Cnoidal Theory Research Articles

Ricci soliton equation with application to a coupled pendula

The Ricci equation and the soliton solutions are obtained in this paper for two coupled pendula in order to optimise its locomotion and structure. The robot is swinging repeatedly like a rope with successive movement steps along a mobile support. The cnoidal theory and a genetic algorithm are used to solve the problem via the Ricci solitons and the pseudospherical reduction of the rheological Zener equations. The Bäcklund transform is applied to Ricci equation to generate pseudo-spherical surfaces.

Wave-induced seabed response in shallow water

To simulate the wave-induced response of coupled pore fluids and a solid skeleton in shallow water, a set of solutions with different formulations (fully dynamic, partly dynamic, and quasi-static) corresponding to each soil behavior assumption is presented. To deal with Jacobian elliptic functions involved in the cnoidal theory, a Fourier series approximation is adopted for expanding the boundary conditions on the seabed surface. The parametric study indicates the significant effect of nonlinearity for shallow water wave, which also enhances the effect of soil characteristics. The investigation of the applicability of reduced formulations reveals the necessity of a partly or even fully dynamic formulation for the wave-induced seabed response problem in shallow water, especially for thickened seabed. The analysis of liquefaction in the seabed indicates that the maximum depth of liquefaction is shallower, and the width of liquefaction is broader under cnoidal wave loading. The present analytical model can provide more reasonable result for the wave-induced seabed response in the range of shallow water wave.

Numerical Modeling of Waves in the Port of Manzanillo, Mexico by Cnoidal Theory

This paper presents the application of one computer program developed to calculate the physical characteristics of the progressive water waves using cnoidaltheory of Keulegan and Patterson: profile of water surface, wave length, horizontal and vertical orbital velocities, horizontal and vertical orbital accelerations, period, and under water surface pressure. The Turbo-Pascal programming language was used. CNOIDAL program requires the following data to be introduced: depth, height and phase angle of waves, as the depth ‘z’ of the particle to be analyzed. In the first part of the program, the Newton-Raphsonmethod is used to determine the wave length. In its second part, the nine physical characteristics of waves, providing of equations with elliptic Jacobeans functions, are calculated: profile of water surface, celerity, wave length, horizontal and vertical orbital velocity, horizontal and vertical orbital acceleration, under water surface pressure,and period. A measurement of some characteristics of the waves in the port of Manzanillo was realized by twelve months in the year of 2013 and 2014 using the dates obtained by the Transportation Mexican Institute (TMI) in this port. Results after application this computer program validate the application region proposal by Littman and Struik and the experimental works of LéMehauté, Divoky and Lyn in the Tetra Tech wave tank, of 32 meters of length, square across section of 1.219 meters and waves generator with a speed control in a rank from 1 to 12seconds of period and with waves ranks 0.244 m/s2≥ d/T2≥ 0.015 m/s2. These results were compared with the calculus of the program proposed in this work for the free surface profile, speeds and accelerations of the particles and it was observed a very good accurate by the Keulegan and Patterson cnoidal theory.

A function to determine wavelength from deep into shallow water based on the length of the cnoidal wave at breaking

Various methods exist to determine the wavelength of shoaling waves. However, many are applicable to only part of the depth range, while most require tedious calculations or graphical resolutions. It is shown here that the wavelength immediately seaward of the breaker, L b, is equal to two-thirds of the deepwater wavelength L o for fully developed waves breaking over a nearly horizontal bottom. It is also given by T w√[ g(0.5 H b + d b)] for any H o/ L o ratio or bottom slope, where T w is the wave period, g the acceleration due to gravity, H b the breaker height, d b the breaker depth, and H o the deepwater wave height. This corresponds to within 4.5% with the wavelength at this depth according to cnoidal theory, for which simplified equations are presented. These relationships yield a continuous function of the wavelength L w at any depth d, viz. L w = {[ L b T w][ g(0.5 H b + d)] 0.5} 0.5, where d has a maximum value of L o / 2.965. At this depth, L w coincides with the deepwater wavelength given by the standard Airy equation gT w 2 / 2π for any wave period. The wave celerity C b just seaward of the breaker, given by L b / T w, is also equal to the surf bore velocity U s = √[ g(0.5 H b + d b)] immediately after wave breaking, thus maintaining continuity into the surf zone.

