ABSTRACT
We multiply equations (3.l0)-(3.14) by -{?c2(de/dt)-lve, c2, hi, hand de/dt respectively, and then add these equations. Using the eikonal equation (3.l5), it may be shown that in the resulting equation the sum of tenns containing P2, W2, rJ2, Xi.2 and S2 is zero. Then, in this equation we take into account the equality WI • (h V S + hi V Ci) = WI . (V P - c2V (?) (this follows from the equation of state P = P({?, S, Cd if we operate on both sides of this equation with the operator WI . V) and express WI and rJI in terms of PI, making use of equation (3.17). As a result, we obtain the transport equation for PI:
c2 c2
Removing the brackets from this equation and uniting the tenns proportional to apI/at, V PI and PI, we express the transport equation in the following form:
apI [ 2 a 3 2-+ 2u . V PI + PI V· U - U . V In(Qc ) - - In(Qc ) at at
where u = co + v. It will be shown below that u is the group velocity of the sound wave, i.e. the velocity of sound energy propagation. Using the dispersion equation (3.16), we express the sum of the two last tenns in square brackets in equation (3.18) in the fonn (o/k)· Vw - u· Vlna, where a = w - k· v is the frequency of the wave in the coordinate system connected with the moving medium. Then, it follows from the equalities
and k = a / C that
Taking these equalities into account, we express the transport equation in its final fonn
2 api + 2u . V PI + PI (_i In(Qc2a) + V . u - U· V In(QC2a)] = O. (3.19) at at
