Bilinear forms through the binary Bell polynomials, N solitons and Bäcklund transformations of the Boussinesq–Burgers system for the shallow water waves in a lake or near an ocean beach

Water waves, one of the most common phenomena in nature, play an important role in the marine/offshore engineering, hydraulic engineering, energy development, mechanical engineering, etc. Hereby, for the shallow water waves near an ocean beach or in a lake, we study a Boussinesq-Burgers system. With respect to the water-wave horizontal velocity and height deviating from the equilibrium position of water, we find out (1) two hetero-Backlund transformations via the Bell polynomials and symbolic computation, and (2) a set of the similarity reductions via symbolic computation, to a known ordinary differential equation, for which we also construct some solutions. The results rely on the oceanic water-wave dispersive power. We hope that our hetero-Backlund transformations and similarity reductions could help the researchers investigate certain modes of the shallow water waves near an ocean beach.

Beholding the shallow water waves near an ocean beach or in a lake via a Boussinesq-Burgers system

A porous-wavemaker theory is developed to analyse small-amplitude surface waves on water of finite depth, produced by horizontal oscillations of a porous vertical plate. Analytical solutions in closed forms are obtained for the surface-wave profile, the hydrodynamic-pressure distribution and the total force on the wavemaker. The influence of the wave-effect parameter C and the porous-effect parameter G, both being dimensionless, on the surface waves and on the hydrodynamic pressures is discussed in detail.

A porous wavemaker theory

People pay attention to the oceans crossing the Solar System: the Earth, Enceladus and Titan. Hereby, on a variable-coefficient nonlinear dispersive-wave system for the shallow oceanic environment, (A) nonlinear-water-wave symbolic computation and Bell polynomials lead to two hetero-Backlund transformations and (B) nonlinear-water-wave symbolic computation gives rise to a similarity reduction, both depending on the variable coefficients representing the wave elevation and surface velocity of the water wave. This paper might be of some use for the future oceanic studies on the Solar System.

Oceanic studies via a variable-coefficient nonlinear dispersive-wave system in the Solar System

Oceans crossing the Solar System attract people’s attention: the Earth, Enceladus, and Titan. On a variable-coefficient nonlinear dispersive-wave system for the shallow oceanic environment, our symbolic computation yields two non-auto-Backlund transformations and auto-Backlund transformations with some solitons, with regard to the wave elevation and surface velocity of the water wave, which depend on the variable coefficients. This paper could be of some use for the future oceanic studies in the Solar System.

Viewing the Solar System via a variable-coefficient nonlinear dispersive-wave system

Scattering of an obliquely incident wave train by two non-identical thin vertical barriers either partially immersed or fully submerged in infinitely deep water was studied by employing Havelock’s expansion of water wave potential and reducing the problem ultimately to the solution of a pair of vector integral equations of the first kind. A one-term Galerkin approximation in terms of a known exact solution of the integral equation corresponding to a single vertical barrier is used to obtain very accurate numerical estimates for the reflection and transmission coefficients. The reflection coefficient is depicted graphically for two different arrangements of the vertical barriers. It is observed that total reflection is possible for some discrete values of the wavenumber only when the barriers are identical, either partially immersed or completely submerged. As the separation length between the two vertical barriers increases, the reflection coefficient becomes oscillatory as a function of the wavenumber, which is due to multiple reflections by the barriers. Also, as the separation length becomes very small, the known results for a single barrier are obtained for normal incidence of the wave train.

Oblique water wave scattering by two unequal vertical barriers

Water wave scattering by multiple thin vertical barriers

The problem of water-wave scattering by two thin vertical plates of unequal lengths submerged beneath the free surface of an infinitely deep water is studied here assuming linear theory. The problem is reduced to a pair of vector integral equations of first kind which are solved approximately by using single-term Galerkin approximation. Very accurate numerical estimates for the reflection and transmission coefficients for different values of the wave number and other parameters are obtained. The numerical results for the reflection coefficient are plotted against the wave number in a number of figures for different configurations of the two plates. It is observed from these figures that the reflection coefficient vanishes for a sequence of values of the wave number only for two identical submerged plates. However, for two non-identical plates, the reflection coefficient never becomes zero, although there exists a few wave numbers at which this becomes small for some particular configurations of the plates. When the two plates become very close to each other, known numerical results for a single plate are deduced.

Water-wave scattering by two submerged thin vertical unequal plates

Scattering of surface waves by two permeable vertical walls having apertures at different depths in uniform finite depth water has been studied assuming linear theory. The problem is reduced to a s...

Oblique wave scattering by two thin non-uniform permeable vertical walls with unequal apertures in water of uniform finite depth

https://iopscience.iop.org/article/10.1088/1873-7005/ab2d4d/pdf

Ranita Roy

Papers

Oblique water wave scattering by two unequal vertical barriers

Water wave scattering by multiple thin vertical barriers

Water-wave scattering by two submerged thin vertical unequal plates

Oblique wave scattering by two thin non-uniform permeable vertical walls with unequal apertures in water of uniform finite depth

Water wave scattering by three thin vertical barriers arranged asymmetrically in deep water