Current effects on resonant reflection of surface water waves by sand bars

1988 ◽  
Vol 186 ◽  
pp. 501-520 ◽  
Author(s):  
James T. Kirby
Keyword(s):  
Water Waves ◽  
Evolution Equations ◽  
Multiple Scale ◽  
Periodic Wave ◽  
Sand Bars ◽  
Linear Waves ◽  
Surface Water Waves ◽  
Wave Trains ◽  
Resonant Reflection ◽  
A Current

The effect of currents flowing across a bar field on resonant reflection of surface waves by the bars is investigated. Using a multiple-scale expansion, evolution equations for the amplitudes of linear waves are derived and used to investigate the reflection of periodic wave trains with steady amplitude for both normal and oblique incidence. The presence of a current is found to shift resonant frequencies by possibly significant amounts and is also found to enhance reflection of waves by bar fields due to the additional effect of the perturbed current field.

Resonant reflection of surface water waves by periodic sandbars

1985 ◽  
Vol 152 ◽  
pp. 315-335 ◽  
Author(s):  
Chiang C. Mei
Keyword(s):  
Water Waves ◽  
Resonance Condition ◽  
Nonlinear Effects ◽  
Sea Bottom ◽  
Strong Reflection ◽  
Weak Reflection ◽  
Surface Water Waves ◽  
Wave Induced ◽  
Incident Waves ◽  
Resonant Reflection

One of the possible mechanisms of forming offshore sandbars parallel to a coast is the wave-induced mass transport in the boundary layer near the sea bottom. For this mechanism to be effective, sufficient reflection must be present so that the waves are partially standing. The main part of this paper is to explain a theory that strong reflection can be induced by the sandbars themselves, once the so-called Bragg resonance condition is met. For constant mean depth and simple harmonic waves this resonance has been studied by Davies (1982), whose theory, is however, limited to weak reflection and fails at resonance. Comparison of the strong reflection theory with Heathershaw's (1982) experiments is made. Furthermore, if the incident waves are slightly detuned or slowly modulated in time, the scattering process is found to depend critically on whether the modulational frequency lies above or below a threshold frequency. The effects of mean beach slope are also studied. In addition, it is found for periodically modulated wave groups that nonlinear effects can radiate long waves over the bars far beyond the reach of the short waves themselves. Finally it is argued that the breakpoint bar of ordinary size formed by plunging breakers can provide enough reflection to initiate the first few bars, thereby setting the stage for resonant reflection for more bars.

scholarly journals The Modified Reductive Perturbation Method as Applied to the Boussinesq Equation

2007 ◽  
Vol 62 (7-8) ◽  
pp. 347-352 ◽  
Author(s):  
Hilmi Demiray
Keyword(s):  
Perturbation Method ◽  
Water Waves ◽  
Present Method ◽  
Evolution Equations ◽  
Expansion Method ◽  
Multiple Scale ◽  
Travelling Wave ◽  
Reductive Perturbation Method ◽  
Kdv Hierarchy ◽  
Reductive Perturbation

In this work, we extended the application of “the modified reductive perturbation method” to long water waves and obtained the governing equations of Korteweg - de Vries (KdV) hierarchy. Seeking localized travelling wave solutions to these evolution equations we have determined the scale parameter g1 so as to remove the possible secularities that might occur. To indicate the effectiveness and the elegance of the present method, we studied the problem of the “dressed solitary wave method” and obtained exactly the same result. The present method seems to be fairly simple and practical as compared to the renormalization method and the multiple scale expansion method existing in the current literature

Formation of sand bars under surface waves

2000 ◽  
Vol 416 ◽  
pp. 315-348 ◽  
Author(s):  
JIE YU ◽  
CHIANG C. MEI
Keyword(s):  
Evolution Equation ◽  
Water Waves ◽  
Layer Structure ◽  
Effective Diffusivity ◽  
Sand Bars ◽  
Surface Water Waves ◽  
Concurrent Processes ◽  
Bed Stress ◽  
Forced Diffusion ◽  
Wave Induced

