Wave Dynamics in the Channels of Variable Cross-Section

Abstract

Dynamics of long sea waves in the channels of variable depth and variable rectangular cross-section is discussed within various approximations -from the shallow water equations to those of nonlinear dispersion theory. General approach permitting to find traveling (non-reflective) waves in inhomogeneous channels is demonstrated within the framework of the shallow water linear theory. The appropriate conditions are determined by solving a system of ordinary differential equations. The so-called self-consistent channel in which the width is connected with its depth in a specific way is studied in detail. Within the linear theory of shallow water, a wave does not reflect from the bottom irregularities. The wave shape remains unchanged on the records of the wave gauges (mareographs) fixed along the channel axis, but it varies in space. Nonlinearity and dispersion lead to the wave transformation in such a channel. Within the framework of the shallow water weakly nonlinear theory, the wave shape is described by the Riemann solution, and the wave breaks (gradient catastrophe) quicker in the zones of decreasing depth. The modified Korteweg -de Vries equation describing evolution of a solitary wave of weak but finite amplitude in a self-consistent channel, the depth of which can vary arbitrary, is derived. Some examples of a solitary wave transformation in such a channel are analyzed (particularly, a soliton adiabatic transformation in the channel with the slowly varying parameters, and a solitary wave fission into the group of solitons after it has passed the zone where the depth changes abruptly. The obtained solutions extend the class of those represented earlier by S.F. Dotsenko and his colleagues.

Reflectionless wave propagation on shallow water with variable bathymetry and current

“…Then, from the flux conservation equation (2.1), it follows that W(x)H 1/2 (x) = const., so that U(x) ∼ W(x). Note that the same relation between the water depth and duct width was obtained by Pelinovsky et al (2017b) to describe global RL solutions for the case when there is no current.…”

“…Such a well-known trick allows one to reduce a linear equation with variable coefficients to one of the reference equations (see e.g. Grimshaw et al 2010a;Pelinovsky et al 2017b;and references therein). After substitution of the expression for ϕ in (2.7), we arrive at the following equation for ψ(x, t):…”

“…Therefore, one of the important questions is whether it is possible to find such conditions when solutions for the profiles are globally defined on the entire x-axis. Such a problem was solved in the case when there is no current (Pelinovsky et al 2017b). We provide the solution for the case when the current is taken into account.…”

“…There is a vast volume of publications on this theme in the past two decades. Here we refer only to the most recent and relevant; others can be found in the cited publications (Didenkulova 2009;Didenkulova, Pelinovsky & Soomere 2009;Dobrokhotov & Tirozzi 2010;Grimshaw, Pelinovsky & Talipova 2010b;Didenkulova & Pelinovsky 2011;Dobrokhotov, Nekrasov & Tirozzi 2011;Dobrokhotov, Medvedev & Minenkov 2013;Pelinovsky et al 2017a;Pelinovsky, Didenkulova & Shurgalina 2017b;Pelinovsky et al 2019). The outcomes of theoretical predictions for shallow-water waves were validated in the numerical modelling (see e.g.…”

“…To solve this problem, we employ the transformation technique that has been used in many publications (see e.g. Grimshaw, Pelinovsky & Pelinovsky 2010a;Pelinovsky et al 2017b;and references therein). Such a technique is a particular case of the general methods of reduction of linear differential equations to some reference equations that can be solved analytically.…”