Asymptotics of frequency of a surface wave trapped by a slightly inclined barrier in a liquid layer

J. H. Videman   ; V. Chiado' Piat   ; S. A. Nazarov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 41, Tome 393 (2011), pp. 46-79

Résumé

We consider the two-dimensional formulation of the problem on an oblique surface wave for an obstacle in the shape of a submerged strip-barrier. If the barrier is vertical, the discrete spectrum of the problem is empty, but for an inclined barrier there appear an eigenvalue below the threshold of the continuous spectrum and the corresponding trapped mode which decays exponentially in the direction, perpendicular to the obstacle. The asymptotics of the eigenvalue is found in the case of a small inclination angle.

Bibliographie

[1] N. Kuznetsov, V. Maz'ya, B. Vainberg, Linear Water Waves, Cambridge University Press, Cambridge, 2002 | MR | Zbl

[2] O. A. Ladyzhenskaya, Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973 | MR

[3] T. H. Havelock, “Forced surface waves on water”, Phil. Mag., 8 (1929), 569–576 | Zbl

[4] S. A. Nazarov, “Prostoi sposob obnaruzheniya lovushechnykh mod v zadachakh lineinoi teorii poverkhnostnykh voln”, Dokl. RAN, 429:6 (2009), 746–749 | MR | Zbl

[5] S. A. Nazarov, “Dostatochnye usloviya poyavleniya lovushechnykh mod v zadachakh lineinoi teorii poverkhnostnykh voln”, Zap. nauchn. semin. POMI RAN, 369, 2009, 202–223 | MR

[6] M. Sh. Birman, M. Z. Solomyak, Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, Izd-vo Leningr. un-ta, L., 1980 | MR

[7] F. Ursell, “Trapping modes in the theory of surface waves”, Proc. Camb. Phil. Soc., 47 (1951), 347–358 | DOI | MR | Zbl

[8] F. Ursell, “Mathematical aspects of trapping modes in the theory of surface waves”, J. Fluid Mech., 183 (1987), 421–437 | DOI | MR | Zbl

[9] R. M. Garipov, “On the linear theory of gravity waves: the theorem of existence and uniqueness”, Arch. Rat. Mech. Anal., 24 (1967), 352–362 | DOI | MR | Zbl

[10] S. A. Nazarov, “Kontsentratsiya lovushechnykh mod v zadachakh lineinoi teorii voln na poverkhnosti zhidkosti”, Matem. sb., 199:12 (2008), 53–78 | DOI | MR | Zbl

[11] F. Ursell, “The effect of a fixed vertical barrier on surface waves in deep water”, Proc. Camb. Phil. Soc., 43 (1947), 374–382 | DOI | MR | Zbl

[12] F. John, “Waves in the presence of an inclined barrier”, Comm. Pure Appl. Math., 1 (1948), 149–200 | DOI | MR | Zbl

[13] D. V. Evans, “Diffraction of water-waves by a submerged vertical plate”, J. Fluid Mech., 40 (1970), 433–451 | DOI | Zbl

[14] P. McIver, “Scattering of water waves by two surface-piercing vertical barriers”, IMA J. Appl. Math., 35 (1985), 339–355 | DOI | MR | Zbl

[15] D. C. Shaw, “Perturbational results for diffraction of water-waves by nearly-vertical barriers”, IMA J. Appl. Math., 34 (1985), 99–117 | DOI | MR | Zbl

[16] X. Yu, A. T. Chwang, “Analysis of wave scattering by submerged circular disk”, J. Eng. Mech., 119 (1993), 1804–1817 | DOI

[17] N. F. Parsons, P. A. Martin, “Scattering of water waves by submerged curved plates and by surface-piercing flat plates”, Appl. Ocean Res., 16 (1994), 129–139 | DOI

[18] L. Farina, P. A. Martin, “Scattering of water waves by a submerged disc using a hypersingular integral equation”, Appl. Ocean Res., 20 (1998), 121–134 | DOI

[19] M. D. Groves, “On the existence of trapped modes in channels of arbitrary cross-sections”, Math. Meth. Appl. Sci., 20 (1997), 521–545 | 3.0.CO;2-2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[20] P. A. Martin, N. F. Parsons, L. Farina, “Interaction of water waves with thin plates”, Mathematical Techniques for Water Waves, Computational Mechanics Publications, ed. B. N. Mandal, 1997, 197–229 | MR | Zbl

[21] C. M. Linton, D. V. Evans, “Trapped modes above a submerged horizontal plate”, Q. J. Mech. Appl. Math., 4 (1991), 487–506 | DOI | MR

[22] N. F. Parsons, P. A. Martin, “Trapping of water waves by submerged plates using hyper-singular integral equations”, J. Fluid Mech., 284 (1995), 359–375 | DOI | MR | Zbl

[23] C. M. Linton, N. G. Kuznetsov, “Non-uniqueness in two-dimensional water wave problems: numerical evidence and geometrical restrictions”, Proc. R. Soc. Lond. A, 453 (1997), 2437–2460 | DOI | MR | Zbl

[24] N. Kuznetsov, P. McIver, C. M. Linton, “On uniqueness and trapped modes in the water-wave problem for vertical barriers”, Wave Motion, 33 (2001), 283–307 | DOI | MR | Zbl