A quantitative theory is described for the formation mechanism of sand bars under surface water waves. By assuming that the slopes of waves and bars are comparably gentle and sediment motion is dominated by the bedload, an approximate evolution equation for bar height is derived. The wave field and the boundary layer structure above the wavy bed are worked out to the accuracy needed for solving this evolution equation. It is shown that the evolution of sand bars is a process of forced diffusion. This is unlike that for sand ripples which is governed by an instability. The forcing is directly caused by the non-uniformity of the wave envelope, hence of the wave-induced bottom shear stress associated with wave reflection, while the effective diffusivity is the consequence of gravity and modified by the local bed stress. During the slow formation, bars and waves affect each other through the Bragg scattering mechanism, which consists of two concurrent processes: energy transfer between waves propagating in opposite directions and change of their wavelengths. Both effects are found to be controlled locally by the position of bar crests relative to wave nodes. Comparison with available laboratory experiments is discussed and theoretical examples are studied to help understand the coupled evolution of bars and waves in the field.

Surface water waves over a shallow canopy

2015 ◽  
Vol 768 ◽  
pp. 572-599 ◽  
Author(s):  
Benlong Wang ◽  
Xiaoyu Guo ◽  
Chiang C. Mei
Keyword(s):  
Water Waves ◽  
Wave Attenuation ◽  
Multiple Scale ◽  
Small Scale ◽  
Vegetation Canopy ◽  
Unit Cells ◽  
Surface Water Waves ◽  
Slow Evolution ◽  
Scale Motion ◽  
Vertical Cylinders

The dynamics of water waves passing over a vegetation canopy is modelled theoretically. To simplify the geometry, we examine a periodic array of vertical cylinders fixed on a slowly varying seabed. The macroscale behaviour of wave attenuation is predicted based on microscale dynamics between plants. Interstitial turbulence is modelled by Reynolds equations with a locally constant eddy viscosity determined by energy considerations. Using the asymptotic method of multiple-scale expansions, the slow evolution of waves is derived by considering the coupling with the small-scale motion in the canopy. After numerical solution of the canonical boundary-value problem in a few unit cells, predictions of macroscale effects such as wave attenuation are made and compared with laboratory experiments. The counteracting effects of shoaling and dissipation are discussed for different vegetation densities.

Non-linear dispersion of water waves

1967 ◽  
Vol 27 (2) ◽  
pp. 399-412 ◽  
Author(s):  
G. B. Whitham
Keyword(s):  
Water Waves ◽  
Wave Train ◽  
Periodic Wave ◽  
Wave Number ◽  
Amplitude Wave ◽  
Linear Dispersion ◽  
Elliptic Case ◽  
Non Linear ◽  
Wave Trains ◽  
Linear Water Waves

The slow dispersion of non-linear water waves is studied by the general theory developed in an earlier paper (Whitham 1965b). The average Lagrangian is calculated from the Stokes expansion for periodic wave trains in water of arbitrary depth. This Lagrangian can be used for the various applications described in the above reference. In this paper, the crucial question of the ‘type’ of the differential equations for the wave-train parameters (local amplitude, wave-number, etc.) is established. The equations are hyperbolic or elliptic according to whetherkh0is less than or greater than 1.36, wherekis the wave-number per 2π andh0is the undisturbed depth. In the hyperbolic case, changes in the wave train propagate and the characteristic velocities give generalizations of the linear group velocity. In the elliptic case, modulations in the wave train grow exponentially and a periodic wave train will be unstable in this sense; thus, periodic wave trains on water will be unstable ifkh0> 1·36, The instability of deep-water waves,kh0> 1·36, was discovered in a different way by Benjamin (1966). The relation between the two approaches is explained.

