By Professor Leonid M. Brekhovskikh, Dr. Oleg A. Godin (auth.)
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Additional resources for Acoustics of Layered Media II: Point Sources and Bounded Beams
Some authors consider rather than a point source, a straight line source parallel to the boundary. This problem is much easier since the reflected and refracted waves are represented as a single integral, both for moving media and media at rest. 80]. 38 1. 06 \ \ 2\\ * * ',* * ..... 02 Fig. 5 for 90 1r/3 as a function of distance from the virtual source. (8) The total reflected field Pr when sound propagates along the flow (
1) can also be estimated by the method of steepest descent. Stationary points for the expression in the exponent can be found from sin 2( '¢ - 'P) = O. Among these points, only one, '¢ = 'P, lies on the integration contour and satisfies the condition qO > O. 6) we find the two first terms of the asymptotic expansion of PI in powers of k- l : PI ~Rllexp(ikR})[Vi(sinOo,'P)-iN/kRd, N = ~ [~ . 7) 2l . 3). 9), as is to be expected. 7) could be also found by the use of the two-dimensional method of stationary phase discussed in Sect.
7r /2 - () ~ 1) when m ~ 1. Indeed, we have V:::::: 1 if 1 - q ~ m- 1 and V = -1 at q = 1. 4 Very Large or Very Small Ratio of Media Densities 21 descent is invalid. 10) N -+ 0 if ()o t= 7r /2 and N -+ 00 if ()o = 7r /2. This discontinuous result has no physical sense. 10) the integral from the first term has the exact value exp(ikRl)/RI. 5) and we obtain Pc = exp(ikRl) (k Rl 2;;:- )1/2 exp (i7r. 4" + lkRI ) P = 2(dq/ds)V(n 2 - q2)q/0 - q2) 1+ 00 -00 2 pes) exp( -l kR lls )ds , (m~ + Vn2 _ q2) -1.
Acoustics of Layered Media II: Point Sources and Bounded Beams by Professor Leonid M. Brekhovskikh, Dr. Oleg A. Godin (auth.)