About the Authors

Finn B. Jensen received the MSc and PhD degrees fromthe Technical University of Denmark (TUD) in 1968and 1971, respectively. From 1969 to 1973, he was anAssistant Professor in the Department of Fluid Mechan-ics at TUD. Since 1973, he has been employed at theNATO Undersea Research Centre, La Spezia, Italy. Hestarted as a research scientist developing numerical mod-els of sound propagation in the ocean, then served for17 years as Head of the Environmental Modelling Groupat NURC, and since 1999 as a Senior Scientist andProject Leader for research related to propagation, rever-beration, and target strength modeling.

William A. Kuperman received his PhD in Physicsfrom the University of Maryland and presently is aProfessor at the Scripps Institution of Oceanography,University of California, San Diego and the director ofits Marine Physical Laboratory. Before coming to UCSDin 1993, he was at the Naval Research Laboratory andthe NATO Undersea Research Centre in La Spezia, Italy.He has done experimental and theoretical research inan assortment of underwater acoustics and signal pro-cessing areas and has spent about 3 years at sea doingexperiments.

F.B. Jensen et al., Computational Ocean Acoustics, Modern Acoustics 773and Signal Processing, DOI 10.1007/978-1-4419-8678-8,c� Springer Science+Business Media, LLC 2011

774 About the Authors

Michael B. Porter graduated from Caltech in 1979 andreceived his PhD from the Northwestern University in1984. He held civil service positions at the Naval OceanSystems Center, the Naval Research Laboratory, and theNATO Undersea Research Centre in Italy. In academia,he was a Professor at the New Jersey Institute of Technol-ogy, and held visiting positions at the Scripps Institutionof Oceanography, and the University of Algarve (Portu-gal). In the commercial sector, he was an Assistant VicePresident and Chief Scientist at SAIC before joining HLSResearch as its President and CEO. His research interestsinclude propagation modeling, target scattering, commu-nications, bioacoustics, and environmental inversion.

Henrik Schmidt is Professor of Mechanical & OceanEngineering at the Massachusetts Institute of Technol-ogy. He received his MS degree from the TechnicalUniversity of Denmark in 1974, and his PhD from thesame institution in 1978. Following a postdoctoral fel-lowship at the Risoe National Laboratory in Denmark, hejoined the NATO Undersea Research Centre in Italy in1982, where he worked on computational ocean acous-tics until he joined the MIT faculty in 1987. ProfessorSchmidt’s research has focused on underwater acousticpropagation and signal processing, in particular on theinteraction of sound in the ocean with seismic waves inthe ocean bottom and the Arctic ice cover.

Name Index

AAbawi, A.T., 582, 609Abramowitz, M., 141, 142, 153, 260, 273, 333Achenbach, J.D., 598, 599, 610Ainslie, M.A., 709, 769Akal, T., 9, 333, 335, 678, 679, 703, 758, 766,

767, 772Aki, K., 216, 231, 391, 452, 634, 659Alford, R.M., 541, 608Alippi, A., 454Alterman, Z., 541, 608Anderson, K.D., 454Anderson, W.L., 272, 333Andreevoi, I.B., 152, 455Andrew, R.K., 57, 64Athanassoulis, G.A., 430, 455

BBabuska, I., 566, 608Backus, G.E., 254, 333Baer, R.N., 426, 454, 465, 509, 528, 529Baggeroer, A.B., 374, 453, 714, 722, 730, 732,

769, 770Bamberger, A., 464, 471, 528Barbagelata, A., 675, 678, 679, 703Bartberger, C.L., 372, 378, 453Baumgartner, G.B., 454Beisner, H.M., 374, 453Belibassakis, K.A., 430, 455Bell, K., 750, 771Bender, C.M., 216, 232Beran, M.J., 666, 703Berenger, J.P., 566, 568, 608, 609Bergman, P.G., 71, 152, 495, 528Berkhous, B.V., 678, 703Berkson, J.M., 333, 335, 703Berliner, M.J., 11, 63Berman, D.H., 465, 528

Biondi, B., 179, 231Bjork, A., 533, 542, 608Bleistein, N., 189, 230Block, S., 645, 660Blottman, J.B., 566, 590, 595, 598, 608Bold, G.E.J., 189, 231Botseas, G., 499, 511, 528, 529, 694, 696, 698,

704Bovio, E., 315, 335Bowlin, J.B., 206, 231Bowman, J.J., 416, 420, 454Boyles, C.A., 372, 453Brekhovskikh, L.M., 41, 43, 50, 51, 64, 67,

115, 133, 138, 152, 153, 221, 232, 410,454, 455

Brent, R.P., 376, 453Brienzo, R.K., 749, 763, 770Brock, H.K., 476, 493, 495, 528Brooke, G.H., 470, 471, 528Brown, M.G., 218, 232Brown, R., 272, 333Browning, D.G., 664, 694, 703Bucker, H.P., 174, 175, 183, 184, 193, 200,

208, 210, 226, 230, 231, 300, 301, 354,

Buckingham, M.J., 426, 429, 454, 678, 703,704

Burenkov, S.V., 443, 455Burnett, D.S., 547, 566, 599, 600, 608, 610Burridge, R., 408, 426, 454

CCakmak, A., 454Campbell, J., 192, 231Canepa, G., 584, 595, 610Cannelli, G.B., 454Carbo-Fite, C., 63Carbone, N.M., 678, 703

775

453, 730, 769

776 Name Index

Carey, W.M., 664, 694, 696, 698, 703, 704,728, 769

Cassereau, D., 758, 771Cavanagh, R.C., 457, 460, 468, 481, 490, 527Cerveny, V., 155, 168, 183, 191, 194, 200,

230, 231Chapman, C.H., 219, 232Chapman, N.R., 34, 63, 372, 379, 453, 454,

465, 528Chapman, R.P., 52, 56, 64Chen, C.F., 511, 529Chew, W.C., 272, 333Chin-Bing, S.A., 511, 528, 529, 546, 608Chuprov, S.D., 133, 152, 444, 455Claerbout, J.F., 463, 464, 511, 527Clark, C.A., 372, 453Clay, C.S., 3, 23, 62, 63, 71, 152, 758, 771Cockrell, K.L., 729, 769Colborn, J.G., 5, 63Colladen, J.-D., 2Collino, F., 566, 608Collins, M.D., 464, 470–472, 474, 487, 498,

508, 511, 517, 528, 529, 532, 607, 621,659, 733, 770

Cook, R., 160, 230Cornuelle, B.D., 3, 63Cornyn, J.J., 206, 231Cox, C.S., 671, 703Cox, H., 662, 703, 723, 750, 752, 754, 769,

771Cray, B.A., 750, 753, 771Cristini, P., 379, 454Cron, B.F., 669, 671, 703Czenszak, S.P., 749, 770

DD’Alembert, J., 1D’Spain, G.L., 11, 63, 133, 153, 220, 232, 443,

455, 681, 686, 704, 750, 752–754, 771Dahlquist, G., 533, 538, 542, 608Dashen, R., 694, 704Davies, B., 379, 454Davis, J.A., 189, 192, 231, 457, 460, 468, 472,

481, 490, 527, 528, 694, 696, 698, 704Dawson, T.W., 277, 281, 334, 571, 609Deane, G.B., 219, 232, 678, 703, 704Deavenport, R.L., 141, 153, 233, 234, 260,

265, 332Debever, C., 749, 771Del Grosso, V.A., 3, 63DelBalzo, D.R., 733, 770Delves, L.M., 378, 453Deschamps, G.A., 180, 231

Diachok, O.I., 694, 696, 704Dicus, R.L., 733, 770DiNapoli, F.R., 141, 153, 233, 234, 260, 265,

307, 332, 335Doolittle, R., 426, 454Doombos, D.J., 453Dougherty, M.E., 633, 659Dowling, D.R., 758, 765, 771, 772Dozier, L.B., 369, 375, 453Drummond, R., 219Dubbelday, P.S., 379, 454Dubra, A., 272, 333Duda, T.F., 206, 231Dushaw, B.D., 3, 63Dzieciuch, M.A., 57, 64, 725, 769Dziewonski, A., 645, 660

EEbbeson, G.R., 33, 34, 63Edmonds, G.L., 750, 771Engquist, B., 463, 464, 471, 528Epifanio, C.L., 678, 704Etter, P.C., 61, 64Evans, R.B., 353, 403, 407, 429, 452, 454,

