scholarly journals Generation, Transformation, and Scattering of Long Waves Induced by a Short-Wave Group over Finite Topography

2011 ◽  
Vol 41 (10) ◽  
pp. 1842-1859 ◽  
Author(s):  
Qingping Zou
Keyword(s):  
Analytical Solutions ◽  
Wave Train ◽  
Short Wave ◽  
Long Waves ◽  
Wave Group ◽  
Wave Groups ◽  
Long Wave ◽  
Wave Orbital Velocity ◽  
Variable Water ◽  
Horizontal Bottom

Abstract Second-order analytical solutions are constructed for various long waves generated by a gravity wave train propagating over finite variable depth h(x) using a multiphase Wentzel–Kramers–Brillouin (WKB) method. It is found that, along with the well-known long wave, locked to the envelope of the wave train and traveling at the group velocity Cg, a forced long wave and free long waves are induced by the depth variation in this region. The forced long wave depends on the depth derivatives hx and hxx and travels at Cg, whereas the free long waves depend on h, hx, and hxx and travel in the opposite directions at . They interfere with each other and generate free long waves radiating away from this region. The author found that this topography-induced forced long wave is in quadrature with the short-wave group and that a secondary long-wave orbital velocity is generated by variable water depth, which is in quadrature with its horizontal bottom counterpart. Both these processes play an important role in the energy transfer between the short-wave groups and long waves. These behaviors were not revealed by previous studies on long waves induced by a wave group over finite topography, which calculated the total amplitude of long-wave components numerically without consideration of the phase of the long waves. The analytical solutions here also indicate that the discontinuity of hx and hxx at the topography junctions has a significant effect on the scattered long waves. The controlling factors for the amplitudes of these long waves are identified and the underlying physical processes systematically investigated in this presentation.

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

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.

Long waves induced by short-wave groups over an uneven bottom

1984 ◽  
Vol 139 ◽  
pp. 219-235 ◽  
Author(s):  
Chiang C. Mei ◽  
Chakib Benmoussa
Keyword(s):  
Group Velocity ◽  
Short Wave ◽  
Long Waves ◽  
Finite Region ◽  
Variable Depth ◽  
Wave Groups ◽  
One Dimensional ◽  
Uneven Bottom ◽  
Short Waves ◽  
Horizontal Bottom

Unidirectional and periodically modulated short waves on a horizontal or very nearly horizontal bottom are known to be accompanied by long waves which propagate together with the envelope of the short waves at their group velocity. However, for variable depth with a horizontal lengthscale which is not too great compared with the group length, long waves of another kind are further induced. If the variation of depth is only one-dimensional and localized in a finite region, then the additional long waves can radiate away from this region, in directions which differ from those of the short waves and their envelopes. There are also critical depths which define caustics for these new long waves but not for the short waves. Thus, while obliquely incident short waves can pass over a topography, these second-order long waves may be trapped on a ridge or away from a canyon.

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

2006 ◽  
Vol 462 (2070) ◽  
pp. 1853-1876 ◽  
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.

scholarly journals Note on the generation of long gravity waves by breaking and shoaling of short-wave groups in gently-sloping beaches: the long-wave similarity parameter

2018 ◽  
Author(s):  
Juan Felipe Paniagua-Arroyave
Keyword(s):  
Gravity Waves ◽  
Short Wave ◽  
Long Waves ◽  
Similarity Parameter ◽  
Wave Groups ◽  
Long Wave ◽  
Wave Conditions

It is proposed a long-wave similarity parameter based on the surf-beat similarity and Ursell parameters. By including the Ursell number, the long-wave similarity allows distinguishing between breaking and shoaling generation of long-waves for conditions rendering similar values of surf-beat similarity. The proposed parameter is tested with three cases of wave conditions published in the literature, with promising results.

Infragravity waves induced by short-wave groups

1993 ◽  
Vol 247 ◽  
pp. 551-588 ◽  
Author(s):  
Hemming A. Schäffer
Keyword(s):  
Surf Zone ◽  
Break Point ◽  
Short Wave ◽  
Radiation Stress ◽  
Wave Groups ◽  
Infragravity Waves ◽  
Short Waves ◽  
Long Wave ◽  
Integrated Conservation ◽  
Infragravity Wave

