Long waves in shallow water over a layer of bingham-plastic fluid-mud—II. Mathematical derivation of long wave equations

Equations of motion are derived for long waves in water of varying depth. The equations are for small amplitude waves, but do include non-linear terms. They correspond to the Boussinesq equations for water of constant depth. Solutions have been calculated numerically for a solitary wave on a beach of uniform slope. These solutions include a reflected wave, which is also derived analytically by using the linearized long-wave equations.

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On long-wave propagation over a fluid-mud seabed

Journal of Fluid Mechanics ◽

10.1017/s0022112007005356 ◽

2007 ◽

Vol 579 ◽

pp. 467-480 ◽

Cited By ~ 25

Author(s):

PHILIP L.-F. LIU ◽

I-CHI CHAN

Keyword(s):

Wave Propagation ◽

Wave Equations ◽

Surface Displacement ◽

Boussinesq Approximation ◽

Fluid Mud ◽

Viscous Boundary Layer ◽

Time Integral ◽

Long Wave ◽

Free Surface Displacement ◽

Viscous Boundary

Using the Boussinesq approximation, a set of depth-integrated wave equations for long-wave propagation over a mud bed is derived. The wave motions above the mud bed are assumed to be irrotational and the mud bed is modelled as a highly viscous fluid. The pressure and velocity are required to be continuous across the water–mud interface. The resulting governing equations are differential–integral equations in terms of the depth-integrated horizontal velocity and the free-surface displacement. The effects of the mud bed appear in the continuity equation in the form of a time integral of weighted divergence of the depth-averaged velocity. Damping rates for periodic waves and solitary waves are calculated. For the solitary wave case, the velocity profiles in the water column and the mud bed at different phases are discussed. The effects of the viscous boundary layer above the mud–water interface are also examined.

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Analytic solution of the linearized shallow-water wave equations for certain continuous depth variations

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000000965 ◽

1975 ◽

Vol 19 (1) ◽

pp. 81-94 ◽

Cited By ~ 5

Author(s):

D. L. Clements ◽

C. Rogers

Keyword(s):

Wave Equation ◽

Shallow Water ◽

Ground Motion ◽

Water Depth ◽

Water Wave ◽

Wave Equations ◽

Analytic Solution ◽

Governing Equations ◽

Long Wave ◽

Shallow Water Wave Equations

AbstractThe linear long-wave equations with (and without) small ground motion are considered. The governing equations are represented in a matrix from and transformations are sought which reduce the system to (for example) a form associated with the conventional wave equation. Integration of the system is then immediate. It is shown that such a reduction may be acheived provided the variation in water depth is specified by certain multi-parameter forms.

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A novel method for travelling wave solutions of fractional Whitham-Broer-Kaup, fractional modified Boussinesq and fractional approximate long wave equations in shallow water

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.3151 ◽

2014 ◽

Vol 38 (7) ◽

pp. 1352-1368 ◽

Cited By ~ 22

Author(s):

S. Saha Ray

Keyword(s):

Shallow Water ◽

Wave Equations ◽

Travelling Wave ◽

Travelling Wave Solutions ◽

Wave Solutions ◽

Long Wave ◽

Novel Method

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Variational Principles for Two Kinds of Coupled Nonlinear Equations in Shallow Water

Symmetry ◽

10.3390/sym12050850 ◽

2020 ◽

Vol 12 (5) ◽

pp. 850

Author(s):

Xiao-Qun Cao ◽

Ya-Nan Guo ◽

Shi-Cheng Hou ◽

Cheng-Zhuo Zhang ◽

Ke-Cheng Peng

Keyword(s):

Shallow Water ◽

Nonlinear Equations ◽

Wave Equations ◽

Variational Principles ◽

Soliton Solutions ◽

Conserved Quantities ◽

Long Wave ◽

Complex Models ◽

Lagrange Functional ◽

Variational Formulations

It is a very important but difficult task to seek explicit variational formulations for nonlinear and complex models because variational principles are theoretical bases for many methods to solve or analyze the nonlinear problem. By designing skillfully the trial-Lagrange functional, different groups of variational principles are successfully constructed for two kinds of coupled nonlinear equations in shallow water, i.e., the Broer-Kaup equations and the (2+1)-dimensional dispersive long-wave equations, respectively. Both of them contain many kinds of soliton solutions, which are always symmetric or anti-symmetric in space. Subsequently, the obtained variational principles are proved to be correct by minimizing the functionals with the calculus of variations. The established variational principles are firstly discovered, which can help to study the symmetries and find conserved quantities for the equations considered, and might find lots of applications in numerical simulation.

