Journal of Mathematical Sciences, VoL 96, No. 4, 1999 THE ACOUSTIC FIELD OF A HIGH-FREQUENCY SOURCE MOVING IN A WAVEGUIDE V. S. Buldyrev, A. V. Sokolov, and A. S. Starkov UDC 534.26 The acoustic field of a source moving at a subsonic velocity in a regular waveguide with perfectly reflecting boundaries is considered. The acceleration of the source is assumed to be small. In a moving coordinate system, the asymptotics of the wave field is obtained. This asymptotics is inapplicable near the critical cross sections, for which the Doppler frequency of the source coincides with the frequency of the waveguide mode under consideration. It is demonstrated that, m this case, the wave field can be represented locally by a special type of integral, which is analyzed by the saddle-point method. Bibliography: 6 titles. 1. In a Cartesian coordinate system (x, y, z) comaected with an immovable media, the problem on a moving point source of oscillations is considered. Let S : ~*= g0(t), g0 = (xo(t), yo(t), zo(t)), be the trajectory of the source and let t be time. Our aim is to find the solution of the equation 02u -A(t)e-~r -~'o(t)), 0 < z <Tr, (1) Au- ~t-- T = subject to the radiation condition and the boundary conditions (2) Here, u is the acoustic potential; A(t) and ((t) are smooth functions, characterizing the dependences of the amplitude and phase of the source on time. The velocity of sound is assumed constant and equal to 1. We choose the unit of length in such a way that the width of the waveguide is equal to 7r. It is assumed that, for t g 0, the source moves uniformly with a subsonic velocity v < 1, and its trajectory is parallel to the x axis; the frequency of the source is constant, and its amplitude is equal to 1, i.e., A(t) = 1, ~(t) = wt, go(t) = (vt, O, zo). For such values of time, the solution of the problem (1), (2) can be found by the Fourier method and is of the form [1] (. w-tk .. u = 2 r ~ exp L'wl--"~-'v2"v 2) E H(~ gmx/(x-vt)2 +Y2(1 v2))sinrazsinmzo, t <_ O, m=l (3) xm = ~/w 2 -m2(1 -- v2)(1 - v2) -1, Rexm > 0 for w > rn~/1-v 2, and Imxrn > 0 for 0 < w < m lx/T-S~- v 2. For the critical frequencies, which correspond to xrn = 0, the related Hankel function in the series (3) must be replaced with ln((x - vt) 2 + y2(1 - v2)). Conditions (1)-(3) define the acoustic potential uniquely. We also assume that, for t > 0, the instantaneous frequency of the source r its velocity r'0 '(~), and amplitude are all slowly varying functions of time, i.e., they depend on the variable st, where e << 1 is a small parameter. In this case, for t > 0, the phase function and the coordinates of the source cem be written in the form 6-1~(~t) and e-l~o (et), respectively. A similar problem on the wave field of a moving source was previously considered in [2, 3]. In contrast to this work, in [2, 3] the vertical component of the velocity was assumed to be small, and, in addition, the phenomenon of wave transformation at critical frequencies was not considered. The investigation of this phenomenon is the main concern of the present paper. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 71-78. Original article submitted No- vember 15, 1996. 1072-3374/99/9604-3327522.00 Kluwer Academic/Plenum Publishers 3327