472, 528, 531, 532, 555, 607, 664, 694,696, 698, 703, 704

Everstine, G.C., 600, 610Evora, V.M., 750, 771Ewing, W.M., 24, 63, 71, 122, 152, 233, 234,

242, 243, 245, 333, 352, 452, 644, 659

FFan, H., 292, 334Fawcett, J.A., 277, 281, 334, 420, 429, 454,

571, 582, 609, 629, 659Feit, D., 568, 609Feit, M.D., 465, 528Felsen, L.B., 281, 334Ferla, C.M., 369, 407, 453, 454, 506–508, 529,

758, 772Ferrari, J.A., 272, 333Ferris, R.H., 640, 659Feshbach, H., 116, 152, 621, 659Feuillade, C., 733, 770Fialkowski, J.M., 52, 54, 64Fialkowski, L.T., 687, 704Fink, M., 758, 771, 772Fisher, F.H., 36, 63Fisher, G.A., 233, 333, 546, 608Fitzgerald, R.M., 624, 630, 659Fizell, R.G., 730, 733, 770Flatte, S.M., 7, 63Fleck, J.A., 465, 528

Name Index 777

Fletcher, C.A.J., 547, 608Flynn, J., 766, 772Fock, V.A., 457, 527Foreman, T.L., 193, 206, 231Forsythe, G.E., 533, 540, 543, 607Fox, W.L.J., 315, 335, 765, 766, 772Francois, R.E., 37, 38, 63Franssens, G.R., 254, 333Frazer, L.N., 277–279, 334, 617, 659, 749, 770Freitag, L.F., 206, 231Fricke, J.R., 534, 541, 543, 545, 608Fried, S., 683, 704Frisk, G.V., 71, 152Fry, W.J., 337, 452Fuchs, K., 233, 234, 256, 332

GGalerkin, B.G., 548, 608Gamow, G., 1, 62Garrison, G.R., 37, 38, 63Gaul, R.D., 664, 703Gauss, R.C., 52, 54, 64Geissler, P., 704Georges, T.M., 226, 232Gerstoft, P., 220, 232, 279, 281, 334, 571, 574,

587–589, 609, 727, 769Gettrust, J.F., 278, 334Giddings, T.E., 568, 609Gilbert, F., 254, 333Gilbert, K.E., 499, 503, 528Gingras, D.F., 733, 770Glattetre, J., 282, 283, 286, 334, 431, 455, 582,

609, 687, 704Glegg, S.A.L., 678, 703Godin, O.A., 43, 50, 64, 67, 71, 115, 133, 152,

410, 454, 584, 610Goh, J.T., 287, 334Goldhahn, R., 729, 769Goncharenko, B.I., 753, 771Gordienko, V.A., 753, 771Gordon, D.F., 175, 231, 372, 451Gottlieb, D., 544, 608Graber, H.C., 664, 703Grachev, G.A., 133, 136, 153Gragg, R.F., 52, 54, 64Gramann, R.A., 11, 63, 752–754, 771Greene, R.R., 463, 471, 479, 480, 482, 527,

528Gubbins, D., 215, 231Guidi, G., 675, 678, 679, 703Gunzberger, M., 544, 608Guthrie, A.N., 630, 659Guthrie, K.M., 359

HHale, F.E., 20, 21, 63Halpern, L., 463, 464, 471, 528Hamilton, E.L., 39, 63, 647, 660Hamming, R.W., 379, 454Hamson, R.M., 733, 770Hardin, R.H., 457, 459, 463, 486, 488, 510,

527Harkrider, D.G., 241, 333Harris, J.H., 52, 64Harrison, C.H., 678, 686, 703, 704, 726, 727,

769Haskell, N.A., 234, 252, 333Haussecker, H., 678, 704Hawker, K.E., 624, 630, 659Hawkes, M., 750, 771Heaney, K.D., 426–429, 443, 454, 455Heitmeyer, R.M., 733, 770Hickman, G., 729, 769Hildebrand, F.B., 196, 231Hill, R.J., 468, 528Hines, P.C., 315, 335Hinich, M.J., 730, 769Hitney, H.V., 379, 454Hobaek, H., 474, 528Hodgkiss, W.S., 220, 232, 681, 686, 704, 727,

733, 749, 750, 758, 763, 764, 766, 767,769–772

Holland, C.W., 314, 315, 317, 335Hollett, R., 315, 317, 335Horrigan, A.A., 133, 137, 153Hovem, J.M., 63, 546, 608Howe, B.M., 3, 57, 63, 64Huang, C.-F., 727, 769Huang, D., 497, 508, 528, 546, 608Hursky, P., 749, 771

IIde, J.M., 337, 452Ihlenburg, F., 566, 599, 600, 608Ilyushin, Ya.A., 753, 771Ingard, K.U., 71, 152, 415, 416, 454, 605, 610,

628, 659Ingenito, F., 285, 334, 374, 388, 413, 417, 421,

453, 454, 580, 609, 663, 678, 687, 688,702–704, 754, 771

JJackson, D.R., 39, 64, 756, 758, 759, 765, 767,

771, 772Jacobson, M.J., 666, 703Jahne, B., 678, 704

778 Name Index

Jardetzky, W.S., 122, 152, 233, 234, 242, 243,245, 333, 352, 452, 659

Jeong, W.-K., 160, 230Jerzak, W., 472, 528Jesus, S.M., 733, 770Johnson, D.H., 716, 720, 769Joly, P., 463, 464, 471, 528Jones, C., 765, 772Jones, D.S., 175, 231Jones, J.P., 771Jones, R.M., 226, 232Joshi, S.M., 664, 703Julian, B.R., 215, 231Junger, M.C., 568, 609

KKaplunov, J.D., 592, 610Karal, F.C., 541, 608Kargl, S.G., 592, 610Kawase, H., 281, 334, 571, 609Keenan, R.E., 174, 230Keller, H.B., 216, 231, 375, 453Keller, J.B., 155, 178, 216, 230, 527Kelly, K.R., 541, 608Kennett, B.L.N., 233, 234, 255, 333Kerman, B.R., 57, 64, 703Kerry, N.J., 233, 234, 255, 333Kessel, R.T., 580, 609Kewley, D.J., 664, 694, 703Kibblewhite, A.C., 671, 703Kim, J., 766, 772Kim, S., 160, 230King, D.B., 192, 231, 472, 528Kinney, W.A., 733, 770Knightly, G.H., 463, 527Knobles, D.P., 664, 703, 749, 770Ko, P., 733, 770Koch, R.A., 372, 453, 733, 770Kolsky, H., 243, 333, 528Kong, J.A., 272, 333Kosloff, D., 545, 608Kosloff, R., 545, 608Kossovich, L.Y., 592, 610Kravtsov, Yu.A., 178, 231Krenk, S., 279, 284, 334, 429, 576, 609Kriegsmann, G.A., 509, 529Krolik, J.L., 729, 733, 749, 769, 770Kutschale, H.W., 233, 234, 251, 254, 306, 332

LLai, H., 750, 752, 753, 771Landau, L.D, 224, 232

Landisman, M., 645, 660Lara-Saenz, A., 63Lasky, M., 2, 62Lee, D., 216, 230, 231, 300, 334, 453, 454,

457, 463, 486, 497, 499, 500, 503, 505,508, 509, 511, 527–529

Lee, J., 285, 309, 311–314, 334, 592, 595, 610Lee, S., 729, 769Lehmer, D.H., 378, 453Leighton, T.G, 7, 63Leonardo da Vinci., 2Leontovich, M.A., 457, 527LePage, K.D., 272, 288, 292, 295, 314, 325,

328, 333–335Levinson, S.J., 372, 453Li, F.-H., 453Lichte, H., 155, 230Lifshitz, L.D., 224, 232Liggett, W.S., 666, 703Lin, Y.-T., 511, 529Lindberg, J.F., 11, 63Lindsay, R.B., 1, 62Lingevitch, J.F., 767, 772Liu, Q.H., 272, 333, 334, 567, 568, 608, 609Liu, Y-C., 173, 230Livingston, E.S, 453Lobkis, O.I., 680, 704Lourtie, I.M.G., 770Lu, I.T., 281, 334Luby, J.C, 11, 63, 750, 752–754, 771Lucifredi, I., 286, 334, 581, 592, 609Ludwig, D., 178, 231Luo, W., 408, 429, 431, 433, 435, 454Lurton, X., 709, 769Lyness, J.N., 378, 453Lyons, A., 310, 312, 335Lysanov, Yu., 41, 43, 51, 64, 133, 138, 153,

