No abstract

539.3On the basis of numerical computations we analyze the echo-signals from a steel spherical shell located near the bottom of water under the action of a quasimonochromatic pulse. We study the effect of the distance between the sphere and the bottom on the echo-pulse.A comprehensive study of the process of scattering of acoustic waves by elastic spherical bodies has been carried out in the literature mainly for the case when the surrounding medium is homogeneous in its acoustic properties. From the practical point of view the. analogous problem of the effect of the bottom near the scatterer on the diffraction of a wave isof considerable:interest: In what follows, on the basis of the method of particular solutions we study the axisymmetric interaction of acoustic waves with an elastic object of spherical shape in the presence of a plane interface between two liquid media.We consider two acoustic half-spaces x 3 > 0 and x 3 < 0 (Ix,,21< oo) with respective densities p, Po and sound velocities c, c o . In the upper half-space x 3 > 0 in a state of equilibrium there is an elastic spherical shell with radii a and b (b = a-h, where h is the thickness of the wall) filled with an acoustic fluid of density p/ and velocity of sound c z . The distance from the origin O of the Cartesian coordinate system xl, x2, x3, on the interface of the two acoustic half-spaces to the center of the shell O t (the origin of the spherical coordinates r, O, r is x ~ (x ~ = a+d ; d being the distance from the point O to the nearest point on the surface of the sphere r = a, 0 = 0). A plane acoustic pressure pulse is incident on the sphere, the front of the wave being parallel to the interface of the two half-spaces x 3 = 0 (Ix,,21< ~o) :where f(t) is the modulation of the pulse at time t. The problem is posed of determining the pressure p~(x, t) in the scattered wave, in particular along the location line 0 = n. To solve it we apply the Fourier transform with respect to time. Then for the Fourier densities (spectral densities) of acoustic pressures in the incident wave, the scattered wave, and the wave refracted to the bottom we shall have on Pine (X, O ) = e -'kx~ = e -ikx~ Z it (2l+ 1) Jl (kr) Pt (cos0),where jl(z) and h~l)(z) are the spherical Bessel and Hankel functions of first kind; Pt(cos0) are the Legendre polynomials; and co is the angular frequency. The solution for the scattered wave is composed of two components: 1) the series for the divergent spherical waves by which we distinguish the effect of the spherical transition; 2) a superposition of planar waves caused by the presence of the lower half-space (bottom). We remark that the second term in formula (3) and the pressure (4) are given in terms of double Fourier integrals, and k= (k,,k~,k3), Imk,_>O,

We consider the problem of emission of sound by an airplane in the stage of its take-off run along the runway of an airdrome under the conditions of wind action. The aircraft engines play the role of the sources of sound. In the problem, they are modeled by point sources moving in air with a constant acceleration parallel to the plane interface between the acoustic and solid elastic half spaces. The solution of the problem is obtained by using the Fourier integral transformation with respect to time and space coordinates and the method of stationary phase. We also perform the numerical analysis of the spacetime distributions of sound pressure and the levels of sound pressure for the case of take-off of a sixengine An-225 "Mriya" aircraft. Formulation of the ProblemAirdromes used for the take-of and landing of the aircrafts are the sites of concentration of strong noise and, hence, the sources of the propagation of acoustic waves to the neighboring areas. Therefore, at present, the problem of acoustic climate near airdromes attracts significant attention of the researchers [8,12,17], in particular, in the fields of measurements of the level of acoustic pressure [7], empirical methods of its evaluation [19,20], in the problems of optimization [5,11], reduction of the aircraft noise [13], and protection against this type of noise [18]. However, the mathematical modeling of acoustic wave processes capable of large-scale evaluation of this kind of hazard to the human health [10] is still poorly developed.In the present work, we make an attempt to construct a mathematical model of this kind. Due to the complexity of the problem, we restrict ourselves to the study of the process of the emission of acoustic energy by aircrafts only in the stage of take-off run.Consider the problem of emission of sound by a starting aircraft. The trajectory of motion of this object consists of four parts. The first of them represents the accelerated motion of the aircraft along the airdrome runway beginning from the initial point of its start. The second part is the accelerated motion in air at certain relatively low heights parallel to the runway. The third part can be described as the motion along a curvilinear path with passing to the fourth part, i.e., to the upward motion with a constant velocity at a certain angle to the runway.In the present work, we study the process of noise generation in the course of the take-off run, i.e., in the first part of the trajectory, because this stage has the largest duration and, as a rule, is realized near the inhabited areas.We replace the engines by point sources of sound with a certain intensity and angular frequency of harmonic vibrations Ω. First, we consider a single source of sound moving in the plane Oxz at a height z = z 0 over

No abstract

Starting from the operator notation for the dyadic Green's function for elastic displacement of the nonstationary theory of elasticity, we propose a method of factoring the mutually orthogonal components of the Green's function in the form of tensor products of operators acting respectively on the coordinates of the points of observation and the source.The Green's function of the dynamic theory of elasticity for an arbitrary space has the following form in operator representation [2]:whereV is the Hamilton• nabla-operator, V 2 is the Laplacian, 1 is a unit tensor, p is the density of the material, CL and CT are the propagation velocities of three-dimensional waves, R = IRI = [x-x'l, r = t-t' > o, x and x' are the radius vectors of the points of observation and the concentrated source respectively, x = {x j}, x' = {x}}, (j = 1, 2, 3), t is time, t' is the time the source is turned on, and 6(z) is the Dirac delta-function.We remark that one of the operators 1A can be expressed in terms of the coordinates x': V = -V'. Then for 1L we decompose the operator in the coordinates VV'The operator IT can be decomposed using the expansion of 1 into a certain series