[25] I. V. Kamotskii, S. A. Nazarov, “Eksponentsialno zatukhayuschie resheniya zadachi o difraktsii na zhestkoi periodicheskoi reshetke”, Matem. zametki, 73:1 (2003), 138–140 | DOI | MR | Zbl

[26] S. A. Nazarov, “Variatsionnyi i asimptoticheskii metody poiska sobstvennykh chisel pod porogom nepreryvnogo spektra”, Sibirsk. matem. zhurnal, 51:5 (2010), 1086–1101 | MR | Zbl

[27] S. A. Nazarov, J. H. Videman, “Existence of edge waves along three-dimensional periodic structures”, J. of Fluid Mech., 659 (2010), 225–246 | DOI | MR | Zbl

[28] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | Zbl

[29] J. Sanchez-Hubert, E. Sanchez-Palencia, Vibration and Coupling of Continuous Systems. Asymptotic Methods, Springer Verlag, Heidelberg, 1989 | MR | Zbl

[30] W. G. Mazja, S. A. Nasarow, B. A. Plamenewski, Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten 1, Akademie-Verlag, Berlin, 1991 | MR

[31] W. Bulla, F. Gesztesy, W. Renrer, B. Simon, “Weakly coupled bound states in quantum waveguides”, Proc. Amer. Math. Soc., 125:8 (1997), 1487–1495 | DOI | MR | Zbl

[32] P. Exner, S. A. Vugalter, “Bound-states in a locally deformed waveguide: the critical case”, Lett. Math. Phys., 39:1 (1997), 59–68 | DOI | MR | Zbl

[33] D. Borisov, P. Exner, R. Gagyl'shin, D. Krejčiřik, “Bound states in weakly deformed strips and layers”, Ann. H. Poincaré, 2 (2001), 553–571 | DOI | MR

[34] V. P. Maslov, “Asimptotika sobstvennykh funktsii uravneniya $\Delta u+k^2u=0$ k kraevymi usloviyami na ekvidistantnykh krivykh i raseyanie elektromagnitnykh voln v volnovode”, Dokl. AN SSSR, 123:4 (1958), 631–633 | Zbl

[35] P. Duclos, P. Exner, “Curvature-induced bound sttes in quantum waveguides in two and three dimensions”, Review Math. Phys., 7:1 (1995), 73–102 | DOI | MR | Zbl

[36] Y. Avishai, D. Bessis, B. G. Giraud, G. Mantica, “Quamtum bound states in open geometries”, Physical Review B, 44:15 (1991), 8028–8034 | DOI

[37] V. V. Grushin, “O sobstvennykh znacheniyakh finitno vozmuschennogo operatora Laplasa v beskonechnykh tsilindricheskikh oblastyakh”, Matem. zametki, 75:3 (2004), 360–371 | DOI | MR | Zbl

[38] R. R. Gadylshin, “O lokalnykh vozmuscheniyakh kvantovykh volnovodov”, Teor. i matem. fizika, 145:3 (2005), 358–371 | DOI | MR | Zbl

[39] A. M. Ilin, Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989 | MR

[40] V. A. Kondratev, “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi ili uglovymi tochkami”, Trudy Moskovsk. matem. obschestva, 16, 1963, 219–292

[41] V. G. Mazya, B. A. Plamenevskii, “O koeffitsientakh v asimptotike reshenii ellipticheskikh kraevykh zadach v oblasti s konicheskimi tochkami”, Math. Nachr., 76 (1977), 29–60 | DOI | Zbl

[42] V. G. Mazya, B. A. Plamenevskii, “Otsenki v $L_p$ i v klassakh Geldera i printsip maksimuma Miranda–Agmona dlya reshenii ellipticheskikh kraevykh zadach v oblastyakh s osobymi tochkami na granitse”, Math. Nachr., 81 (1978), 25–82 | DOI | Zbl

[43] S. A. Nazarov, “Polinomialnoe svoistvo samosopryazhennykh ellipticheskikh kraevykh zadach i algebraicheskoe opisanie ikh atributov”, Uspekhi matem. nauk, 54:5 (1999), 77–142 | DOI | MR | Zbl

[44] S. A. Nazarov, B. A. Plamenevskii, Ellipticheskie zadachi v oblastyakh s kusochno gladkoi granitsei, Nauka, M., 1991

[45] S. A. Nazarov, “Properties of spectra of boundary value problems in cylindrical and quasicylindrical domains”, Sobolev Spaces in Mathematics, v. II, International Mathematical Series, 9, ed. Maz'ya V., Springer, New York, 2008, 261–309 | DOI | MR

[46] M. Van-Daik, Metody vozmuschenii v mekhanike zhidkosti, Mir, M., 1967

[47] J. Hadamard, “Mémoire sur le problème d'analyse relatif à l'éuilibre des plaques élastiques encastrées”, ØE uvres, 2 (1968), 515–631

[48] S. A. Nazarov, “Stsenarii kvazistaticheskogo rosta treschin pri slabom iskrivlenii i izlome”, Prikladnaya matem. i mekhanika, 72:3 (2008), 507–525 | MR