221, 232

MMaguer, A., 315, 335Makris, N.C., 285, 334, 420, 454, 664, 678,

692, 703, 704, 729, 769Mallick, S., 277, 279, 334, 617, 659Malm, N., 566, 590, 595, 598, 608Maranda, B.H., 629, 659Marburg, S., 703Marple, L.,720, 769Marshall, J.R., 56, 64Marston, P.L., 592, 610Matar, O.B., 568, 609May, J., 372, 453

Name Index 779

McDaniel, S.T., 300, 334, 457, 466, 486, 497,500, 505, 508, 527–529

McDonald, B.E., 426–429, 454, 622, 659McGirr, R.W., 192, 200, 210, 231Medwin, H., 3, 62, 71, 152Mellen, R.H., 307, 335Mercer, J.A., 57, 64Mersenne, M., 1Metzger, K., 3, 63Michalopoulou, Z.-H., 749, 770, 771Miklowitz, J., 129, 152, 240, 333, 471, 528Miller, B.E., 307, 335, 655, 660Mindlin, R.D., 655, 660Mitchell, S.K., 372, 453, 749, 770Moler, C.B., 211, 231Monk, P., 566, 567, 608Montagner, J.-P., 772Morse, P.M., 71, 116, 152, 415, 416, 454, 605,

610, 621, 628, 659Moura, J.M.F., 770Mueller, G., 233, 234, 256, 332Muir, T.G., 474, 528Muller, G., 191, 194, 231Munk, W., 71, 152, 356, 453, 694, 704, 725,

769Murphy, E.L., 189, 231Murphy, J.E., 546, 608, 611, 659

NNaugolnykh, K., 10, 63Nehorai, A., 11, 63, 750, 771Nero, R.W., 52, 54, 64Neumann, P., 317, 335Newman, A.V., 374, 453Newmark, N.M., 541, 569, 608Newton, I., 1Nghiem-Phu, L., 465, 528, 611, 659Nicholas, M., 749, 771Nielsen, P.L., 517, 529Nolde, E.V., 592, 610Nolet, G., 230Nolte, L.W., 733, 770Northrup, J., 4, 5, 63Novikov, B.K., 10, 63Nutile, D.A., 624, 630, 659Nuttall, A.H., 750, 753, 771

OOberhettinger, F., 666, 670, 703Officer, C.B., 71, 152Ogilvy, J.A., 51, 64Olson, A.H., 233, 333, 546, 608

Oppenheim, A.V., 262, 266, 309, 333, 613,614, 659, 769

Orcutt, J.A., 233, 333, 546, 608, 671, 703Orlov, Yu.I., 178, 231Orris, G.J., 749, 771Orszag, S.A., 216, 232Osler, J., 317, 335Ostrovsky, L., 10, 63Outing, D.A., 472, 517, 528Owen, M.M., 723, 769Ozard, J.M., 733, 742, 770

PPace, N.G., 310, 312, 335Paldi, E., 11, 63, 750, 771Papadakis, J.S., 527Papadakis, P., 529Pappert, R.A., 379, 454Parent, G.B., 666, 703Parvulescu, A., 758, 763, 771Pecholcs, P.I., 749, 770Pedersen, M.A., 175, 211, 231, 372Pekeris, C.L., 70, 118, 125, 152, 233, 234,

245, 251, 333, 337, 452Perciante, D., 272, 333Perkins, J.S., 509, 529, 687, 704, 742, 743,

749, 770, 771Pernod, P., 568, 609Perozzi, P.J., 216, 231Pierce, A.D, 2, 62, 408, 454, 568, 599, 600,

604, 609, 610, 623, 624, 659Pilipetsky, N.F., 758, 771Popov, M.M., 183, 231Porter, R.P., 767, 772Post, R.F., 337, 452Potter, J.R, 678, 704Pouliquen, E, 310, 312, 315, 335Prada, C., 758, 771Preobrazhensky, V., 568, 609Press, F., 71, 122, 152, 233, 234, 242, 243,

245, 333, 352, 452, 644, 659Pryce, J.D., 384, 454Psencık, I., 183, 191, 194, 231Pythagoras, 1

QQuijano, J.E., 729, 769

RRanz-Guerra, C., 63Rauch, D., 645, 646, 660

780 Name Index

Rayleigh, J.W.S., 129, 152Readhead, M.L., 678, 704Reiss, E.L., 367, 369, 375–377, 392, 395, 452,

453Rekdal, T., 179, 231Revie, R., 133, 137, 153Richards, P.G., 216, 231, 391, 393, 452, 634,

659Richardson, A.M., 733, 770Richardson, M.D., 39, 63, 64, 608Richter, J.H., 379, 454Riley, J.P., 226, 232Ritcey, J.A., 765, 766, 772Roberts, B.G., 208, 231Rogers, A., 314, 335Rosenberg, A.P., 458, 527Ross, D., 57, 64, 665, 703Rouseff, D., 673, 703, 729, 765–767, 769, 772Roux, P., 680–684, 686, 704, 758, 764, 772Rudenko, O.V., 10, 63

SSabra, K.G., 680, 681, 683, 686, 704, 767, 772Sarkissian, A., 580, 609Schafer, R.W., 262, 266, 309, 333, 613, 614,

659Schmalfeldt, B., 645, 646, 660Schmidt, R.O., 721, 769Schneider, H.G., 216, 232Schultz, M.H., 230, 453Schuster, G.T., 281, 334, 571, 609 (PM: Spelt

as “Shuster” in page 281. Please checkand change.)

Sellschopp, J., 6, 63Senior, T.B.A., 416, 420, 454Seong, W., 281, 334, 546, 608Sethian, J.A., 160, 230Shaffer, J.D., 624, 630, 659Shang, E.C., 733, 770Sheer, E.K., 722, 732, 769Sheppard, C.V., 372, 453Sherman, C.H., 669, 671, 703Shirron, J.J., 566, 568, 608, 609Shkunov, V.V., 758, 771Siderius, M., 609, 686, 704, 726, 727, 749,

767, 769, 771, 772Siegmann, W.L., 472, 509, 511, 517, 528, 529Simmen, J., 453Simmons, G., 272, 333Simmons, V.P., 36, 63Simons, D.G., 609, 678, 703Smith, G.D., 533, 538, 540, 543, 607Smith, K.B., 372, 453

Smith, L.C., 281, 334, 571, 609Snoek, M., 675, 678, 679, 703Solomon, L.P., 211, 231Song, H.C., 443, 445, 455, 727, 758, 764, 766,

767, 769, 772Spiesberger, J.L., 3, 63, 206, 231St. Mary, D.F., 463, 527Stakgold, I., 338, 351, 386, 452Stearns, S.D., 717, 769Stegun, I.A., 141, 142, 153, 260, 273, 333Stepanishen, P., 580, 609Stephen, R.A., 257, 333, 534, 541, 590, 608,

633, 659Stephens, R.W.B., 452Sternberg, R.L., 453Stevenson, M., 764, 766, 767, 772Stickler, D.C., 354, 372, 453Stoll, R.D., 63, 608Sturm, F., 509–511, 519–524, 529, 641–644,

659Sturm, J., 2

TTang, D.J., 673, 703Tango, G., 234, 245, 257, 333, 616, 633, 645,

659Tao, J., 568, 609Tappert, F.D., 156, 216, 230, 231, 369, 375,

453, 457–459, 463, 465, 476, 486, 488,490, 495, 496, 510, 527, 528, 611, 659

Taroudakis, M.I., 429, 431, 454, 529Tesei, A., 566, 584, 590, 595, 598, 608, 610Thode, A.M., 443, 445, 455, 681, 686, 704,

767, 772Thomas, S.J.L., 133, 137, 153Thomson, D.J., 465, 479, 528Thomson, W.T., 234, 252, 333Thorp, W.H., 36, 63Thurston, R.N., 610Timoshenko, V.I., 10, 63Tindle, C.T., 189, 219, 231, 232, 359, 372,

