Surface waves in basins of variable depth. (English) Zbl 0588.76021

The linearized boundary-value problem for surface waves of frequency \(\omega\) in a closed basin of variable depth is reduced to a non-self- adjoint partial differential equation in the plane of the free surface. The corresponding variational form (which does not provide a definite upper or lower bound) for the eigenvalue \(\kappa =\omega^ 2/g\) is constructed. A self-adjoint partial differential equation, for which the variational form is the Rayleigh quotient (which provides an upper bound to \(\kappa)\), also is constructed; it offers significant advantages vis- à-vis the non-self-adjoint formulation, but at the expense of a more complicated operator. Three relatively simple variational approximations are constructed, two for a class of basins with sloping sides and the third for basins for which the variation of the depth relative to its mean is small.


76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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[1] Goldsbrough, Proc. R. Soc. Lond. 130 pp 157– (1930)
[2] Chrystal, Trans. R. Soc. Edinb. 41 pp 599– (1905) · JFM 36.0807.02
[3] Wehausen, Encyclopedia of Physics 9 pp 624– (1960)
[4] Vint, Proc. Lond. Math. Soc. 22 pp 1– (1923)
[5] Storchi, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 12 pp 544– (1952)
[6] Storchi, 1st Lombardo Sci. Lett. Rend. Cl. Sci. Mat. Nat. 13(82) pp 95– (1949)
[7] Sen, Proc. Lond. Math. Soc. 26 pp 363– (1927)
[8] Rayleigh, Phil. Mag. 47 pp 566– (1899)
[9] Poisson, Mém. de l’Institut 3 pp 1– (1820)
[10] Rayleigh, Proc. Lond. Math. Soc. 4 pp 357– (1873)
[11] Lundberg, Z. angew. Math. Mech. 64 pp 1– (1984)
[12] Proudman, Proc. Lond. Math. Soc. 14 pp 240– (1915) · JFM 45.1402.02
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