Numerical wave tank based on a σ‐transformed finite element inviscid flow solver

AbstractInviscid two‐dimensional free surface wave motions are simulated using a σ‐transformed finite‐element model based on potential theory for irrotational, incompressible fluid flow over a flat fixed bed. The free surface boundary condition is fully non‐linear, and so non‐linear effects up to very high order can be modelled. The σ‐transformation involves linear stretching of the mesh between the bed and free surface. This has two major advantages. First, remeshing due to the moving free surface is avoided. Second, the mesh nodes are aligned vertically, allowing a high order calculation of the free surface vertical velocity component to be implemented without smoothing, except for very steep waves. The model however is further restricted to non‐overturning, non‐breaking waves because of the uniqueness of the σ‐transformation. Excellent agreement is obtained with analytical and alternative numerical data for small amplitude free sloshing in a rectangular tank and forced sloshing in a horizontally base‐excited rectangular tank. At higher amplitudes, non‐linear effects are evident in the simulations by the present numerical model. The model is also able to reproduce steep progressive waves due to a wave‐maker in agreement with Stokes 5th theory, second‐order shallow water waves in agreement with cnoidal theory, and focused wave groups that match the experimental measurements acquired by Baldock et al. [A laboratory study of non‐linear surface waves on water. Phil. Trans. R. Soc. Lond. A 1996; 354: 649–676]. Copyright © 2003 John Wiley & Sons, Ltd.

On the largest wave height in water of constant depth

Experimental evidence of the fact that, both in the laboratory and in the field, the largest wave height to water depth ratio realisable for oscillatory waves propagating in water of constant depth is about 0.55, has been published recently (Nelson, 1985); (Nelson, 1987); (Nelson, 1994). This paper presents various theoretical approaches to estimate this maximum value. In particular, the higher approximations of the Stokes and cnoidal theories give a much higher limiting wave height, close to 0.78 h, which is commonly used in engineering practice. However, the inclusion of higher harmonics, generated by a wave-maker paddle, into the analysis provides maximum wave height less than ≈ 0.6 h, which is in good agreement with observations.

COUPLING STOKES AND CNOIDAL WAVE THEORIES IN A NONLINEAR REFRACTION MODEL

An efficient numerical model is presented for calculating the refraction and shoaling of finite-amplitude waves over an irregular sea bottom. The model uses third-order Stokes wave theory in relatively deep water and second-order cnoidal wave theory in relatively shallow water. It can also be run using combinations of lower-order wave theories, including a pure linear wave mode. The problem of the connection of Stokes and cnoidal theories is investigated, and it is found that the use of second-order rather than first-order cnoidal theory greatly reduces the connection discontinuity. Calculations are compared with physical model measurements of the height and direction of waves passing over an elliptical shoal. The finite-amplitude wave model gives better qualitative and quantitative agreement with the measurements than the linear model.

Application of Stokes, Cnoidal, and Fourier Wave Theories

A consistent framework for the selection and application of higher‐order steady wave theories is presented. Fifth‐order formulations for cnoidal (shallow water) and the corrected Stokes (deep water) wave theories are reviewed, in addition to Fourier approximation theory (deep, transitional, and shallow water). All three theories are developed in a standardized fashion with respect to coordinate transformations, notation, and presentation of results, so as to facilitate their application in engineering practice. A coflowing uniform current is accommodated by all three theories, which is essential in maintaining consistency at higher orders. The cnoidal theory has been specifically extended to include current to fifth order. Consideration is given to the calculation of integral parameters, forces and moments from the O'Brien‐Morison equation, in addition to field velocities, accelerations, and pressures. Comparative predictions from the three theories for several depth and current conditions illustrate characteristic features, predictive capabilities, and limitations of the separate theories.

Intercomparison of near-bottom kinematics by several wave theories and field and laboratory data

This paper summarizes a study to identify the most suitable theories (models) for representing the near-bottom kinematics due to waves and/or currents. The basis for the evaluation of the various models included comparison of models with field and laboratory data and intercomparison of various models. Wave theories evaluated included two versions of the linear wave theory, the Stokes II, III and V order theory, the Cnoidal theory, the Extended Velocity Potential theory and the Stream Function theory.

A NUMERICAL MODEL FOR REFRACTION OF LINEAR AND CNOIDAL WAVES

A numerical model for refraction of linear and cnoidal waves over an arbitrary bottom is presented. The model, which is based on the ray theory of refraction (11), utilizes linear theory in deep and intermediate water and cnoidal theory in shallow water. The model permits one to determine nearshore wave properties using a nonlinear wave theory. Nearshore wave properties such as wave height, celerity, and wave angle determined using the linear-cnoidal refraction model may vary considerably from those determined using a linear refraction model. In general, cnoidal waves refract less, travel faster, and shoal higher than linear waves. This result can have a considerable effect on coastal engineering design, particularly in the areas of coastal structure design and longshore transport computations.