379, 453, 454, 474, 516, 528, 529, 678,703

Tolstoy, A., 426, 454, 509, 529, 732, 733, 770Tolstoy, I., 22, 23, 63, 71, 143, 146, 152, 372,

453, 619, 659Tourin, A., 772Tracey, B., 291, 334Traer, J, 727, 769Trefethen, L.N., 463, 528Treitel, S., 541, 608Troiano, L., 315, 317, 335

Name Index 781

Tsang, L., 272, 333Turkel, E., 544, 608

UUrban, H.G, 10, 63, 709, 769Urick, R.J., 2, 3, 10, 22, 25–27, 29, 30, 36, 37,

54, 57, 62–64, 709, 710, 713, 769Uslenghi, P.L.E., 416, 420, 454

VVeksler, N.D., 602, 610Veljkovic, I., 580, 584, 609Vichnevetsky, R., 454Vidale, J., 160, 230Vidmar, P.J., 189, 231, 733, 770Vilmann, O., 279, 334Virovlyansky, A.L., 224, 225, 232

WWagstaff, R.A., 694, 696, 704, 728, 769Wales, S.C., 694, 696, 704Ward, R.W., 541, 608Wasow, W.R., 533, 540, 543, 607Watson, W.H., 200, 210, 231Weaver, R.L., 680, 704Weinberg, H., 174, 216, 230, 408, 426, 454Wenz, G.M., 58, 59, 64Westervelt, P.J., 10, 63Weston, D.E., 133, 137, 153Westwood, E.K., 189, 231, 372, 379, 453, 472,

508, 517, 528, 529, 749, 770Wetton, B.T.R., 470, 471, 528Whitaker, R.T., 160, 230White, D., 457, 460, 468, 481, 490, 527Whitham, G.B., 619, 659

Widrow, B., 717, 769Wilkinson, J.H., 453Williams, A.O., 337, 452Williams, N.J., 664, 703Wilson, G.A., 733, 770Wilson, G.R., 11, 63, 752–754, 771Wilson, J.D., 664, 692, 703Wilson, J.H., 694, 704Wilson, O.B., 10, 63Woodhouse, J.H., 377, 453Worcester, P.F., 3, 63, 71, 152, 725, 769Worzel, J.L., 24, 63Wright, E.B., 465, 528Wu, F., 758, 771Wunsch, C., 71, 152Wurmser, D., 52, 54, 64

YYang, T.C., 733, 770Yefet, A., 566, 567, 608

ZZakarauskas, P., 733, 770Zala, C.A., 742, 770Zampolli, M., 517, 529, 566, 580, 584, 590,

595, 598–600, 608–610Zel’dovich, B.Y., 758, 771Zeng, Y.Q., 568, 609Zeskind, R.M., 723, 769Zhang, X., 39, 64Zhang, Z.Q., 272, 334Zhou, J.-X., 39, 64, 374, 453Zienkiewicz, O.C., 547, 548, 552, 557, 558,

563, 569, 570, 608Zurk, L.M., 729, 769

Subject Index

AAcoustic lens, 512, 513, 515Adaptive beamforming

eigenvector decomposition, 721minimum variance (MV), 716, 719–720

Adaptive integration, 259, 260, 279–280, 332Adiabatic approximation, 403, 407–410,

412–413, 426, 441–443, 445Afternoon effect, 3Airy phase, 125, 150, 637, 639Aliasing, 259, 262–268, 275, 277, 323,

612–617Ambient noise, 57–60, 429, 661–703, 726,

728–729See also Noise

Ambient noise spectra, 57, 58Ambiguity

function, 732, 735, 741–743, 745–748,761, 767

surface, 735–746, 748, 749volume, 745–747, 750

Angular limitationof PEs, 466, 473, 476, 483

Arctic propagation, 7, 25, 27–28, 305–309,571, 585, 654, 655

Array gain, 705–708, 712, 714–715, 723,728–729, 752

Array processing, 705, 718, 722, 725, 733,753, 767

Artificial bottomin PEs, 472, 533

Attenuation, 130–132, 243–244, 496–497conversion of units, 37in mode solutions, 687in PE solutions, 515, 519, 524in ray solutions, 178–179in seawater, 36–38in sediments, 43, 131, 317, 494, 644in WI solutions, 243–244

loss tangent, 35, 131, 151, 243, 244perfectly matched layer, 566–568quality factor, 151, 244

BBackscattering

at rough seafloor, 281at rough sea surface, 51, 571from volume inhomogeneities, 54–57,

292–296, 316, 319Bartlett beamformer, 716, 720, 732, 738Beam

focusing, 512, 515propagation, 180, 512–516splitting, 512–514

Beamformingadaptive, 716–721, 767for vector sensor, 750–754linear (Bartlett), 715time delay, 681–683, 724–726, 763with multiple constraints, 722with white-noise constraints, 723–724

BEM: Boundary Element MethodBottom loss, 27, 29, 38–50, 102, 132, 305,

428, 429, 642, 672Bottom scattering, 54Boundary conditions

acoustic halfspace, 191, 382, 398–399, 606Dirichlet, 148, 397–398, 580, 599, 600elastic halfspace, 281, 330, 354, 367, 377,

382, 385, 399–400, 571, 587natural, 148, 542, 552, 561, 565, 566, 573,

575–577Neumann, 148, 283, 286, 398, 431, 557,

580, 599, 600Boundary Element Method (BEM)

boundary-element equations, 573–576,578, 579

783

784 Subject Index

Boundary Element Method (BEM) (cont.)boundary-integral equation, 571–573coupled domains, 576–579

Branch cutEJP, 122, 123, 125, 127, 150, 352Pekeris, 125, 127, 148, 234, 352, 354, 355

Branch line integral, 123, 125–127, 353, 354,628, 667

Branch point, 123, 132, 270, 309, 355Broadband modeling, 611–659Bulk modulus, 7, 8, 69, 151, 237, 244

CCauchy’s theorem, 106, 268, 352Caustic

cusped, 648false, 179, 211–213

Cell methodsc-linear, 206, 208–212, 226n2-linear, 206–208, 211, 226

Chapman–Harris formula, 52Characteristic equation, 121, 124, 125, 132,

150, 151, 331, 353, 361, 375–377Coherent transmission loss

for modes, 22, 340for rays, 169–173

Completeness relation, 343, 419Complex source point method, 180–182Continuous spectrum, 43, 116, 125, 127, 259,

279, 287, 297, 300, 301, 343, 349, 403,616, 662, 667, 672, 673, 676–678, 731,734

Continuous Wave (CW), 13, 114, 312, 570,611, 612, 624, 645, 650, 651, 658, 716

Contour integrationfor Pekeris waveguide, 118–133, 138, 352,

355for pressure-release waveguide, 106

Convergence gain, 22Convergence zone (CZ), 16, 20–24, 32, 62,

306, 379, 411, 413, 433, 440, 516, 526,652, 654, 738, 744, 746, 755

Convergence zone spacingin the Atlantic, 22in the Mediterranean, 21, 22

Correlation functionof noise field (NCF), 664–666, 671,

678–686, 726Coupled modes, 139, 281, 403–409, 411,

429–435, 437, 506, 507, 531, 532, 555Crank–Nicolson integration, 472

Critical angle, 25, 41, 42, 45, 50, 62, 96, 98,100, 136, 170, 230, 313–316, 355, 359,498, 512, 516, 590, 634, 672, 677, 678,760, 764

Critical depth, 16Cross-spectral density

of noise field, 664, 667, 668, 671Cross-Spectral Density Matrix (CSDM), 715,

718–721, 723, 724, 752, 753CSDM: Cross-Spectral Density MatrixCutoff frequency

in Pekeris waveguide, 125, 350, 636in pressure-release waveguide, 111in rigid-bottom waveguide, 337in shallow-water duct, 29, 30, 130in surface duct, 26

CW: Continuous WaveCycle distance, 16, 221–223, 359, 451, 738Cylindrical spreading, 14, 15, 61, 75, 113, 181,

263, 268, 355, 634, 668CZ: Convergence Zone

DDamping matrix, 562, 563, 570Decibel (dB), 13, 43, 52, 707–709, 711–713Deep scattering layer, 56Deep sound channel, 4, 5, 15, 16, 24–25, 36,

