Numerical comparison of two asymptotic methods for solving wave diffraction problems in smooth inhomogeneous media

Abstract

UDC 534.1;621.371 The methods of Maslov's canonical operator (MCO) and Gaussian beam summation (GBS), which are used to describe the diffraction structures of wave fields in complex inhomogeneous media, are numerically compared. The analysis is performed on a sample calculation of the amplitude of an acoustic wave field propagating in an inhomogeneous sound channel. It is shown that, on the axis of the sound channel, the results of calculation by the MCO and GBS methods agree within graphical accuracy. However, far from the axis, application of GBS method encounters definite difficulties. On the basis of the results of the comparison, it is concluded that the MCO method better reveals the mechanism of the diffraction field formation in the focal region. In the short wave approximation of wave diffraction and propagation problems in complex inhomogenous media, the task often arises to describe the field in the focal region, where the traditional ray (geometric) approach is incorrect. As the practice of such calculations shows, focal regions of dissimilar form are a similar phenomenon. They really exist in the ionosphere and tropospheric channels of radio communication, in underwater acoustic channels, in fiber optics, etc. This makes the task of the detailed description of the field in them very topical. At the present time, the field in the focal region is most fully described in the framework of methods for plotting asymptotic solutions, using rapidly oscillating integrals, and by the methods of parabolic equations (PE). Among the first group of such methods, we shall examine the means of plotting solutions based on Maslov's canonical operstor (MCO) [i, 2] including the topological classification of the focal regions and the local description of the field in them [3-7], since the practice of solving similar problems recommends just this method to us. We will compare our numerical solution with a solution plotted by a new and promising method -- the method of Gaussian beam summation (GBS) [8-11]. The present work compares the methods of MCO and GBS with respect to the final result of their application in a concrete problem. This will allow a more complete answer to the question of how effective the examined methods are in the applied problems, since the theoretical evaluations of their effectiveness in the literature often have a rather abstract character and are uncorrelated. We made use of the results of [8, 9] as a completed and fully documented GBS solution of the problem, which studied the propagation of monochromatic sound waves, excited by a point source located on the axis of a symmetric acoustic wave guide in a plane-laminar medium. i. Statement of the Problem. Asymptotic Solution in the MCO Form. We shall examine Helmholtz's equation, which we shall write in the symbology of [8, 9]: