Boussinesq-type Wave Equations Research Articles

An Unorthodox Arrangement of Boussinesq-Type Wave Equations for Accurate and Robust Numerical Treatment

A set of Boussinesq-type wave equations with enhanced dispersion characteristics is presented for accurate, efficient, and robust numerical treatment. New arrangement uses three different velocity variables simultaneously in order to keep continuity and momentum equations in simplest conservation forms while improving the dispersion characteristics. This approach allows us to retain all the nonlinear contributions with minimum number of terms. Spatial and time-dependent variations of the seabed are fully accounted for and the effect of external free surface pressure is included. A numerical scheme based on finite differences is developed, and various well-known experimental cases are simulated for testing the performance of the proposed set of equations. Comparisons of simulations with measurements reveal quite satisfactory agreements and, hence, bolster confidence in the wave model.

Complex solutions to the higher-order nonlinear boussinesq type wave equation transform

Complex solutions to the higher-order nonlinear boussinesq type wave equation transform

Dispersive and propagation of shallow water waves as a higher order nonlinear Boussinesq-like dynamical wave equations

The propagation of small amplitude long capillary–gravity waves on the surface of shallow water described by the higher order nonlinear Boussinesq type wave equation. This model is arising in mathematical physics, shallow water waves, fluid dynamics and fluid flow. We applied the modified Jacobi elliptic function expansion method to obtained exact solutions in the type of solitary wave and elliptic functions solutions. These solutions helps the mathematicians and physicians to understand the physical phenomena of this model. This technique can be utilized on other models to launch further exclusively novel solutions for other categories of nonlinear PDEs occurring in mathematical Physics.

On solutions of a Boussinesq-type equation with displacement-dependent nonlinearity: A soliton doublet

In this paper the permanent profile waves governed by a Boussinesq-type wave equation are analysed. The model involves displacement-type nonlinearities and dispersion terms. Physically such a model equation describes longitudinal waves (density change) in biomembranes which have an internal structure composed by lipid molecules. The possible solutions are constructed and analysed. The phase plane analysis and numerical simulation reveal a novel phenomenon: the possible existence of a soliton doublet.

On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes

Boussinesq-type wave equations involve nonlinearities and dispersion. In this paper a Boussinesq-type equation with displacement-dependent nonlinearities is presented. Such a model was proposed by Heimburg and Jackson for describing longitudinal waves in biomembranes and later improved by Engelbrecht, Tamm and Peets taking into account the microinertia of a biomembrane. The steady solution in the form of a solitary wave is derived and the influence of nonlinear and dispersive terms over a large range of possible sets of coefficients demonstrated. The solutions emerging from arbitrary initial inputs are found using the numerical simulation. The properties of emerging trains of solitary waves are analysed. Finally, the interaction of solitary waves which satisfy the governing equation is studied. The interaction process is not fully elastic and after several interactions radiation effects may be significant. This means that for the present case the solitary waves are not solitons in the strict mathematical sense. However, like in other cases known in solid mechanics, such solutions may be conditionally called solitons.

Numerical modeling of subaerial and submarine landslide-generated tsunami waves—recent advances and future challenges

Landslide-generated waves (LGWs) are among natural hazards that have stimulated attentions and concerns of engineers and researchers during the past decades. At the same period, the application of numerical modeling has been progressively increased to assess, control, and manage the risks of such hazards. This paper represents an overview of numerical studies on LGWs to explore associated recent advances and future challenges. In this review, the main landslide events followed by an LGW hazard are scrutinized. The uncertainty regarding landslide characteristics and the lack of data concerning generated tsunami properties highlights the necessity of probabilistic analysis and numerical modeling. More than 53 % of landslides show the slide length larger than about 20 times of the slide thickness. This fact justifies the popular application of depth-averaged equations (DAEs) for landslides’ motion simulations. Such models are reviewed and tabulated based on their mathematical, numerical, and conceptual approaches. A landslide is generally treated as a homogeneous, mixture, or a multi-phase fluid with different rheologies. The Coulomb type rheology is the most-used rheology applied in more than 70 % of landslide models. Some of the recent studies are considering the effects of multi-phase nature, dynamic changes of rheological parameters, and grain-size segregation of the landslide on its deformations. The numerical tools that model LGWs are also reviewed, categorized, and examined. These models conceptualize a landslide as a general rigid LGW (R-LGW) or deformable LGW (D-LGW) mass. The rigid slide assumption is mainly applied in the LGW models with a focus on the accurate simulation of the wave propagation stage, particularly by means of higher order Boussinesq-type wave equations (BWEs). The majority of D-LGW models solve either the Navier–Stokes equations (NSEs) for a multi-phase (landslide material, water, and air) flow or the shallow water equations (SWEs) for a two-layer (a layer of granular material moving beneath a layer of water) flow. NSEs are more comprehensive models but less robust than DAEs. The key effect of dispersion in LGWs, which are typically important in intermediate and even deep water wave domains, challenges researchers to apply higher order BWEs instead of SWEs in two-layer models. Regarding numerical approaches, Lagrangian’s are more robust than Eulerian’s, but they have been rarely applied due to their high computational demands for real cases. The remaining challenges are reviewed as the necessity of probabilistic analysis to assess the risk of the related hazards more accurately for both past and potential LGW hazards; further thorough laboratory-scale experiments and field data measurements to have accurate and detailed benchmark data; providing RS/GIS-based worldwide hazard map for potential LGWs and compiled database for occurred events; extending BWEs for granular flows and DAEs with non-hydrostatic corrections; and economizing the computational costs of models by advanced techniques like parallel processing and GPU accelerators.

On the role of nonlinearities in the Boussinesq-type wave equations

Longitudinal mechanical waves in biomembranes are described by a Boussinesq-type wave equation. It is shown that in this case the nonlinearities are of a different type compared with conventional models of solids. The governing equation analysed in this paper is the improved Heimburg–Jackson model with two dispersive terms. The soliton-type solutions of such a wave equation are found and analysed. The existence of solitons depends on the ratio of nonlinear terms and the width of solitons is governed by dispersive terms.

2D solitons in Boussinesq equation with dissipation

We investigate numerically the stationary solutions of 2D Boussinesq type wave equations with square and cubic nonlinearity. In the model equation dissipation is added and we investigate the physical properties of the problem. To investigate and study the problem we implement the Christov spectral technique in $$L^2(-\infty ,\infty )$$ . The method was found to be accurate and computationally efficient in other works of the author, mainly in 1D problems.

Dispersion-Correction Finite Difference Model for Simulation of Transoceanic Tsunamis

A finite difference numerical model, which can correctly consider dispersion effect of waves over a slowly varying water depth, is developed for the simulation of tsunami propagation. The present model employs a linear Boussinesq-type wave equation that can be solved more easily than typical Boussinesq equations. In the present model numerical dispersion is minimized by controlling the dispersion-correction parameter determined by the time step, grid size and local water depth. In order to examine the applicability of the present model to dispersive waves, the propagation of tsunamis is simulated for an initial water surface displacement of Gaussian shape for the cases of several constant water depths and a submerged circular shoal. The numerical results are compared with analytical solutions or numerical solutions of linearized Boussinesq equations. The comparisons show that satisfactory agreement is obtained.

NOTES AND CORRESPONDENCE Constituent Boussinesq Equations for Waves and Currents

Abstract The Boussinesq long-wave equations in constituent form are generalized to encompass both the irrotational long-wave dynamics and the rotational current dynamics. For irrotational long waves, the generalized equations are shown to represent both the weakly nonlinear and fully nonlinear Boussinesq-type wave equations given previously in the literature. The generalized equations are of additional interest in that they are effectively model equations for a weakly nonhydrostatic wave/current system, in which the conventional hydrostatic ocean current/wave model is a limiting case. The derived equations are related to the surface flow variables that are accessible to synoptic surface wave/current measurements such as those available from radar remote sensing.