61, 440Depth-dependent Green’s function, 87–92,

296–301, 320–323, 625–628, 667–668,672

for fluid halfspace, 80, 87, 92, 93, 320for n2-linear profile layer, 177for pressure-release waveguide, 111for unbounded medium, 77

Depth excess, 16, 22Depth-separated wave equation, 85, 86, 105,

140, 143, 144, 258, 526WKB approximation, 143–145

Detection threshold, 709–713, 729DGM: Direct Global Matrix methodDifferential equations

classification of, 532–533Diffracted arrival, 648–649, 654Direct Global Matrix method (DGM), 121,

245, 251–255, 257, 258, 321, 331Direct global matrix solution, 245–251, 546Directivity index, 707–708, 711–714Dirichlet boundary condition, 148, 599Discrete methods

boundary elements, 532, 570

Subject Index 785

finite differences, 531, 532, 545, 550, 562,564, 570, 571, 580–581,585, 590,600–601

finite elements, 531, 532, 545, 550, 562,564, 570, 571, 585, 590

Discrete spectrum, 42, 43, 125, 127, 128, 132,259, 279, 297, 300, 301, 349, 662, 672,673, 676, 677, 731

Discretization error, 284, 537, 538, 542Dispersion

geometrical, 636intrinsic, 636

Dispersion curvesfor Pekeris waveguide, 636for pressure-release waveguide, 112–113for seismic interface wave, 643–647

Dispersion relation, 150, 345, 400, 452, 461Displacement potential

definition, 68–69Doppler shift

in waveguide, 623–633normal mode representation, 628–630numerical example for propagating ice

fractures, 654–657wavenumber integral representation,

625–628Double-duct profile, 21Dynamic ray tracing, 168–169, 183

EEigenfunction, 110, 338, 341, 345, 351, 357,

359, 364, 367, 374, 376, 382–384, 387,410, 428, 432, 442, 445, 446, 473–474,526, 554–556, 629, 668, 757

Eigenray, 107, 169, 170, 172, 177, 179, 187,199, 200, 213–216, 218, 220, 226, 229,617

complex, 176Eigenray finding, 187, 200, 213–216, 226

interpolation, 214–215iteration, 215the bending method, 215–216the continuation method, 216

Eigenvalue, 137, 147, 338–339, 346, 351–352,355, 360–361, 364–369, 371–379,383–387, 389, 392–394, 396, 410, 428,445, 446, 449–451, 471, 473–476,520–521, 526, 527, 544, 628, 629, 669,720, 721, 724, 757–758

Eigenvector-decomposition beamformer,720–721

Eikonal equation, 159–163, 187EJP: Ewing, Jardetzky and Press

Elasticityin mode solution, 391–402in PE solution, 470–472

Energy conservationin PEs, 506–508

Energy densityof plane wave, 12, 752

Energy leakage, 654Euler’s method, 201, 227Evanescent modes, 108, 109, 346, 349, 471,

475Evanescent spectrum, 89–91, 100, 108, 109,

127, 128, 322–323, 568, 621, 671, 676,677

Ewing, Jardetzky and Press (EJP), 122, 123,125, 127, 150, 234, 352

FFast field approximation, 260–261Fast Field Program (FFPs), 60, 216, 233, 234,

260, 265–267, 272–277, 279, 320–323,332, 469, 470, 612, 620, 621, 635

Fast Fourier Transform (FFT), 260, 265, 266,269, 272–274, 276, 277, 279, 323, 332,490, 525, 612–616, 621, 731

Fathometerpassive, 686, 726–727

FDM: Finite Difference MethodFDTD: Finite-Difference Time-DomainFEM: Finite Element MethodFermat’s principle, 195–196, 223–225FFP: Fast Field ProgramFFP integration

along real axis, 266, 267, 279with contour offset, 332

FFT: Fast Fourier TransformFigure of Merit (FOM), 712, 714Filon integration, 278–279, 332Finite Difference Method (FDM)

convergence and stability, 537–538difference approximations, 165, 363,

534–538, 606sol’n of acoustic wave equation, 538–545

Finite-difference solutionof modal equation, 362of parabolic equation, 501, 541of wave equation, 532See also IFD solution

Finite-Difference Time-Domain (FDTD), 568Finite Element Method (FEM)

coupled fluid-elastic domains, 565–566mathematical derivation, 547–551perfectly matched layer, 566–568

786 Subject Index

Finite Element Method (FEM) (cont.)sol’n of acoustic wave equation, 551–561sol’n of elastic wave equation, 562–564steady-state solution, 566time recurrence, 569–570

Flexural wave, 309, 589, 590, 657FOM: Figure of MeritFourier synthesis, 66, 72, 432, 458, 566, 569,

611–618, 621, 627, 628, 633, 638, 645,649, 652, 657, 658, 663, 680, 755, 759,762, 763

Fourier transform, 71, 75, 77, 85, 90, 91, 93,111, 116–118, 233, 258, 262, 265, 270,273, 280, 289, 290, 292, 457, 476, 479,487–489, 494, 506, 525, 541, 612–615,618–620, 624–627, 630, 666, 680, 702,705, 715, 725, 755, 762, 763, 768

Fredholm integral equation, 572Frequency integration, 244, 280–281, 627

complex, 616–617

GGalerkin’s method, 548, 549, 553–556, 558,

569Gaussian beams

in free space, 180–181tracing, 174, 181–185, 194, 212, 216, 218,

618Gauss’s theorem, 164Geoacoustic model, 38, 39, 189Geoacoustic properties, 39Geometric beams, 173–175, 184Global propagation (Perth–Bermuda),

426–429Green’s function

depth-dependent form, 85, 87–92, 105,106, 110, 115, 125, 126, 128, 148, 244,296–301, 320–323, 329, 330, 332, 554,625–628, 665, 667, 668, 672

for source in fluid halfspace, 80–81for source in unbounded medium, 77general form, 78–80, 85modal expansion, 343–344reciprocity principle, 77, 129, 130

Green’s identity, 147, 553Green’s theorem, 78–81, 85, 87, 147, 149, 413,

415, 417, 532, 544, 566, 571, 573, 582Group slowness, 135, 220, 221, 441, 442, 444Group velocity, 111–113, 125, 135, 136, 150,

345, 390–391, 438, 443, 452, 616, 629,636, 637, 639, 658

HHamiltonian

Fermat’s principle, 223–225Hankel function

asymptotic form, 74–75, 87, 99, 108, 260,273, 342, 418, 436, 459, 695–696,701–702

Hankel transform, 86–88, 90, 98–99, 110, 119,122, 123, 127, 217, 219, 234, 236, 239,243, 257–258, 260, 264, 272–277, 283,285, 286, 292, 296, 323, 619–620, 625,628

Head wave, 100, 316–319, 616, 633–635, 658Helmholtz equation, 74–78, 240, 341, 425,

533, 625depth-separated form, 216derivation of, 75, 217, 458, 459, 465, 467,

469, 663–664for point source, 89, 282in Cartesian coordinates, 73, 84, 86,

158–159, 624in cylindrical coordinates, 74, 86, 235–236,

283, 408, 458, 541, 619–620in spherical coordinates, 74–75, 84integral transform solution, 86–87, 131,

235–236sol’n for deep-ocean waveguide, 140sol’n for fluid halfspace, 86sol’n for homogeneous medium, 71–83,

89, 181, 476, 482, 500, 541, 606sol’n for Pekeris waveguide, 102–103sol’n for pressure-release waveguide, 527sol’n for rigid-bottom waveguide, 102–103sol’n for two fluid halfspaces, 94

Helmholtz–Kirchhoff integral theorem, 592,595, 604

Hooke’s law, 69, 237, 241–242, 564, 599Horizontal refraction, 423–427, 510, 512,

519–520, 522, 641, 643, 644

IIFD: Implicit Finite DifferenceIFD solution of parabolic equation, 505

error analysis, 503–505numerical implementation, 505

Image method, 81, 103–105, 107, 109Impedance

acoustic, 13, 14, 81–82, 544Implicit Finite Difference (IFD), 497, 498,

500–503, 505, 506Incoherent transmission loss

for modes, 340–341for rays, 172

Subject Index 787

Index of refraction, 3, 141, 162, 175, 223complex, 493, 497, 505effective, 426, 495