On the modulation of water waves in the neighbourhood of kh ≈ 1.363

1977 ◽  
Vol 357 (1689) ◽  
pp. 131-141 ◽  
Keyword(s):  
Schrödinger Equation ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Multiple Scales ◽  
Wave Interaction ◽  
Periodic Wave ◽  
Stokes Wave ◽  
Modulation Equation ◽  
Wave Trains ◽  
The Stability

In 1967, T. Brooke Benjamin showed that periodic wave-trains on the surface of water could be unstable. If the undisturbed depth is h , and k is the wavenumber of the fundamental, then the Stokes wave is unstable if kh ≥ σ 0 , where σ 0 ≈ 1.363. The instability is provided by the growth of waves with a wavenumber close to k . This result is associated with an almost resonant quartet wave interaction and can be obtained by examining the cubic nonlinearity in the nonlinear Schrodinger equation for the modulation of harmonic water waves: this term vanishes at kh = cr0. In this paper the multiple-scales technique is adapted in order to derive the appropriate modulation equation for the amplitude of the fundamental when kh is near to σ 0 . The resulting equation takes the form i A T - a 1 A ζζ - a 2 A | A | 2 + a 3 A | A | 4 + i( a 4 | A | 2 A ζ - a 5 A (| A | 2 ) ζ ) - a 6 Aψ T = 0 where ψ ζ = | A | 2 , and the a i are real numbers. [Coefficients a 3 - a 6 are given on kh ≈ 1.363 only.] This equation is uniformly valid in that it reduces to the classical non-linear Schrödinger equation in the appropriate limit and is correct when a 2 = 0, i.e. at kh = σ 0 . The equation is used to examine the stability of the Stokes wave and the new inequality for stability is derived: this now depends on the wave amplitude. If the wave is unstable then it is expected that soli to ns will be produced: the simplest form of soliton is therefore examined by constructing the corresponding ordinary differential equation. Some comments are made concerning the phase-plane of this equation, but more analytical details are extracted by treating the new terms as perturbations of the classical Schrodinger soliton. It is shown that the soliton is both flatter (symmetrically) and skewed forward, although the skewing eventually gives way to an oscillation above the mean level.

scholarly journals Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

2021 ◽  
Vol 5 (3) ◽  
pp. 88
Author(s):  
Supaporn Kaewta ◽  
Sekson Sirisubtawee ◽  
Sanoe Koonprasert ◽  
Surattana Sungnul
Keyword(s):  
Exact Solutions ◽  
Fiber Optics ◽  
Soliton Solution ◽  
Water Waves ◽  
Evolution Equations ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
New Exact Solutions

The core objective of this article is to generate novel exact traveling wave solutions of two nonlinear conformable evolution equations, namely, the (2+1)-dimensional conformable time integro-differential Sawada–Kotera (SK) equation and the (3+1)-dimensional conformable time modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation using the (G′/G2)-expansion method. These two equations associate with conformable partial derivatives with respect to time which the former equation is firstly proposed in the form of the conformable integro-differential equation. To the best of the authors’ knowledge, the two equations have not been solved by means of the (G′/G2)-expansion method for their exact solutions. As a result, some exact solutions of the equations expressed in terms of trigonometric, exponential, and rational function solutions are reported here for the first time. Furthermore, graphical representations of some selected solutions, plotted using some specific sets of the parameter values and the fractional orders, reveal certain physical features such as a singular single-soliton solution and a doubly periodic wave solution. These kinds of the solutions are usually discovered in natural phenomena. In particular, the soliton solution, which is a solitary wave whose amplitude, velocity, and shape are conserved after a collision with another soliton for a nondissipative system, arises ubiquitously in fluid mechanics, fiber optics, atomic physics, water waves, and plasmas. The method, along with the help of symbolic software packages, can be efficiently and simply used to solve the proposed problems for trustworthy and accurate exact solutions. Consequently, the method could be employed to determine some new exact solutions for other nonlinear conformable evolution equations.

scholarly journals Multiple stopbands and wavefield asymmetry of surface water waves in non-Bragg structures

AIP Advances ◽  
2021 ◽  
Vol 11 (1) ◽  
pp. 015215
Author(s):  
Joshua-Masinde Kundu ◽  
Ting Liu ◽  
Jia Tao ◽  
Jia-Yi Zhang ◽  
Ya-Xian Fan ◽  
...  
Keyword(s):  
Surface Water ◽  
Water Waves ◽  
Bragg Structures ◽  
Surface Water Waves
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