A theoretical model for infragravity waves generated by incident short-wave groups is developed. Both normal and oblique short-wave incidence is considered. The depth-integrated conservation equations for mass and momentum averaged over a short-wave period are equivalent to the nonlinear shallow-water equations with a forcing term. In linearized form these equations combine to a second-order long-wave equation including forcing, and this is the equation we solve. The forcing term is expressed in terms of the short-wave radiation stress, and the modelling of these short waves in regard to their breaking and dynamic surf zone behaviour is essential. The model takes into account the time-varying position of the initial break point as well as a (partial) transmission of grouping into the surf zone. The former produces a dynamic set-up, while the latter is equivalent to the short-wave forcing that takes place outside the surf zone. These two effects have a mutual dependence which is modelled by a parameter K, and their relative strength is estimated. Before the waves break, the standard assumption of energy conservation leads to a variation of the radiation stress, which causes a bound, long wave, and the shoaling bottom results in a modification of the solution known for constant depth. The respective effects of this incident bound, long wave and of oscillations of the break-point position are shown to be of the same order of magnitude, and they oppose each other to some extent. The transfer of energy from the short waves to waves at infragravity frequencies is analysed using the depth-integrated conservation equation of energy. For the case of normally incident groups a semi-analytical steady-state solution for the infragravity wave motion is given for a plane beach and small primary-wave modulations. Examples of the resulting surface elevation as well as the corresponding particle velocity and mean infragravity-wave energy flux are presented. Also the sensitivity to the variation of input parameters is analysed. The model results are compared with laboratory experiments from the literature. The qualitative agreement is good, but quantitatively the model overestimates the infragravity wave activity. This can, in part, be attributed to the neglect of frictional effects.

scholarly journals WAVE GROUPINESS AS A SOURCE OF NEARSHORE LONG WAVES

1986 ◽  
Vol 1 (20) ◽  
pp. 38 ◽  
Author(s):  
Jeffrey H. List
Keyword(s):  
Wave Amplitude ◽  
Cross Correlation ◽  
Surf Zone ◽  
Low Frequency ◽  
Long Waves ◽  
Progressive Wave ◽  
Wave Groups ◽  
Long Wave ◽  
Cross Correlation Analysis ◽  
Limited Usefulness

Data from a low energy swell-dominated surf zone are examined for indications that observed low frequency motions are simply group-forced bounded long waves. Time series of wave amplitude are compared to filtered long wave records through cross-spectral and cross-correlation analysis. These methods are found to have limited usefulness until long waves are separated into seaward and shoreward components. Then a clear picture of a rapidly shoaling bounded long wave emerges, with a minimum of nearly one fourth of the long wave amplitude being explainable by this type of motion close to shore. Through the zone in which waves were breaking, and incident wave amplitude variability decreased by 50%, the contribution from the bounded long wave continued to increase at a rate much greater than a simple shoaling effect. Also present are clear signs that this amplified bounded long wave is reflected from a position close to the shoreline, and is thus released from wave groups as a free, offshore-progressive wave.

A stochastic model of sea-surface roughness. II

1991 ◽  
Vol 435 (1894) ◽  
pp. 405-422 ◽  
Keyword(s):  
Wave Energy ◽  
Short Wave ◽  
Long Waves ◽  
Scale Model ◽  
Short Waves ◽  
Long Wave ◽  
Proportional Rate ◽  
The Mean ◽  
Sheltering Effect ◽  
Increasing Function

A two-scale model of a wind-ruffled surface is developed which includes (1) modulation of the short waves by orbital straining in the long waves, (2) dissipation of short-wave energy by breaking, and (3) regeneration of the short-wave energy by the wind. For simplicity the long waves are at first assumed to be uniform. It is shown that the character of the surface is governed by the parameter Ω = (β/σγKA ), where β is the proportional rate of short-wave growth due to the wind, σ , K and A are the long-wave frequency wavenumber and amplitude, and γ = 2.08. When Ω < 1 the short waves break over only part of the long-wave surface. When Ω ≽ 1 they break everywhere. The mean-square steepness s 2 ¯ of the short waves is an increasing function of β/σ , but a decreasing function of the long-wave steepness AK . The phase angle between s 2 ¯ and the long-wave elevation η is an increasing function of Ω . The correlation between s 2 ¯ and η is largest when Ω ≪1, but tends to 0 as Ω → 1. The simple model is extended to the case when the long-wave amplitude A has a Rayleigh probability density. To take account of the ‘sheltering ’ effect of high waves we compute the case when any two successive waves have a bivariate Rayleigh density. The application of the model to laboratory and field data is discussed.

Transmission of Long Waves Induced by Short-Wave Groups Through a Composite Breakwater

2001 ◽  
Vol 43 (2) ◽  
pp. 83-97 ◽  
Author(s):  
Akter Hossain ◽  
Wataru Kioka ◽  
Toshikazu Kitano
Keyword(s):  
Short Wave ◽  
Long Waves ◽  
Wave Groups

On the stability of crystal lattices VII. Long-wave and short-wave stability for the face-centred cubic lattice

1942 ◽  
Vol 38 (1) ◽  
pp. 61-66 ◽  
Author(s):  
S. C. Power
Keyword(s):  
Short Wave ◽  
Long Waves ◽  
Three Dimensions ◽  
Linear Lattice ◽  
Crystal Lattices ◽  
Cubic Lattice ◽  
Wave Stability ◽  
Long Wave ◽  
The Face ◽  
The Stability

It is shown that the theorem stated in Born's paper, and proved for the case of a linear lattice of N equal particles under certain restrictions concerning the forces between the particles, that macroscopic stability (stability for long waves) implies microscopic stability, may be extended to three dimensions for the particular case of a face-centred cubic lattice, where the effects of all neighbours, other than the first twelve neighbours, are neglected.I take this opportunity of expressing my sincere thanks to Prof. Born for much valuable advice.

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