By Professor Leonid M. Brekhovskikh, Dr. Oleg A. Godin (auth.)
Acoustics of Layered Media II offers the speculation of sound propagation and mirrored image of round waves and bounded beams in layered media. it really is mathematically rigorous yet whilst care is taken that the actual usefulness in functions and the good judgment of the speculation usually are not hidden. either relocating and desk bound media, discretely and regularly layered, together with a range-dependent setting, are taken care of for numerous different types of acoustic wave resources. precise appendices supply additional heritage at the mathematical methods.
This moment version displays the amazing fresh growth within the box of acoustic wave propagation in inhomogeneous media.
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Extra info for Acoustics of Layered Media II: Point Sources and Bounded Beams
7) by replacing V1 by V2 (q - qd/ 2 . 7), this contribution gives the ray acoustics result for a reflected field plus corrections of the order of 1/kR 1 in the amplitude and phase. The second stationary point at l,b = l,bl appears when B(l,bd > o(l,bd. 15) + cos(l,b - cp)[l + (0')2(1 + 2cot 20) - o"cot oj} . Let IBo - o(cp)1 ~ 1. Then lu(cp)1 » 1. 10), Il,bl - cpl « 1. Hence IBo - o(cp)1 ;::::0 IB(l,bd - O(l,bl) + [B'(l,bd - o'(l,bd](cp -l,bdl « 1. This contradiction shows that the two cases considered above, IBo -o(cp)1 « 1 and lu(cp)1 » 1, IU(l,bdl » 1, cover the entire problem.
The asymptotics of integrals of this type are considered in the Appendix A. 58) we find + i/kr) exp(ikr)/r i(m 2 - 1)(1 + i/kr) 2imn2(1 + i/k1r) exp(ik1r)/r -30(k-2 k- 2) + k1rm2(1 - n 2) - in 2(m 2 - 1)(1 + i/k1r) +r + 1 . 12) This result was obtained first by D. 54]. Here, the first term represents the wave propagating in the upper medium, the second term is due to propagation in the lower one. 12) is rather valuable because of its universality, as well as its simplicity. It may be used to check more complicated asymptotics obtained when the source is above the boundary.
1) equals 1. 3) the integration path must lie in the region It I < 1. To obey this condition we choose the path going around the point q = 1 in quadrant IV along the halfcircle of the radius which is large enough, and then again return to the path T (Fig. 2). Since the integrand has no singularities on the upper sheet, such deformation of the integration path is permissible. In the integrals obtained, the integration path can be transformed into the real axis without any influence on the value of the integral.