c = d/a << 1, where F is a certain positive smooth function. Suppose that a harmonic acoustic pressure wavemoves from infinity to the scattering body, where A0 is the amplitude, l is the direction of incidence, x = (xl, x2, z) are Cartesian coordinates with origin at the center of the obstacle, (.,-) is the inner product, k is the wave number, and we omit the time dependence exp(iwt).The scattered field Pp(x) satisfies the Helmholtz equation outside the surface S, and the boundary condition for it follows from the condition of absence of oscillation in the velocity on the surface S:where O/On is the derivative in the direction of the outward normal to the surface S. Here the field Pp(X) satisfies the Sommerfeld radiation condition, from which it follows thatwhere f(k; I, u) is the complex amplitude of the scattering and v = x/R is the direction of observation. We consider the following inverse problem: From the experimentally measured function f(k; 1) = f(k; 1, -1) determine the shape of the surface S. We assume that the orientation of the vector 1 relative to the axis of revolution of the body is known and prescribed in directions that are not close to tangential. We determine the frequency range from the condition ekd << 1. We remark that the problem of remote determination of the shape of ideal (acoustically rigid or soft) scatterers from acoustic waves has been the object of investigation by many authors [2][3][4][5][6]. The question of the uniqueness of the solution of inverse boundary-value problems is discussed in [9][10][11].Sinces assuming that the distribution of the scattered wave along the z-axis on the surface S is the same as that of the incident wave (1), we obtain from Eqs.
We consider a rectangular plate in a plane with dimensions 2a j, j = 1, 3, made of an orthotropic material. We choose a Cartesian coordinate system with origin at the geometric center of the plate in such a way that the axes are parallel to its sides. The plate is loaded on its faces by uniformly distributed normal and tangential forces. We solve the problem by a method proposed in [1], under the assumption that the plate is in a condition of cylindrical bending. With this form of the strain it is assumed that the displacements in the direction of one of the coordinate axes, for example the Ox 2 -axis, are constant: u 2 = const. Then the stress-strain state of the plate, as one can easily verify, is two-dimensional. The equilibrium equation, Hooke's law, and the Cauchy relations assume the form: b55We assume that the displacement vector field in the plate can be described by the relation u --I1 oil F , where F (F t, F 3) is an unknown vector-valued function; D is the matrix of differential operators:Substituting (6) into the geometric relations (4), we obtain (3) i, j = 1, 3.E o is the stiffness matrix. In the case of an orthotropic body referred to a (5)
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