Elevation and velocity measurements of laboratory shoaling waves

Measurements of wave elevation and orbital velocity in the shoaling, breaking, and bore regime of single‐frequency laboratory waves show that third‐order Stokes theory, when energy flux is conserved, predicts the wave height change and harmonic growth in the regime where the Ursell number Ur = (H/ h)/(kh)2 is 0(1) or less. Shoreward of the Stokes region and up to the breakpoint, harmonic amplitudes are well described by the cnoidal theory. It is shown theoretically that a smooth transition regime exists between Stokes and cnoidal regions for waves which eventually break by plunging. The wave profile asymmetry about the vertical plane observed in near‐breaking waves and bores is due to slow changes of phase of the harmonics relative to the primary wave as the wave train shoals. By contrast, only asymmetry about the horizontal plane is possible in the Stokes and cnoidal wave theories, since these classical solutions allow no relative phase shifts between harmonics. Velocity measurements made with hot‐film anemometers show that ‘unorganized’ fluctuations at the bottom under breaking waves are of the order of half the rms amplitude of the wave‐induced ‘organized’ flow. The correlation between surface elevation and bottom velocity under breakers and bores suggests that turbulence contributes more strongly to the unorganized flow at the bottom under plunging than under spilling waves.

Horizontal particle velocities in long waves

A general expression is derived for the horizontal particle velocities in long waves. The derivation is based on a polynomial expansion in the vertical coordinate z which results in an expression for u in terms of the surface elevation η and its derivatives. The result is valid for high waves but is in the preliminary form not directly applicable. It is shown, however, that for waves of relatively small amplitude to depth ratio the result corresponds to the well‐known cnoidal wave theory. The horizontal velocities predicted by that theory are then critically examined, and it is shown that there are conflicting properties, which imply that no cnoidal theory can be satisfactory for waves higher than 30–40% of the depth of water. Instead, a simple formula based on a parabolic truncation of the above‐mentioned polynomial in z is derived for the velocity under the crest of arbitrary high waves. The results are compared both with experiments and with stream function theory, and, particularly, the latter gives remarkably good agreement even for the highest waves.

EXPERIMENTAL INVESTIGATIONS OF PERIODIC WAVES NEAR BREAKING

The results presented are from a series of experiments of periodic waves approaching breaking on a gently sloping beach. The recorded surface profiles are compared with a cnoidal theory taking into account the effect of the bottom slope. In both theory and experiments a skewness of the profile is observed. A fair agreement is found within certain limits of the wave height to water depth ratio and when the slope is sufficiently gentle. The change in water depth over a wave length must be sufficiently small. The detailed analysis of the surface profiles including the skewness element shows good agreement with the cnoidal theory in cases where the wave height variation is as predicted by cnoidal wave shoaling. The recorded surface profiles together with the recorded phase velocities are further used in computing the wave energy flux based on the theoretical relations from linear and cnoidal theories. Reasonable constancy is observed over the whole length covered by the experiments. Finally the recorded wave set-down is compared with the results of linear and cnoidal theories. When approaching breaking neither of the theories can apparently predict the recorded water level changes.

A high-order cnoidal wave theory

A method is outlined by which high-order solutions are obtained for steadily progressing shallow water waves. It is shown that a suitable expansion parameter for these cnoidal wave solutions is the dimensionless wave height divided by the parameter m of the cn functions: this explicitly shows the limitation of the theory to waves in relatively shallow water. The corresponding deep water limitation for Stokes waves is analysed and a modified expansion parameter suggested.Cnoidal wave solutions to fifth order are given so that a steady wave problem with known water depth, wave height and wave period or length may be solved to give expressions for the wave profile and fluid velocities, as well as integral quantities such as wave power and radiation stress. These series solutions seem to exhibit asymptotic behaviour such that there is no gain in including terms beyond fifth order. Results from the present theory are compared with exact numerical results and with experiment. It is concluded that the fifth-order cnoidal theory should be used in preference to fifth-order Stokes wave theory for wavelengths greater than eight times the water depth, when it gives quite accurate results.

Wave Runup around Large Circular Cylinder

The wave runup around a large circular cylinder is considered. Predictions based on both sinusoidal and cnoidal wave diffraction theories are summarized and compared with experimental data obtained under conditions close to the shallow water range. The cnoidal theory is found to predict the measured runup profiles more closely than does the linear theory, although the measured maximum runups are still significantly greater than either prediction.

The wave height variation for regular waves in shoaling water

The theory of cnoidal wave shoaling has previously been connected to data in deeper water (described by sinusoidal wave theory) by assuming continuity in energy flux E f at the matching point between the two theories (Svendsen and Brink-Kjær, 1972) resulting in a small shift in wave height H at the matching point. In this paper we assume directly continuity in wave height and tables are given from which the wave height variation can be determined when the wave is specified at any water depth (including deep water). It is also shown that a deep water limit for cnoidal waves exists. The nature of this limit and the behaviour of the waves close to it are analysed. Experimental data for waves with no free second harmonic components are compared with a shoaling model based on linear theory for h/L 0 > 0.10 and cnoidal theory for h/L 0 < 0.10 . The agreement is very good for deep water wave steepnesses H 0 /L 0 up to 3–4%. For larger steepnesses the linear theory fails to predict the variation for h/L 0 > 0.10 .

CNOIDAL WAVES IN SHALLOW WATER

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.