+ 2. +

Here, VJ. = (vy,vz) and 'VJ. = (alay,alaz). Equation (2.86) describes the propagation of sound waves in a moving medium in the small-angle approximation. This equation may be derived directly from equation (2.75) if, in the latter equation, we express p in the form p = A exp(ikox) and then neglect the term

2.4.3 Parabolic equation

Equation (2.86) is rather complicated. We will show that for short wavelengths, most terms in this equation can be ignored. To show this, let us consider the order of magnitude of terms in this equation. We have

Here, M = vlco is the Mach number and v = I'l - 'loll'lo characterizes the variation in the density. Note that, almost always, v « 1. Furthermore, in equation (2.86), aAlax '"" AI I because the variation of A along the x-axis is caused by inhomogeneities of the medium. Finally, let us estimate the order of magnitude of I'V 1. A I. The phase factor of the sound pressure p is of the order of magnitude given by exp[iko(x cos e + e . r sin e»), where e is the unit vector in the direction of the projection of the wave vector ko of the sound wave on the yz-plane. Therefore, in the small-angle approximation, A contains the phase factor exp(ikoBe. r), so that I'VJ.Allko ~ Amax(llkol.Bm). Here, Bm is the maximum value of the angle e between the direction of wave propagation and the x-axis.