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This study investigates numerically the interaction between a Gaussian pulse and variable topography using the linear Euler equations. The impact of topography variation on the amplitude and behavior of the wave pulse is examined through numerical simulations and statistical analysis. On one hand, we show that for slowly varying topographies, the incoming pulse almost retains its shape, and little energy is transferred to the small reflected waves. On the other hand, we demonstrate that for rapidly varying topographies, the shape of the pulse is destroyed, which is different from previous studies.

Stagnation points beneath solitary gravity-capillary waves in the weakly nonlinear weakly dispersive regime in a sheared channel with finite depth and constant vorticity are investigated. A Korteweg-de Vries equation that incorporates the surface tension and the vorticity effects is obtained asymptotically from the full Euler equations. The velocity field in the bulk fluid is approximated which allow us to compute stagnation points in the solitary wave moving frame. We show that stagnation points bellow the crest of elevation solitary waves exist for large values of the vorticity and Bond numbers less than a critical value that depends on the vorticity. Remarkably, the positions of these stagnation points do not depend on the surface tension. Besides, we show that when there are two stagnation points located at the bottom of the channel, they are pulled towards the horizontal coordinate of the solitary wave crest as the Bond number increases until its critical value.

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.

This work presents the results of a direct numerical simulation of the nonlinear free surface evolution of a finite-depth fluid with a linear shear flow under the action of horizontal electric fields. The method of time-dependent conformal transformation for the description of the combined effects of the electric fields and constant vorticity is generalized for the first time. The simulation results show that strong shear flow co-directed in the wave propagation direction leads to the formation of large-amplitude surface waves, and, for some limiting vorticity value, a wave breaking process with the formation of an air bubble in the liquid is possible. The oppositely directed shear flow can cause the retrograde motion of a surface wave (wave propagation in the opposite direction to the linear wave speed). The simulations conducted taking into account the electro-hydrodynamic effects demonstrate that a high enough external horizontal electric field suppresses these strongly nonlinear processes, and the surface waves tend to preserve their shape.

In this work we investigate the propagation of rotational solitary waves over a submerged obstacle in a vertically sheared shallow water channel with constant vorticity. In the weakly nonlinear regime, the problem is formulated in the forced Korteweg-de Vries framework. The initial value problem for this equation is solved numerically using a Fourier pseudospectral method with integrating factor. Solitary waves are taken as initial data, and the interaction wave-current-topography is analysed. We identify three types of regimes according to the intensity of the vorticity. A rotational solitary wave can bounce back and forth over the obstacle remaining trapped for large times, it can pass over the obstacle without reversing its direction or the wave can be blocked, i.e., it bounces back and forth above the obstacle until reaching a steady state. Such behaviour resembles the classical damped spring-mass system.

Nonlinear steady rotational waves in a low-pressure region are investigated. The problem is formulated in a simplified canonical domain through the use of a conformal mapping, which flattens the free surface. Steady waves are computed numerically using a Newton’s method and classified into three types. Besides, our results indicate that there is a region in which steady waves do not exist. The thickness of this region is compared with the one predicted by the weakly nonlinear, weakly dispersive regime.