scholarly journals SHOALING OF CNOIDAL WAVES

1972 ◽  
Vol 1 (13) ◽  
pp. 18 ◽  
Author(s):  
I.A. Svendson ◽  
O. Brink-Kjaer
Keyword(s):  
Wave Train ◽  
Wave Theory ◽  
Cnoidal Wave ◽  
Sinusoidal Wave ◽  
Cnoidal Waves ◽  
Numerical Examples ◽  
Sloping Bottom

An equation is derived which governs the propagation of a cnoidal wave train over a gently sloping bottom. The equation is solved numerically, the solution being tabulated in terms of fH (Eq. 47) as a function of Ei = (Etr/pg) 1/3/gT2 and hi = h/gT2. Results are compared with sinusoidal wave theory. Two numerical examples are included.

scholarly journals CNOIDAL WAVES IN SHALLOW WATER

1964 ◽  
Vol 1 (9) ◽  
pp. 1
Author(s):  
Frank D. Masch
Keyword(s):  
Shallow Water ◽  
Power Transmission ◽  
Wave Train ◽  
Wave Theory ◽  
Finite Amplitude ◽  
Long Waves ◽  
Practical Application ◽  
Cnoidal Waves ◽  
Water Power ◽  
The Waves

The propagation of long waves of finite amplitude in water with depth to wavelength ratios less than about one-tenth and greater than about one-fiftieth can be described by cnoidal wave theory. To date little use has been made of the theory because of the difficulties involved in practical application. This paper presents the theory necessary for predicting the transforming characteristics of long waves based on cnoidal theory. Basically the method involves calculating the power transmission for a wave train m shallow water from cnoidal theory and equating this to the deep water power transmission assuming no reflections or loss of energy as the waves move into shoaling water. The equations for wave power have been programmed for the range of cnoidal waves, and the results are plotted in non-dimensional form.

Self-Acceleration in the Parameterization of Orographic Gravity Wave Drag

2010 ◽  
Vol 67 (8) ◽  
pp. 2537-2546 ◽  
Author(s):  
John F. Scinocca ◽  
Bruce R. Sutherland
Keyword(s):  
Gravity Wave ◽  
Middle Atmosphere ◽  
Wave Train ◽  
Leading Edge ◽  
Wave Theory ◽  
Wave Drag ◽  
Tuning Parameter ◽  
Mean Flow ◽  
Propagating Waves ◽  
The Impact

Abstract A new effect related to the evaluation of momentum deposition in conventional parameterizations of orographic gravity wave drag (GWD) is considered. The effect takes the form of an adjustment to the basic-state wind about which steady-state wave solutions are constructed. The adjustment is conservative and follows from wave–mean flow theory associated with wave transience at the leading edge of the wave train, which sets up the steady solution assumed in such parameterizations. This has been referred to as “self-acceleration” and it is shown to induce a systematic lowering of the elevation of momentum deposition, which depends quadratically on the amplitude of the wave. An expression for the leading-order impact of self-acceleration is derived in terms of a reduction of the critical inverse Froude number Fc, which determines the onset of wave breaking for upwardly propagating waves in orographic GWD schemes. In such schemes Fc is a central tuning parameter and typical values are generally smaller than anticipated from conventional wave theory. Here it is suggested that self-acceleration may provide some of the explanation for why such small values of Fc are required. The impact of Fc on present-day climate is illustrated by simulations of the Canadian Middle Atmosphere Model.

Effects of Cnoidal Wave-Induced Seepage on Vertical Breakwater Resting on Permeable Elastic Seabed

2014 ◽  
Vol 716-717 ◽  
pp. 284-288
Author(s):  
Jian Kang Yang ◽  
Hua Huang ◽  
Lin Guo ◽  
Jing Rong Lin ◽  
Qing Yong Zhu ◽  
...  
Keyword(s):  
Shallow Water ◽  
Wave Theory ◽  
Cnoidal Wave ◽  
Load Prediction ◽  
Wave Load ◽  
Seepage Pressure ◽  
Theoretical Investigations ◽  
Nonlinearity Effect ◽  
Wave Nonlinearity ◽  
Wave Induced

Theoretical investigations on cnoidal waves interacting with breakwater resting on permeable elastic seabed are presented in this paper. Based on the shallow water reflected wave theory and Biot consolidation theory on wave-induced seepage pressure, the analytical solutions to first order cnoidal wave reflection and wave-induced seepage pressure are obtained by the eigenfunction expansion approach. Numerical results are presented to show the effects of depth of water, breakwater geometry on cnoidal wave-induced seepage uplift force and overturning moment. Compared with Airy wave theory, in certain shallow water conditions, the shallow water wave theory can more effectively illustrate wave nonlinearity effect in wave load prediction.

