Parabolic approximation of nonlocal boundary conditions in ocean acoustics

References

• Bleinstein , N. and Handelsman , R.A. 1975 . “ Asymptotic expansions of integrals ” . New York : Dover Publ. Inc .
   Google Scholar
• Edelstein , S.L. 1990 . Asymptotic behavior of the modes of a smoothly irregular waveguide with permeable boundaries . Soviet Phys. Acoust , 36 ( 2 ) : 193 – 195 .
   Google Scholar
• Edelstein , S.L. 1995 . On the Helmholtz equation in a strip with penetrable boundary . Func. Anal. Appl , 28 ( 3 ) : 223 – 225 .
   Web of Science ®Google Scholar
• Edelstein , S.L. 1992 . Operator analogs of VKB-type estimates and the solvability of boundary-value problem . Mat. Zametki , 51 ( 4 ) : 124 – 131 .
   Google Scholar
• Edelstein , S.L. 1992 . Asymptotic splitting of boundary value problem for differential equations in Banach spaces . Russian Acad. Sci. Dokl. Math , 45 ( 2 ) : 379 – 381 .
   Google Scholar
• Gradshteyn , I.S. and Ryzhik , I.M. “ Tables of integrals, Series and products ” .
   Google Scholar
• Lebedev , N.N. 1972 . “ Special functions and their applications ” . New York : Dover Publ. Inc .
   Google Scholar
• Lee , D. , Botseas , D. and Papadakis , J.S. 1981 . Finite-difference solution to the parabolic wave equation . J. Acoust. Soc. Am , 70 : 795 – 800 .
   Google Scholar
• Lee , D. and Botseas , D. 1982 . An implicit finite-difference computer model for solving the parabolic equation . NUSC Tech. Report 6659 ,
   Google Scholar
• Lee , D. 1995 . Parabolic equation development in recent decade . J.Comp. Acoust , 3 ( 2 ) : 95 – 173 .
   Google Scholar
• Lee , D. and Papadakis , J.S. 1980 . Numerical solution of the parabolic wave equation: An ordinary differential eqution approach . J. Acoust. Soc. Am , 68 : 1482 – 1488 .
   Web of Science ®Google Scholar
• Makrakis , G.N. 1996 . Asymptotic study of the elastic bottom effects in underwater sound propagation . preprint
   Google Scholar
• Papadakis , J.S. Impedance formulation of the bottom boundary condition for the parabolic equation model in underwater acoustics . NORDA Parabolic Equation Workshop, NORDA Tech. Rep ,
   Google Scholar
• Papadakis , J.S. “ Impedance bottom boundary conditions for the parabolic type approximations in underwater acoustics, in Proc. PE Workshop ” . In U.S. Naval Research Laboratory Edited by: Ching-Bing , S.A. , King , D. , Davis , J.A. and Evans , R. 265
   Google Scholar
• Papadakis , J.S. and Mayfield , B. 1986 . IFD+IMPEDANCE Impedance boundary conditions along the bottom interface incorporated in the implicit finite difference code for solving the parabolic equation . IACM Report ,
   Google Scholar
• Papadakis , J.S. and Pelloni , B. 1996 . A method for the treatment of a slopping seabottom in the parabolic approximation . J. Comp. Acoust , 4 ( 1 ) : 89 – 100 .
   Web of Science ®Google Scholar
• Papadakis , J.S. , Taroudakis , M.I. , Papadakis , P.J. and Mayfield , B. 1980 . A new method for a realistic treatment of the seabottom in the parabolic approximation . J. Acoust. Soc. Am , 92 ( 4 ) : 2030 – 2038 .
   Google Scholar
• pierce , A. and Lee , D. 1993 . The influence of the reference wavenumber in computational ocean acoustics . J. Comp. Acoust , 1 ( 1 ) : 77 – 90 .
   Google Scholar
• Polyanskii , E.A. 1970 . On a boundary-value problem for Schroedinger equation . USSR J. Math. Phys , 10 ( 3 ) : 251 – 256 .
   Google Scholar
• Polyanskii , E.A. 1972 . The connection between the solutions of the Helmholtz equation and those of Schroedinger type equation . USSR J. Math. Math. Phys , 12 ( 1 ) : 318 – 329 .
   Google Scholar
• Tappert , R.D. “ The parabolic approximation mathod, in Wave Propagation and Underwater Acoustics ” . In Springer-Verlaf, Berlin, Chap. Edited by: Keller , J.B. and Papadakis , J.S. 224 – 287 .
   Google Scholar
• Weder , R. 1991 . Spectral and scattering theory for wave propagation in perturbed stratified media . Appl. Math. Sci , 87
   Google Scholar
• Xu , Y. 1990 . The propagating solution and far-field patterns for acoustic harmonicwaves in a finite depth ocean . Appl. Anal , 35 : 129 – 151 .
   Google Scholar