Tsunami waveguides

S. V. Sukhinin ; V. S. Yurkovskiǐ

Sibirskij žurnal industrialʹnoj matematiki, Tome 14 (2011) no. 4, pp. 111-124.

Résumé

We show that the submerged ridges that are periodic (in one direction), chains of submerged mountains and islands possess waveguide properties for long waves. We state a boundary value problem whose generalized eigensolutions describe the generalized eigenwaves on the fluid surface localized near the structure. We show the existence of generalized eigensolutions and eigenwaves localized near a periodic structure above or below the surface. We prove that in a neighborhood of zero there exists a pass band. We study resonance phenomena near periodic in one direction tsunami waveguides and show that two types of resonance phenomena may occur in these problems: spatially localized and synchrophasotronic.

Détail

Keywords: waveguide, periodic structures, spectral properties of the Laplacian
Mots-clés : tsunami.

Bibliographie

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

[2] Jones D. S., “The eigenvalues of $\Delta u+\lambda u=0$ when the boundary condition are given on semi infinite domains”, Proc. Cambridge Phil. Soc., 49 (1953), 668–684 | DOI | MR | Zbl

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

[4] Garipov R. M., “Neustanovivshiesya volny nad podvodnym khrebtom”, Dokl. AN SSSR, 161:3 (1965), 547–550 | Zbl

[5] Lavrentev M. A., Shabat B. V., Problemy gidrodinamiki i ikh matematicheskie modeli, Nauka, M., 1973 | MR

[6] Bichenkov E. I., Garipov R. M., “Rasprostranenie voln na poverkhnosti tyazheloi zhidkosti v basseine s nerovnym dnom”, Prikl. mekhanika i tekhn. fizika, 1969, no. 2, 21–26

[7] Nalimov V. I., Plotnikov P. I., “Neregulyarnye zadachi na sobstvennye znacheniya i effekt volnovoda”, Dinamika sploshnoi sredy, 23, 1975, 132–151

[8] Babich V. M., Bilyi I. Ya., “O volnovodnykh svoistvakh podvodnogo gornogo khrebta”, Mekhanika zhidkogo gaza, 1979, no. 3, 152–156 | MR

[9] Masayuki Oikawa, Junkichi Satsuma, Nobio Yajima, “Shallow water waves propagating along undulation of bottom surface”, J. Phys. Soc. Japan, 37:2 (1974), 511–517 | DOI | MR

[10] Sukhinin S. V., “Effekt volnovoda”, Annotatsii dokladov 6 Vsesoyuz. s'ezda po teoreticheskoi i prikladnoi mekhanike, Tashkent, 1986, 537–538

[11] Sukhinin S. V., “Effekt volnovoda”, Prikl. mekhanika i tekhn. fizika, 1989, no. 2, 92–101 | MR

[12] Bonnet-Ben Dhia A.-S., Joly P., “Mathematical analysis of guided water waves”, SIAM J. Appl. Math., 53:6 (1993), 1507–1550 | DOI | MR | Zbl

[13] Sukhinin S. V., “Effekt shepchuschei poverkhnosti”, Prikl. matematika i mekhanika, 63:6 (1999), 923–937 | MR

[14] Sukhinin S. V., “Effekt volnovoda odnomerno periodicheskoi pronitsaemoi struktury”, Prikl. mekhanika i tekhn. fizika, 31:4 (1990), 77–85 | MR

[15] Sukhinin S. V., “Volnovodnoe, anomalnoe i shepchuschee svoistva periodicheskoi tsepochki prepyatstvii”, Sib. zhurn. industr. matematiki, 1:2 (1998), 175–198 | MR | Zbl

[16] Nikulin V. V., Shafarevich I. R., Geometrii i gruppy, Nauka, M., 1983 | MR

[17] Bagavantam S., Venkataraiudu T., Teoriya grupp i ee primenenie k fizicheskim problemam, Izd-vo inostr. lit., M., 1959

[18] Lyubarskii G. Ya., Teoriya grupp i ee primenenie v fizike, Fizmatgiz, M., 1958 | MR

[19] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1976 | MR

[20] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki, v. 4, Mir, M., 1982