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Long wave generation by the shoaling and breaking of transient wave groups on a beach

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2005.1642 ◽

2006 ◽

Vol 462 (2070) ◽

pp. 1853-1876 ◽

Cited By ~ 33

Author(s):

T.E Baldock

Keyword(s):

Shallow Water ◽

Water Waves ◽

Surf Zone ◽

Laboratory Data ◽

Short Wave ◽

Long Waves ◽

Transient Wave ◽

Wave Groups ◽

Spatial Gradients ◽

Long Wave

This paper presents new laboratory data on the generation of long waves by the shoaling and breaking of transient-focused short-wave groups. Direct offshore radiation of long waves from the breakpoint is shown experimentally for the first time. High spatial resolution enables identification of the relationship between the spatial gradients of the short-wave envelope and the long-wave surface. This relationship is consistent with radiation stress theory even well inside the surf zone and appears as a result of the strong nonlinear forcing associated with the transient group. In shallow water, the change in depth across the group leads to asymmetry in the forcing which generates significant dynamic setup in front of the group during shoaling. Strong amplification of the incident dynamic setup occurs after short-wave breaking. The data show the radiation of a transient long wave dominated by a pulse of positive elevation, preceded and followed by weaker trailing waves with negative elevation. The instantaneous cross-shore structure of the long wave shows the mechanics of the reflection process and the formation of a transient node in the inner surf zone. The wave run-up and relative amplitude of the radiated and incident long waves suggests significant modification of the incident bound wave in the inner surf zone and the dominance of long waves generated by the breaking process. It is proposed that these conditions occur when the primary short waves and bound wave are not shallow water waves at the breakpoint. A simple criterion is given to determine these conditions, which generally occur for the important case of storm waves.

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Free and forced components of shoaling long waves in the absence of short wave breaking

Journal of Physical Oceanography ◽

10.1175/jpo-d-20-0214.1 ◽

2021 ◽

Author(s):

Stephanie Contardo ◽

Ryan J. Lowe ◽

Jeff E. Hansen ◽

Dirk P. Rijnsdorp ◽

François Dufois ◽

...

Keyword(s):

Shallow Water ◽

Wave Breaking ◽

Phase Relationship ◽

Wave Model ◽

Short Wave ◽

Long Waves ◽

Wave Group ◽

Wave Groups ◽

Long Wave ◽

Complete Representations

AbstractLong waves are generated and transform when short-wave groups propagate into shallow water, but the generation and transformation processes are not fully understood. In this study we develop an analytical solution to the linearized shallow-water equations at the wave-group scale, which decomposes the long waves into a forced solution (a bound long wave) and free solutions (free long waves). The solution relies on the hypothesis that free long waves are continuously generated as short-wave groups propagate over a varying depth. We show that the superposition of free long waves and a bound long wave results in a shift of the phase between the short-wave group and the total long wave, as the depth decreases prior to short-wave breaking. While it is known that short-wave breaking leads to free long generation, through breakpoint forcing and bound wave release mechanisms, we highlight the importance of an additional free long wave generation mechanism due to depth variations, in the absence of breaking. This mechanism is important because as free long waves of different origins combine, the total free long wave amplitude is dependent on their phase relationship. Our free and forced solutions are verified against a linear numerical model, and we show how our solution is consistent with prior theory that does not explicitly decouple free and forced motions. We also validate the results with data from a nonlinear phase-resolving numerical wave model and experimental measurements, demonstrating that our analytical model can explain trends observed in more complete representations of the hydrodynamics.

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