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Analysis of the Nonlinear Shallow Water Equations Over Nonplanar Topography
Author(s) -
Antuono Matteo,
Brocchini Maurizio
Publication year - 2010
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2009.00464.x
Subject(s) - breaking wave , waves and shallow water , nonlinear system , planar , geology , bathymetry , boundary value problem , shore , shallow water equations , forcing (mathematics) , boundary (topology) , position (finance) , kondratiev wave , geometry , mechanics , mathematical analysis , geophysics , physics , mathematics , wave propagation , oceanography , optics , climatology , computer science , quantum mechanics , computer graphics (images) , finance , economics
The role played by the beach bottom profile on coastal inundation phenomena is analyzed here by means of approximate analytical solutions of the nonlinear shallow water equations (NSWEs) over uneven bottoms. These are obtained by only using the assumptions of small waves at the seaward boundary and small topographic forcing. Our work, built on the Carrier and Greenspan [1] hodographic transformation and on the solution of the boundary value problem (BVP) for the NSWEs proposed by Antuono and Brocchini [2], focuses on the propagation of nonlinear non‐breaking waves over quasi‐planar beaches. Since the terms associated with the perturbed bottom only appear in the second‐order perturbed solutions, the breaking conditions for the planar‐beach bathymetry also predict well the breaking occurring on the nonplanar beaches analyzed here. The most important results, concerning the shoreline position and the near‐shoreline velocity, are given for both pulse‐like and periodic input waves propagating over two types of nonplanar bathymetries. The solution proposed here is a fundamental benchmark for any numerical and theoretical analyzes concerned with estimates of wave run‐up on beaches of complex shape.