By Professor Leonid M. Brekhovskikh, Dr. Oleg A. Godin (auth.)

ISBN-10: 3662027763

ISBN-13: 9783662027769

ISBN-10: 366202778X

ISBN-13: 9783662027783

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Additional info for Acoustics of Layered Media II: Point Sources and Bounded Beams

Example text

Using this fact and reasoning as in Sect. 7) 1m {qp} ;:::0. Calculating the pole residue, we obtain finally I Pr -_ exp(ikR ) RI + [ ( . 70 - [8'k~: x U(qs, qp) q;r' iN + 0 ( -1- )] -- kRI k2 Rr exp {i [kRI (qp sinO. 8) . 9) where N = -2Tf(1 + Tf cos Bo)(cos Bo + Tf)-3 ,I. 8) must be modified. 4, 12), we obtain 24 1. Reflection and Refraction of Spherical Waves Pr= exp(ikRl) (. 10) Im{u}20, where correction terms of the order O(l/kRl) are omitted. 10) we have a smooth function of u, whereas the passage method gives a discontinuous factor 27r1/2U when the pole is far enough from the passage path.

We assume M < 1. Then the points q = -I, q = q2 are not touched in the process of integration. 6 Reflection from Moving Medium 33 Vt and Vz have no singularities at q = ql. We denote by PI or P2 and 411 or 412 those components of Pr and 41 which originate from the terms Vt or Vz(q - ql)1/2 in the reflection coefficient. First consider Pl. 2) analyzed by the method of steepest descent. 2) has only one stationary point at q = qo(,¢) where qO = r cos('P - '¢)/ R('¢) , R('¢) == [r2 cos2('P - '¢) + (z + zd]1/2.

This result, however, is erroneous. 58] cannot be used as was done by Stickler. The asymptotics for Pr could be constructed by the reference integrals method. 2). 59], but it is rather difficult. 30 1. Reflection and Refraction of Spherical Waves However, the direct numerical calculation of the reflected field, using integral representation, is rather efficient. , their ability to treat different problems in the same manner. For example, it is worth mentioning the numerical treatment of the integral Fourier expansion of the sound field over the horizontal coordinates.

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


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