Integral representationsfor homogeneous elastic layer, 239–242for homogeneous fluid layer, 235, 237–238for n2-linear fluid layer, 238–239

Integral transformsfor axisymmetric problems, 291for plane problems, 84–85

Intensityof plane wave, 12–15, 52

Intensity striation, 134, 136Interface wave, 43, 91, 127, 248, 255, 287,

297, 474, 633, 643–647, 674–675, 677,678

Interferenceconstructive, 19, 43, 81, 82, 171, 450, 451,

482, 629, 766destructive, 19, 81, 171, 175, 482modal, 113–114, 127, 132, 137, 297, 522,

641Intromission angle, 42, 62, 98Invariant

of waveguide propagation, 133Invariant embedding solution, 255Inverse iteration, 365, 367–368, 379–380, 446

JJacobian, 166–167

relation to phase, 227–228

KKMAH index, 178, 205–206

LLagrange extrapolation, 542Lambert’s law, 54Lame constants, 240, 243, 326, 329, 392, 471,

562, 599Lateral wave, 100, 309, 316Leaky precursor, 654Limiting ray, 175, 176, 178Linear beamforming, 714–716Lloyd mirror pattern, 17–20, 81, 91, 93, 171,

172, 359, 479, 483Loss tangent, 35, 131, 151, 243–244

MMass matrix, 562, 563, 570Matched Field Processing (MFP), 260, 276,

329, 705, 730–750, 761, 763, 767,769

numerical examples, 329, 732MCM: Multiple Constraints MatchingMDL: Minimum Detectable LevelMethod of canonical problems, 72, 167Method of characteristics, 160MFP: Matched Field ProcessingMinimum Detectable Level (MDL), 728–730Minimum-variance beamformer, 716, 717,

720, 722, 732, 735, 738Minimum Variance Distorsionless Response

(MVDR), 716, 719, 723, 724Minimum Variance processor (MV), 716, 717,

719–724, 732, 734–741, 743–746, 748,749

Mixed layer, 3–4, 25, 26, 375Modal dispersion

in Pekeris waveguide, 125–126in pressure-release waveguide, 111, 114

Modal equationin cylindrical geometry, 338, 339in plane geometry, 341

Modal excitation, 108, 123, 127, 150, 631Modal interference, 113–114, 127, 132, 137,

267, 297, 522, 641Modal losses

as a perturbation, 385–389Mode coupling, 408, 413, 429–438, 531–532

adiabatic approximation, 409–410around seamount (3-D), 429–438single-scatter approximation, 408, 531–532

Mode cycle distance, 359, 451, 738Mode expansion

for Pekeris waveguide, 118–121for pressure-release waveguide, 111, 124for rigid-bottom waveguide, 337of the Green’s function, 291–292, 343–344

Mode solution for 3-D environment, 423–438horizontal refraction equations, 423–426

Mode solution for elastic medium, 241,243–244, 282, 306, 308, 325–326, 329,391–402, 458, 470, 474, 541, 543, 544,561, 565, 598–599, 671

Mode solution for range-dependentenvironment

adiabatic modes, 407–412coupled modes, 403–409, 411, 412one-way coupled modes, 407–408, 411

Mode theoryfor 3-D environment, 662, 745

788 Subject Index

Mode theory (cont.)for range-dependent environment, 402–413mathematical derivation, 158–175, 688

Modesbottom-bounce, 359, 360evanescent, 108–109, 346, 349, 471, 475for Munk profile, 213, 358, 360, 375, 437,

449in Pekeris waveguide, 124–126, 151, 297,

355, 356, 527in pressure-release waveguide, 527in rigid-bottom waveguide, 337leaky, 127, 309, 354, 355, 359, 360, 376,

378, 399, 404normalization of, 371–373orthogonality of, 110, 768propagating, 108, 110–111, 113, 114, 127,

266, 346, 347, 367, 382, 419, 445, 468,516, 519, 522, 527, 641, 650

virtual, 122, 125, 297, 300, 652waterborne, 300, 359, 360, 677

Multiple-constraints beamforming, 722Multiple Constraints Matching (MCM), 716,

717, 722, 732, 733, 735–741, 743–746Munk profile, 212, 213, 356–360, 374, 437,

439, 449, 450MV: Minimum Variance processorMVDR: Minimum Variance Distorsionless

Response

NNCF: Noise Correlation FunctionNeumann boundary condition, 148, 557, 580,

599, 600Neumann stability analysis, 540Newmark’s method, 541, 569, 570Noise

in continental-slope environment, 691, 693,698

in elastic waveguide, 674–678in fluid waveguide, 671–674in Gulf Stream environment, 742–743in homogeneous halfspace, 669–671in seamount environment, 691–692

Noise Correlation Function (NCF), 665,678–686, 726

Noise fieldsextracting time-domain Green’s functions,

678–686intensity distribution, 691, 692spatial correlation, 661, 662, 673, 674, 757vertical directivity, 672, 681, 693, 694,

696, 698, 734, 735

Noise modeling in 3-D oceansadiabatic-mode representation, 687–690parabolic equation representation, 694

Noise modeling in stratified oceansmathematical derivation, 663–665normal-mode representation, 667–669wavenumber-integral representation,

666–667Noise notch, 673, 694, 696Noise sources

correlation function, 664–666, 671, 680,682, 683

normalized strength, 663, 667, 668, 670,733

Noise spectrain elastic waveguide, 674–678

Normal modesSee Modes

Normalizationof modes, 371–373

Numerical discretizationEuler’s method, 201Runge–Kutta method, 200–202

Numerical evaluation of frequency integral,259, 280, 287, 612

Numerical evaluation of wavenumber integral,106, 262–263, 268, 284, 320

wavenumber discretization, 262–265Numerical examples of FD/FE solutions

scattering by Arctic ice features, 584–590Numerical examples of MFP solutions

Arctic environment, 738, 7423-D environment, 741–750shallow-water environment, 734–738

Numerical examples of mode solutionsglobal propagation (3-D), 226, 426–429Gulf Stream problem (3-D), 423–425ideal, rigid-bottom waveguide, 337Munk profile, 358, 360, 375, 437, 450propagation around seamount (3-D),

429–438warm-core eddy, 411–413

Numerical examples of noise modelingcontinental-slope environment, 6913-D Gulf Stream environment, 423, 424,

691, 7473-D seamount environment, 431, 691elastic waveguide, 674–678fluid waveguide, 671–674

Numerical examples of PE solutions, 458, 471,474, 483, 484, 487, 506, 524

beam propagation, 512–515propagation in 2-D wedge, 515–516

Subject Index 789

propagation in 3-D wedge, 517–523propagation over elastic bottom, 472, 517,

518propagation over seamount, 514, 516propagation through ocean front, 32–33

Numerical examples of pulse solutionsacoustic emission from ice fractures,

654–657Arctic propagation, 305–309deep-water propagation, 647–651mode dispersion in waveguide, 636–640propagation in 3-D wedge, 640–643seismic interface waves, 643–647surface-duct propagation with leakage,

652–654the head-wave problem, 633–635

Numerical examples of ray solutionsBalearic Sea, 162, 171Dickins seamount, 192global propagation (3-D), 226, 426–429parabolic bathymetry profile, 195See also Ray diagram

Numerical examples of WI solutionsArctic propagation, 305–309beam propagation, 317bottom reverberation, 571Bucker waveguide, 300–301elastic-bottom waveguide, 297–300scattering from volume inhomogeneities,

292–296target scattering, 260, 285–287, 313–314

Numerical Hankel transform, 243, 258, 272Numerical solution of depth-separated wave

equation, 85–88, 105, 140, 141, 143,144, 233, 236, 238, 526, 546, 554, 606,625

direct global matrix approach, 245–251invariant embedding approach, 255–257propagator matrix approach, 251–255

Numerical solution of frequency integral, 244,280–281, 612–613

Numerical solution of modal equationfinite-difference methods (FDMs),

361–372, 376, 534–545inverse iteration, 367–368layer methods, 372–373root finders, 375–379shooting methods, 373–375Sturm sequences, 446

Numerical solution of PEsFD/FE methods, 60, 487, 497split-step Fourier method, 465, 487–490,

