Wave beam with minimum divergence in a smoothly inhomogeneous medium

Abstract

We consider acoustic wave beams propagating in a smoothly inhomogeneous medium along a given reference ray. It is shown that in the aberration-free approximation, a Gaussian beam diverges with distance (on the average) less than any other beam with the same initial width. This result has been obtained by solving a variational problem that is similar to the well-known quantummechanical problem of seeking the quantum state with minimum uncertainty (coherent state). An example of the beam with minimum divergence in the realistic model of a deep-ocean acoustic waveguide is considered. An approximate analytical estimate for the amplitudes of normal modes forming the beam is obtained. STATEMENT OF THE PROBLEMDuring acoustic monitoring of the underwater sound-channel inhomogeneities, the problem of selected illumination of separate parts of the water bulk often arises. For example, it may be required that the initial field on the aperture of the radiating antenna should be chosen in such a way that the radiation is focused into the vicinity of a given observation point [1] or the maximum (path-average) density of sound energy is reached inside a fixed interval of depths [2].In this paper, we consider the problem of choosing the initial field of the antenna for the formation of a narrow wave beam, which can propagate over large distances along a given reference ray without notable spreading. Actually, we discuss the possibility of creating an underwater searchlight illuminating the vicinity of the chosen beam up to distances of the order of hundreds of kilometers. Although we are speaking of field formation in an underwater sound channel, the expression for the initial field of a minimum-divergence beam, which is the main result of this paper, is general and applies not only for waveguides.Neglecting the horizontal refraction of waves, which is fairly weak in a deep sea, we consider the two-dimensional model of an underwater sound waveguide with constant sound density and velocity field c(x, z), where x is the horizontal distance and z is the depth. The z axis is oriented vertically upward, and the free water surface is located at the level z = 0. Focusing only on waves trapped by the waveguide, we assume that the sea depth is formally infinite. Moreover, for simplicity, we will consider the beams which propagate in the water bulk and do not reach the water surface. Thus, the surface effect actually will not be taken into account in our analysis.We assume that a monochromatic sound field with complex amplitude u(x, z) is radiated by a vertical antenna located along the x = 0 line. Our purpose is the optimal choice of the initial field on the antenna u(0, z) = u 0 (z)( 1 )