Linear modes for channels of constant cross-section and approximate Dirichlet-Neumann operators. (English) Zbl 1454.76030
Summary: We study normal modes for the linear water wave problem in infinite straight channels of bounded constant cross-section. Our goal is to compare semi-analytic normal mode solutions known in the literature for special triangular cross-sections, namely isosceles triangles of equal angle of \(45^{\circ}\) and \(60^{\circ}\), see [H. Lamb, Hydrodynamics. 6th ed. Cambridge: Cambridge University Press (1932; JFM 58.1298.04); H. M. Macdonald, Proc. Lond. Math. Soc. 25, 101–111 (1894; JFM 25.1479.01), A. G. Greenhill, “Wave motion in hydrodynamics”, Am. J. Math. 9, No. 2, 97–112 (1887; doi:10.2307/2369329); B. A. Packham, Q. J. Mech. Appl. Math. 33, 179–187 (1980; Zbl 0442.76013)], and [M. D. Groves, Q. J. Mech. Appl. Math. 47, No. 3, 367–404 (1994; Zbl 0868.76013)], to numerical solutions obtained using approximations of the non-local Dirichlet-Neumann operator for linear waves, specifically an ad-hoc approximation proposed in [R. M. Vargas-Magaña and P. Panayotaros, Wave Motion 65, 156–174 (2016; Zbl 1467.76019)], and a first-order truncation of the systematic depth expansion by W. Craig et al. [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 461, No. 2055, 839–873 (2005; Zbl 1145.76325)]. We consider cases of transverse (i.e. 2-D) modes and longitudinal modes, i.e. 3-D modes with sinusoidal dependence in the longitudinal direction. The triangular geometries considered have slopping beach boundaries that should in principle limit the applicability of the approximate Dirichlet-Neumann operators. We nevertheless see that the approximate operators give remarkably close results for transverse even modes, while for odd transverse modes we have some discrepancies near the boundary. In the case of longitudinal modes, where the theory only yields even modes, the different approximate operators show more discrepancies for the first two longitudinal modes and better agreement for higher modes. The ad-hoc approximation is generally closer to exact modes away from the boundary.
MSC:
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 35Q35 | PDEs in connection with fluid mechanics |
Keywords:
linear Whitham-Boussinesq water wave model; variable topography; pseudodifferential operator; semi-analytic normal mode solutionCitations:
Zbl 0442.76013; Zbl 0868.76013; Zbl 1145.76325; Zbl 1467.76019; JFM 58.1298.04; JFM 25.1479.01References:
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