523

Numerical solution of ray equations, 156, 202,214, 227

c-linear cell method, 200, 206, 208–212,226, 318

direct integration, 200–206, 211, 226n2-linear cell method, 146, 176, 177,

206–208, 211, 213, 226Numerical solution of wavenumber integral

adaptive integration, 259, 279–280, 332Fast Hankel transform, 264, 272–277FFP integration, 234, 261, 266, 267, 269,

279, 332Filon integration, 278–279, 332trapezoidal rule integration, 269, 276–279,

332Numerical stability

of direct global matrix (DGM) solution,121, 234, 248–251

of invariant embedding solution, 234, 256of propagator matrix solution, 234,

253–254Numerov’s method, 369, 450Nyquist sampling criterion, 266, 615, 616

OOptimum frequency

in the Arctic, 27in shallow water, 30–32, 388, 389

Orthogonality of modes, 110, 339, 386, 432,768

PParabolic equation (PE), 457–527

angular limitation of, 466–470energy conservation in, 506–508, 527for 3-D environment, 510, 691, 741,

744–748for elastic medium, 7, 470–472, 474–476,

517, 518for horizontal interface, 498–500for variable density, 495–496generalized form, 460–462generalized high-angle form, 464in the time domain, 621–623mathematical derivation, 458–470nonlinear, 622, 623phase error in, 466–470standard form, 458–459starting fields, 472–486wide-angle form, 463–466, 468, 470, 479,

482, 483, 486, 498, 503, 506, 511

790 Subject Index

Paraxial approximation, 66, 459, 462, 623Partial discretization

in finite-element method, 551PC: Phase ConjugationPE: Parabolic EquationPE forms

Claerbout, 463, 464, 466–469, 481–483,486, 498, 501, 506, 507, 511, 512

elastic, 470–472Greene, 463, 468, 469, 471, 479–483, 485,

486, 494, 498, 501, 516LOGPE, 465Pade, 464Tappert, 459, 462Thomson–Chapman, 482, 483, 485, 506,

507, 516PE starting fields

Gaussian source, 476–486, 494, 523generalized Gaussian source, 480–482Greene’s source, 479, 481, 483, 485,

486modal starter, 469, 473–474self starter, 474–476, 486spectral properties of, 479, 482–486Thomson’s source, 479, 480, 483, 486

Pekeris waveguide, 103, 110, 118–133, 137,138, 148, 150, 151, 222, 261, 263, 264,266, 267, 270, 271, 275, 297, 331, 332,349, 350, 352, 354–356, 372, 379,420–422, 438, 439, 445, 449, 468, 527,606, 607, 636, 637, 639–641, 764

modal solution, 124Perfectly Matched Layer (PML), 566–568,

579, 591–597Phase Conjugation (PC), 758–767Phase error

in PEs, 463, 465–470, 483, 486, 506Phase shift

at caustic, 177–178upon reflection, 41, 42, 96–98

Phase slowness, 136, 221, 442, 444Phase velocity, 111, 135, 303, 345, 438, 443,

444, 467, 520, 629, 636, 657Plane wave

energy density, 11–12intensity, 12

Plane-wave beamformingadaptive, 716–721linear (Bartlett), 714–716with multiple constraints, 722with white-noise constraints, 723–724

PML: Perfectly Matched LayerPoisson sum, 116–118

Pressure-release waveguide, 111, 124, 127,527, 682

mode solution, 107–110ray solution, 103

Probability of detection, 709, 710, 712, 713,729, 754

Propagationcharacteristic paths, 15–17convergence zone, 20–24, 306, 359–360,

411, 516, 652–654in 2-D wedge, 515–516in 3-D wedge, 517–523, 640–643in deep sound channel, 15, 16, 24–25, 36in deep water, 17–28, 495, 647–651in the Arctic, 25, 27–28, 305–309, 571in shallow water, 9, 28–32, 42, 45, 341,

388, 486in surface duct, 15, 25–26, 306, 379,

652–654over elastic bottom, 472, 512, 517, 518over seamount, 32–35, 192, 516through ocean front, 32–33

Propagator matrix solution, 234, 251–256, 321Prufer transforms, 380–385Pulse modeling in time domain

parabolic equations, 621–623ray methods, 617–619spectral integral techniques, 619–621

Pulse modeling via frequency domainFourier synthesis, 612–617

QQR algorithm, 365Quality factor, 151, 244

RRadiation condition, 70, 76, 78, 79, 88, 91, 94,

106, 107, 119, 122, 142, 148, 243, 253,340, 342, 343, 406, 432, 472, 493, 505,533, 537, 539, 542–545, 561, 566,570–572, 579, 581, 599

Range dependence, 32, 60, 149, 208, 379, 411,412, 460, 461, 467, 494

Raydisplacement, 99, 100, 189loop length, 22, 23, 359trajectory, 158, 160, 161, 214, 215, 220,

223, 316travel time, 160, 168, 173, 195, 199, 428turning point, 100, 145, 146, 315, 358, 359,

648, 651, 652, 725See also Eigenray

Subject Index 791

Ray anomalycaustic, 175–178false caustic, 211–213shadow zone, 175–178

Ray caustic, 522, 641Ray coordinates, 160, 163, 218Ray diagram

for Arctic profile, 27for convergence-zone propagation, 21for Dickins seamount, 192, 193for leaky surface duct, 652, 653for Mediterranean profile, 162for Munk profile, 212, 651for n2-linear profile, 176, 177, 179, 184for parabolic bathymetry profile, 195for propagation over seamount, 34for propagation through ocean front, 33for shallow-water propagation, 29for SOFAR-channel propagation, 24for surface-duct propagation, 26

Ray equationsnumerical solution, 200–216r(z)-form, 186–187s-form, 160–161� -form, 185–186for stratified media, 196–198z(r)-form, 186–187

Ray expansionfor pressure-release waveguide, 103, 104

Ray family, 648, 651Ray invariant, 220–224Ray series, 159, 619Ray theory

in 3-D, 225–226in the time domain, 617–619mathematical derivation, 158–175region of validity, 178–179sol’n of eikonal equation, 160–163sol’n of transport equation, 163–165via the WKB approximation, 232, 219–220

Ray tube, 164–166, 168, 174, 175, 226, 617,618

Ray typesrefracted bottom-reflected (RBR), 17, 28,

162refracted refracted (RR), 17, 162refracted surface-reflected (RSR), 18, 173,

17, 162surface-reflected bottom-reflected (SRBR),

17, 162Rayleigh reflection coefficient, 41, 43Rayleigh roughness parameter, 51Rayleigh wave, 392, 400, 644Ray–mode analogy, 118, 359, 450, 520, 738

Real Fast Fourier Transform (RFFT), 615, 617Receiver Operating Characteristic (ROC),

709–711, 713Recipe

for mode code, 445–446for PE code, 523–525for ray code, 226–227for WI/FFP code, 320–323

Reciprocityof displacement potentials, 128–130of Green’s function, 77–78, 147–149, 321,

626, 763, 764in layered media, 128–130of parabolic equations, 465in Pekeris waveguide, 128–129, 148of ray solution, 147, 199–200

Reciprocity principle, 77, 129–130, 147–148,465

for Green’s function, 77, 129, 147–148Recognition differential, 712Reflection

at fluid–fluid interface, 40–43, 96, 157,189, 515

at fluid–solid interface, 43–45at half-wavelength layer, 48–50at layered fluid halfspace, 39, 45–50at layered solid halfspace, 39, 97, 98, 102at quarter-wavelength layer, 48at rough interface, 51

Reflection examplesdifferent bottom types, 39, 46, 101hard bottom, 95–97, 100, 101soft bottom, 95, 97–98

Reflectivity method, 233, 256, 257Reflectivity zone, 256, 257Refraction index, 141, 162, 175, 185, 187,

206, 207, 223, 238, 373, 426, 458, 462,465, 466, 487, 490, 491, 493–495, 497,505, 510, 524–526

Reverberationfrom rough interfaces, 139, 281, 287–292,

531RFFT: Real Fast Fourier TransformRichardson extrapolation, 365, 368–369, 446,

450Riemann sheet, 122Rigid-bottom waveguide, 124, 337

modal solution, 385, 449Robin boundary condition, 398–399ROC curves, 709–711, 713ROC: Receiver Operating CharacteristicRoot-finding

analytic estimates, 378bisection, 367, 375–377

792 Subject Index

Root-finding (cont.)brute-force search, 377complex plane, 378–379continuation methods, 378deflation, 377

