Acoustics of Layered Media II: Point Sources and Bounded by Professor Leonid M. Brekhovskikh, Dr. Oleg A. Godin (auth.)

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 truly is mathematically rigorous yet even as care is taken that the actual usefulness in purposes and the common sense of the idea usually are not hidden. either relocating and desk bound media, discretely and constantly layered, together with a range-dependent atmosphere, are taken care of for varied sorts of acoustic wave assets. designated appendices offer additional history at the mathematical methods.
This moment variation displays the extraordinary contemporary 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

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E. kr » m 2 » l. It is clear from symmetry considerations that the condition kr » m- 2 » 1 must be fulfilled if m « l. 12) is valid at rather moderate kr, since, as it can be easily proved, it coincides with the exact results if m = 1, nand r are arbitrary or n = 1 and m and r are arbitrary. Indeed, when m = 1 or n = 1, 9 is a linear function of q2 and gl is that of u 2. 58). 9-11) obtained by the method of steepest descent. Let us return, however, to the problem of the field of the source located above a weak boundary.

AjV . 5) . J!. = (-7fi/2)H6 1\u). 3) can be neglected. 7) . 1). 7). We introduce the substitution (u 2 + t 2 ) 1/2 - U = 8 2 . 8) U . In the case we are interested in (7f /2 - eo « 1) 86 = 2kR1 sin 2 (i _ e~) « u = kR1 sin eo. 5). 7) is elementary. We have finally: _exp(ikR 1 ) {m-l imkR1(n 2 -1)[ r= W2( f I)]} R + ( )2 l+v 7fwe er w+ , 1 m+l m+l Prw 37fi) VInI:D . (7f = exp ( 4 2kR1 sm "4 - 2eo) . 6), in terms of Fresnel integrals with real arguments. 3) is in powers of kRI(n2 -1). 17,18). 9).

The asymptotics for P2 contain an additional term with the factor exp[ikR('lj;)+u 2/2] = exp[ikR( 'lj;) cos(B-8)] if lui» 1, Re {u} < o. The expression in the exponent has a stationary point at 'lj; = 'lj;l. 5) we obtain the equation for 'lj;l sin('lj;l - 'P) = [cot 8('l/Jl) COS('l/Jl - 'P) - cot Bo]8'('lj;d . 10) When lui ;S 1 we have I(8/8'lj;) Dy(u)1 c:o:' 18u/8'lj;1 ;S IkR('lj;)ll/2. 8) can be assumed to be changing slowly compared to the exponent if 'l/J is not too close to the stationary point 'l/J = 'P.

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