Stability of periodic waves in shallow water

1974 ◽  
Vol 66 (1) ◽  
pp. 81-96 ◽  
Author(s):  
P. J. Bryant
Keyword(s):  
Shallow Water ◽  
Wave Train ◽  
Periodic Wave ◽  
Fundamental Wave ◽  
Cnoidal Waves ◽  
Margin Of Stability ◽  
Fundamental Wavelength ◽  
Stokes Waves ◽  
Wave Trains ◽  
Permanent Shape

Waves of small but finite amplitude in shallow water can occur as periodic wave trains of permanent shape in two known forms, either as Stokes waves for the shorter wavelengths or as cnoidal waves for the longer wavelengths. Calculations are made here of the periodic wave trains of permanent shape which span uniformly the range of increasing wavelength from Stokes waves to cnoidal waves and beyond. The present investigation is concerned with the stability of such permanent waves to periodic disturbances of greater or equal wavelength travelling in the same direction. The waves are found to be stable to infinitesimal and to small but finite disturbances of wavelength greater than the fundamental, the margin of stability decreasing either as the fundamental wave becomes more nonlinear (i.e. contains more harmonics), or as the wavelength of the periodic disturbance becomes large compared with the fundamental wavelength. The decreasing margin of stability is associated with an increasing loss of spatial periodicity of the wave train, to the extent that small but finite disturbances can cause a form of interaction between consecutive crests of the disturbed wave train. In such a case, a small but finite disturbance of wavelength n times the fundamental wavelength converts the wave train into n interacting wave trains. The amplitude of the disturbance subharmonic is then nearly periodic, the time scale being the time taken for repetitions of the pattern of interactions. When the disturbance is of the same wavelength as the permanent wave, the wave is found to be neutrally stable both to infinitesimal and to small but finite disturbances.

scholarly journals BREAKWATER ARMOR DISPLACEMENT THRESHOLDS: A POSSIBLE CORRELATION WITH CUMULATIVE WAVE ENERGY

1984 ◽  
Vol 1 (19) ◽  
pp. 186
Author(s):  
Daniel L. Behnke ◽  
Frederic Raichlen
Keyword(s):  
Wave Energy ◽  
Wave Train ◽  
Wave Length ◽  
Simple Wave ◽  
Wave Theory ◽  
Irregular Waves ◽  
Reconstruction Technique ◽  
Regular Wave ◽  
Regular Waves ◽  
Three Dimensional Model

An extensive program of stability experiments in a highly detailed three-dimensional model has recently been completed to define a reconstruction technique for a damaged breakwater (Lillevang, Raichlen, Cox, and Behnke, 1984). Tests were conducted with both regular waves and irregular waves from various directions incident upon the breakwater. In comparison of the results of the regular wave tests to those of the irregular wave tests, a relation appeared to exist between breakwater damage and the accumulated energy to which the structure had been exposed. The energy delivered per wave is defined, as an approximation, as relating to the product of H2 and L, where H is the significant height of a train of irregular waves and L is the wave length at a selected depth, calculated according to small amplitude wave theory using a wave period corresponding to the peak energy of the spectrum. As applied in regular wave testing, H is the uniform wave height and L is that associated with the period of the simple wave train. The damage in the model due to regular waves and that caused by irregular waves has been related through the use of the cumulative wave energy contained in those waves which have an energy greater than a threshold value for the breakwater.

scholarly journals Discrete spiral modes in disk galaxies: Some numerical examples based on density wave theory

1977 ◽  
Vol 74 (11) ◽  
pp. 4726-4729 ◽  
Author(s):  
G. Bertin ◽  
Y. Y. Lau ◽  
C. C. Lin ◽  
J. W.- K. Mark ◽  
L. Sugiyama
Keyword(s):  
Density Wave ◽  
Wave Theory ◽  
Disk Galaxies ◽  
Numerical Examples ◽  
Density Wave Theory

On the excitation of edge waves on beaches

1977 ◽  
Vol 79 (2) ◽  
pp. 273-287 ◽  
Author(s):  
A. A. Minzoni ◽  
G. B. Whitham
Keyword(s):  
Shallow Water ◽  
Incident Wave ◽  
Water Wave ◽  
Wave Train ◽  
Wave Theory ◽  
Edge Waves ◽  
Shallow Water Theory ◽  
Shallow Water Approximation ◽  
Complete Discussion ◽  
Standing Edge Waves

The excitation of standing edge waves of frequency ½ω by a normally incident wave train of frequency ω has been discussed previously (Guza & Davis 1974; Guza & Inman 1975; Guza & Bowen 1976) on the basis of shallow-water theory. Here the problem is formulated in the full water-wave theory without making the shallow-water approximation and solved for beach angles β = π/2N, where N is an integer. The work confirms the shallow-water results in the limit N [Gt ] 1, shows the effect of larger beach angles and allows a more complete discussion of some aspects of the problem.

Directional illumination analysis using beamlet decomposition and propagation

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. S147-S159 ◽  
Author(s):  
Ru-Shan Wu ◽  
Ling Chen
Keyword(s):  
Wavelet Transform ◽  
Wave Theory ◽  
Image Space ◽  
Low Velocity ◽  
Numerical Examples ◽  
Depth Migration ◽  
Local Angle ◽  
Spatial Axis ◽  
Local Image

We evaluate directional illumination (DI) and acquisition-aperture efficacy through wave theory-based beamlet decomposition of the wavefield. Beamlet decomposition (wavelet transform along spatial axis) provides localizations in both space and direction of a wavefield. We introduce the image conditions in beamlet domain and local angle domain and then define the local image matrix (LIM). We calculate the DI in the image space for a given source or a group of sources by decomposing Green’s functions into local angle domain at image points. Acquisition-aperture efficacy (AAE) matrix and acquisition dip-response (ADR) vector can be defined to quantify the efficacy of an acquisition configuration for a given subsurface point. As numerical examples, we calculate the DI maps and ADR maps for high- and low-velocity lens models and for the SEG-EAGE 2D salt model. We further investigate the influences of acquisition geometry and overlaying structures on the quality of prestack depth migration image for the subsalt area of the SEG-EAGE model. We find that the ADR maps for different dip angles have good correlation with the image qualities of the corresponding reflectors. DI analysis can be used in the aperture correction for image amplitude in local angle domain for wave theory-based migration methods.

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