Roughness perturbation operators, 324–328Rounding error, 538Runge–Kutta method, 200–202, 226, 374

SScattering, 51–55, 413–422, 590–597

at the seafloor, 16–17, 28, 51, 281at the sea surface, 51–53, 57by Arctic ice features, 584–590Chapman–Harris formula, 52column strength, 55farfield computations, 604–605from buried spherical shell, 594–595from half-buried spherical shell, 595, 597from object buried in the seabed, 309from object in waveguide, 413–422from rippled seabed, 310–313from rough interfaces, 290from spherical shell on the seabed,

590–594from volume inhomogeneities, 292–296Lambert’s law, 54virtual source concept (VSC), 579–584

Scattering cross section, 51–52, 315, 316Scattering strength, 52–56, 296Scholte wave, 297, 299, 300, 330, 400, 402,

644, 645Secular equation, 353, 396Sediment stratification, 8Seismic interface wave, 91, 127, 248, 287,

297, 643–647, 674Seismic profile, 9Seismogram, 219, 257, 309, 310, 648, 649Semicoherent transmission loss

for rays, 172–173Separation of variables, 1, 84, 138–140, 281,

338, 349, 466Shadow zone, 25, 175–178, 183, 184, 359,

413, 433–434, 520, 522, 641Shallow water propagation, 9, 28, 42, 45, 388

variability of, 29Shape function

in finite-element method, 547, 550, 551Shear speed

in sediments, 38, 39, 45, 298, 401, 645Signal excess, 712, 728

Signal processing, 11, 57, 139, 260, 678, 694,711, 712, 716, 722, 725, 727, 733, 746,750, 759, 766, 767

Simulation and stimulation, 754Slowness, 135, 136, 220–224, 440–442, 444Snapshot, 379, 586, 612, 633–635, 640, 718,

752, 755–757Snell’s law, 15, 16, 22, 25, 40–43, 47–48, 144,

154, 157, 158, 198–199, 210, 220–221,223, 224, 283, 440, 441, 462, 515, 725,726

SOFAR channel, 24, 61, 155Sommerfeld–Weyl integral, 88–89, 286, 669Sonar equation

active, 705, 713–714passive, 705, 711–713

Sonar performance prediction, 172, 486–487,711

Sound propagation models, 50, 60–61Sound speed

generic profiles, 4, 672in seawater, 13, 15, 37in seawater w. bubbles, 7, 8in sediments, 39, 42, 391microstructure variability, 6–7Munk profile, 357, 375, 433, 434, 437, 438n2-linear profile, 176, 177profile interpolation, 140–141, 190, 202pseudo-linear profile, 141, 302

Source strengthof point source, 76, 82, 238of surface noise source, 663

Spectral domainscontinuous, 127discrete, 127evanescent, 96, 97, 110, 127for elastic waveguide, 127for Pekeris waveguide, 127radiating, 96, 127

Spectrum level, 13, 734Spherical spreading, 14, 15, 19, 22, 29, 75,

276, 639, 648Split-step Fourier algorithm, 465, 487–490,

523error analysis, 488, 502–505numerical implementation, 505

Spreading loss, 130, 512cylindrical, 14, 15, 75, 113spherical, 14, 75

Square-root operator, 461–465, 467–468, 501Feit–Fleck splitting, 465Pade expansion, 464, 471, 487, 510rational-linear expansion, 463Taylor expansion, 461, 510, 511

Subject Index 793

Stability error, 538Starting fields

See PE starting fieldsStationary phase method, 99, 107, 112, 219,

220Stiffness matrix, 562–564, 580–582, 595, 607Stoneley wave, 297, 644Striation, 134, 136Sturm–Liouville problem, 137, 338, 339, 349,

353, 360–361, 365, 380, 445, 554,667–668

Sturm sequence, 365–367, 371, 375–376, 446Sturm’s method, 365–367Surface duct, 3, 4, 15, 25–26, 32, 33, 179, 306,

379, 411, 652–654, 671, 674, 738, 741Surface scattering, 25, 27, 52–53, 57,

388–389, 458, 531

TTarget scattering

in waveguide, 285, 416–417Target strength, 285, 592, 593, 596, 597, 713Taylor series, 201, 362, 363, 461, 462,

490–491, 499, 504, 510, 511, 535, 537TDGF: Time Domain Green’s FunctionTime-delay beamforming, 724–726Time Domain Green’s Function (TDGF),

104–105, 613, 678–686Time recurrence

in finite-element method, 569–570Time reversal, 758–767Time Reversal Mirror (TRM), 758–764, 766Topology matrix, 248, 559–560, 607Transducers, 10–11, 441, 444Transmission loss, 14–15, 30, 81–83

coherent, 169–171, 341incoherent, 171–172, 340–341, 388, 389semicoherent, 172–173

Transport equation, 159, 163–167, 218, 227Trapezoidal rule integration, 269, 276–279,

332Trial function, 575–577, 606

in finite-element method, 546, 548, 550,551, 553–556, 558, 562, 569

TRM: Time Reversal MirrorTunneling

See Energy leakageTurning point, 100, 145–147, 210, 218,

220–221, 315, 358, 359, 374, 375, 441,648, 651, 652, 725–726

UUnits, 11, 13, 35, 37, 39, 56, 496, 728, 751

VVariational formulation

for fluid-elastic domains, 598–604Variational principle, 532, 545, 547, 549Vector sensor, 10, 11, 402

beamforming, 750–754Velocity potential

definition, 68, 413Virtual Source Concept (VSC), 579–584Volume scattering, 15, 51–57, 315, 713VSC: Virtual Source Concept

WWave equation

depth-separated form, 85–88, 105, 140,141, 143, 144, 233, 236, 238, 245, 251,252, 258–259, 318, 324, 546, 554, 625

derivation, 83, 623for displacement potential, 68–69for particle velocity, 67–68for pressure, 67–69for velocity potential, 68in frequency domain, 71, 86in time domain, 77, 458, 536, 537, 551,

621–623linear, 67–71, 83, 569nonlinear, 66one-way form, 460, 462, 465, 466, 472,

479, 483, 500, 508, 621solution of, 60, 69–72, 75, 83, 86, 175,

233, 245, 251–252, 258, 259, 458, 546,547, 551, 612, 732, 733

See also Helmholtz equationWavefront, 90, 109, 156, 158, 160, 166, 189,

191, 358, 450, 561, 634, 681–682, 716,760

extracted from noise data, 682–686Waveguide

deep ocean, 139–147ideal, pressure release, 527ideal, rigid bottom, 337Pekeris, 103, 110, 118–133, 137, 138, 148,

150, 261, 271, 275, 297, 331, 332, 349,350, 352, 354–356, 372, 379, 420–422,438, 439, 445, 468, 636, 637, 639–641,764

Waveguide invariant, 133–139, 438–445,440–441

for isovelocity waveguide, 118, 222, 674

794 Subject Index

Waveguide invariant (cont.)for n2-linear refracting waveguide, 223for range-dependent environments,

443–445ray representation, 220–223

Waveguide propagation, 118, 133, 140,416–417, 579, 590, 600, 613

continuous spectrum, 42–43, 662discrete spectrum, 42–43, 662evanescent spectrum, 89–91, 108, 109, 662radiating spectrum, 89–91, 93

Wavenumber Integration (WI), 107, 110, 122,126, 233–332, 407, 420, 473, 531, 571,580, 582, 584, 588, 590, 619–620, 624,628, 633, 661, 665, 735

aliasing in, 262–265along real axis, 266–268, 270, 277in 3-D, 282–284integral representations for homogeneous

elastic layer, 240–242

integral representations for homogeneousfluid layer, 237–238

integral representations for n2-linear fluidlayer, 238–239

in the time domain, 619–620mathematical derivation, 235–244with contour offset, 270

Weak interfaces, 194–195Weighted residuals method, 547–548

Galerkin’s approach, 548White-Noise constraint (WN) processor,

723–724WI: Wavenumber IntegrationWKB approximation

to depth-separated wave equation, 144to modal eigenfunctions, 357–358

WKB: Wenzel, Kramers and BrillouinWrap-around, 244, 259, 262, 265, 266, 268,

270–271, 275, 276, 614, 617Wronskian, 199–200, 351–353, 447, 449