Geometrical theory of sound propagation in the atmosphere

Abstract

This paper deals with the geometry of sound waves and rays propagated in a moving inhomogeneous medium with particular reference to the atmosphere. The theory developed is, essentially, a generalization of that in geometrical optics for the propagation of light in an inhomogeneous isotropic medium to' the case where the medium is in motion. Differential equations defining the wavefront and rays of a sound propagation in an inhomogeneous moving medium are derived, and then transformed by expressing the unit normal of the wavefront in terms of certain trace velocities. With the equations in this form, an integral is immediately obtainable for sound propagated in the atmosphere by assuming that the velocity of sound and wind-velocity vector are functions of height only. From this solution the law of refraction, the condition for total reflection of a sound ray, and the equations for the wave-front at any time are found. By way of example, the theory is applied to a simple velocity of sound v. height relation, and expressions are obtained for the range and time at which the abnormal sound propagation from a ground-level source returns to earth. Numerical results calculated from these formulae are found to be in accord with observations that have been made on this phenomenon.