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Keywords:
time-harmonic velocity potential; uniqueness theorem; Helmholtz equation; Neumann’s eigenvalue problem for Laplacian; integral equation method; weighted Hölder spaces; velocity potential; uniqueness; Neumann’s eigenvalue problem; Laplacian; linearized problem; radiation; scattering; time-harmonic water wave; vertical shell
time-harmonic velocity potential; uniqueness theorem; Helmholtz equation; Neumann’s eigenvalue problem for Laplacian; integral equation method; weighted Hölder spaces; velocity potential; uniqueness; Neumann’s eigenvalue problem; Laplacian; linearized problem; radiation; scattering; time-harmonic water wave; vertical shell
Summary:
The uniqueness theorem is proved for the linearized problem describing radiation and scattering of time-harmonic water waves by a vertical shell having an arbitrary horizontal cross-section. The uniqueness holds for all frequencies, and various locations of the shell are possible: surface-piercing, totally immersed and bottom-standing. A version of integral equation technique is outlined for finding a solution.
References:
[1] C. J. R. Garrett: Bottomless harbours. J. Fluid Mech. 43 (1970), 433–449. DOI 10.1017/S0022112070002495 | Zbl 0212.60101
[2] A. Hulme: Some applications of Maz’ja’s uniqueness theorem to a class of linear water wave problems. Math. Proc. Camb. Phil. Soc. 95 (1984), 511–519. DOI 10.1017/S0305004100061417 | MR 0727091 | Zbl 0546.76031
[3] F. John: On the motion of floating bodies, II. Comm. Pure Appl. Math. 3 (1950), 45–101. DOI 10.1002/cpa.3160030106 | MR 0037118
[4] N. G. Kuznetsov: Plane problem of the steady-state oscillations of a fluid in the presence of two semiimmersed cylinders. Math. Notes Acad. Sci. USSR 44 (1988), 685–690. MR 0972199
[5] N. G. Kuznetsov: Uniqueness of a solution of a linear problem for stationary oscillations of a liquid. Diff. Equations 27 (1991), 187–194. MR 1109387 | Zbl 0850.76059
[6] N. Kuznetsov, P. McIver: On uniqueness and trapped modes in the water-wave problem for a surface-piercing axisymmetric body. Q. J. Mech. Appl. Math. 50 (1997), 565–580. DOI 10.1093/qjmam/50.4.565 | MR 1488385
[7] M. J. Lighthill: Two-dimensional analyses related to wave-energy extraction by submerged resonant ducts. J. Fluid Mech. 91 (1979), 253–317. DOI 10.1017/S0022112079000148 | Zbl 0403.76025
[8] V. G. Maz’ya: Solvability of the problem on the oscillations of a fluid containing a submerged body. J. Soviet Math. 10 (1978), 86–89. DOI 10.1007/BF01109726
[9] V. G. Maz’ya: Boundary integral equations. Encyclopaedia of Math. Sciences 27 (1991), 127–222. DOI 10.1007/978-3-642-58175-5_2 | MR 1098507
[10] V. G. Maz’ya, B. A. Plamenevskii: Schauder estimates of solutions of elliptic boundary value problems in domains with edges on the boundary. Elliptic boundary value problems AMS Transl. 123 (1984), 141–169.
[11] V. G. Maz’ya, J. Rossmann: Über die Asymptotic der Lösungen elliptischer Randwertaufgaben in der Umgebung von Kanten. Math. Nachr. 138 (1988), 27–53. DOI 10.1002/mana.19881380103 | MR 0975198
[12] M. McIver: An example of non-uniqueness in the two-dimensional linear water-wave problem. J. Fluid Mech. 315 (1996), 257–266. DOI 10.1017/S0022112096002418 | MR 1403701 | Zbl 0869.76010
[13] P. McIver, M. McIver: Trapped modes in an axisymmetric water-wave problem. Q. J. Mech. Appl. Math. 50 (1997), 165–178. DOI 10.1093/qjmam/50.2.165 | MR 1451065
[14] M. J. Simon: Wave-energy extraction by a submerged cylindrical resonant duct. J. Fluid Mech. 104 (1981), 159–187. DOI 10.1017/S0022112081002875 | Zbl 0463.76021
[15] M. J. Simon, N. G. Kuznetsov: On uniqueness of the water-wave problem for a floating toroidal body. Appl. Math. Report 96/1 (1996), Department of Mathematics, University of Manchester, England.
[16] M. J. Simon, F. Ursell: Uniqueness in linearized two-dimensional water-wave problem. J. Fluid Mech. 148 (1984), 137–154. DOI 10.1017/S0022112084002287 | MR 0778139
[17] J. R. Thomas: The absorption of wave energy by a three-dimensional submerged duct. J. Fluid Mech. 104 (1981), 189–215. DOI 10.1017/S0022112081002887 | Zbl 0455.76007
[18] F. Ursell: Surface waves on deep water in the presence of a submerged circular cylinder, I. Proc. Camb. Phil. Soc. 46 (1950), 141–152. DOI 10.1017/S0305004100025561 | MR 0033714 | Zbl 0035.42201
[19] F. Ursell: Some unsolved and unfinished problems in the theory of waves. Wave Asymptotics, Camb. Univ. Press, Cambridge, 1992. MR 1174354 | Zbl 0817.76008
[20] B. R. Vainberg, V. G. Maz’ya: On the problem of the steady state oscillations of a fluid layer of variable depth. Trans. Moscow Math. Soc. 28 (1973), 56–73. MR 0481654
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