A Numerical Study of Linear Long Water Waves over Variable Topographies using a Conformal Mapping

Abstract

In this work we present a numerical study of surface water waves over variable topographies for the linear Euler equations based on a conformal mapping and Fourier transform. We show that in the shallow-water limit the Jacobian of the conformal mapping brings all the topographic effects from the bottom to the free surface. Implementation of the numerical method is illustrated by a MATLAB program. The numerical results are validated by comparing them with exact solutions when the bottom topography is flat, and with theoretical results for an uneven topography.

Gaussian Pulses over Random Topographies for the Linear Euler Equations

“…These results ties with the study of Nachbin and Solna [20]. We notice that the changing in the dynamic presented in this work is different from the ones reported by Flamarion and Ribeiro-Jr [11]. In this article, the authors showed that a monochromatic rapidly varying topography barely affects the shape of the pulse, while a slowly varying topography causes a considerable variation in amplitude and an increasing of reflected waves.…”

“…In this article, the authors showed that a monochromatic rapidly varying topography barely affects the shape of the pulse, while a slowly varying topography causes a considerable variation in amplitude and an increasing of reflected waves. This occurs because only monochromatic topographies are considered in [11]. Therefore, our study shows that when the topography is modeled by a series of Fourier modes the wave field changes considerably.…”

“…The formulation presented here is identical to the one described in [11], hence we will briefly summarize the main steps.…”

“…In addition, the time advance of the system (3.3) is computed through the Runge-Kutta fourth order method (RK4) with time step ∆t. The numerical method proposed here is implemented in MATLAB and it consists of a variation of the code available in [11].…”