Showing papers on "Cnoidal wave published in 1981"

Some exact solutions to the nonlinear shallow-water wave equations

Abstract: These exact solutions correspond to time-dependent motions in parabolic basins. A characteristic feature is that the shoreline is not fixed. It is free to move and must be determined as part of the solution. In general, the motion is oscillatory and has the appropriate small-amplitude limit. For the case in which the parabolic basin reduces to a flat plane, there is a solution for a flood wave. These solutions provide a valuable test for numerical models of inundating storm tides.

Near-limiting gravity waves in water of finite depth

Abstract: Progressive, irrotational gravity waves of constant form exist as a two-parameter family. The first parameter, the ratio of mean depth to wavelength, varies from zero (the solitary wave) to infinity (the deep-water wave). The second parameter, the wave height or amplitude, varies from zero (the infinitesimal wave) to a limiting value dependent on the first parameter. For limiting waves the wave crest ceases to be rounded and becomes angled, with an included angle of 120°. Most methods of calculating finite-amplitude waves use either a form of series expansion or the solution of an integral equation. For waves nearing the limiting amplitude many terms (or nodal points) are needed to describe the wave form accurately. Consequently the accuracy even of recent solutions on modern computers can be improved upon, except at the deep-water end of the range. The present work extends an integral equation technique used previously in which the angled crest of the limiting wave is included as a specific term, derived from the well known Stokes corner flow. This term is now supplemented by a second term, proposed by Grant in a study of the flow near the crest. Solutions comprising 80 terms at the shallow-water end of the range, reducing to 20 at the deep-water end, have defined many field and integral properties of the flow to within 1 to 2 parts in 106. It is shown that without the new crest term this level of accuracy would have demanded some hundreds of terms while without either crest term many thousands of terms would have been needed. The practical limits of the computing range are shown to correspond, to working accuracy, with the theoretical extremes of the solitary wave and the deep-water wave. In each case the results agree well with several previous accurate solutions and it is considered that the accuracy has been improved. For example, the height: depth ratio of the solitary wave is now estimated to be 0.833 197 and the height: wavelength ratio of the deep-water wave to be 0.141063. The results are presented in detail to facilitate further theoretical study and early practical application. The coefficients defining the wave motion are given for 22 cases, five of which, including the two extremes, are fully documented with tables of displacement, velocity, acceleration, pressure and time. Examples of particle orbits and drift profiles are presented graphically and are shown for the extreme waves to agree very closely with simplified calculations by Longuet-Higgins. Finally, the opportunity has been taken to calculate to greater accuracy the long-term Lagrangian-mean angular momentum of the maximum deep-water wave, according to the recent method proposed by Longuet-Higgins, with the conclusion that the level of action is slightly above the crest.

Elevation and velocity measurements of laboratory shoaling waves

Abstract: 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.

On the theory of nonlinear wave-wave interactions among geophysical waves

Abstract: The problem of nonlinear wave-wave interactions is reformulated, in a Eulerian framework, for two classical geophysical systems: barotropic Rossby waves and internal gravity waves on a vertical plane. The departure of the dynamical fields from the equilibrium state is expanded in the linear-problem eigenfunctions, using their properties of orthogonality and completeness. The system is then completely described by the expansion amplitudes, whose evolution is controlled by a system of equations (with quadratic nonlinearity) which is an exact representation of the original model equations. There is no a priori need for the usual multiple-time-scale analysis, or any other perturbation expansion, to develop the theory. These or other approximations (like truncation of the expansion basis or the Boltzmann equation for a stochastic description) can, if desired, be performed afterwards.The evolution of the system is constrained mainly by the conservation of energy E and pseudo-momentum P, properties related to time and space homogeneity of the model equations. Conservation of E and P has, in turn, some interesting consequences: (a) a generalization of Fjortoft's theorem, (b) a class of exact nonlinear solutions (which includes the system of one single wave), and (c) conservation of E and P in an arbitrarily truncated system (which is useful in the development of approximations of the problem).The properties of all possible resonant triads are shown and used to estimate the order of magnitude of off-resonant coupling coefficients.The results are used in two different problems: the stability of a single wave (maximum growth rates are evaluated in both the strong- and weak-interactions limits) and the three-wave system. The general solution (for any initial condition and for both the resonant and off-resonant cases) of the latter is presented.

A Direct Approach to Multi-Periodic Wave Solutions to Nonlinear Evolution Equations

Abstract: The direct method of calculating the multi-periodic wave solutions by Nakamura are applied to the following equations; the Korteweg-de Vrics equation, the Toda equation, the Sawada-Kotera equation and the model equation for shallow water waves, and are obtained three-periodic wave solutions expressed in terms of Riemann's θ-function. All parameters determining the characters of the wave are obtained numerically.

Solitary wave solutions for a model of the two-way propagation of water waves in a channel

Abstract: Bona and Smith (6) have suggested that the coupled system of equations has the same formal justification as other Boussinesq-type models for the two-way propagation of one-dimensional water waves of small but finite amplitude in a channel with a flat bottom. The variables u and η represent the velocity and elevation of the free surface, respectively. Using the energy invariant they show that for a restricted, but nevertheless physically relevant, class of initial data, the system (1·1) has solutions which exist for all time, and that in such circumstances the wave height is bounded solely in terms of the initial data.

Convergence of periodic wavetrains in the limit of large wavelength

Abstract: The Korteweg-de Vries equation was originally derived as a model for unidirectional propagation of water waves. This equation possesses a special class of traveling-wave solutions corresponding to surface solitary waves. It also has permanent-wave solutions which are periodic in space, the so-called cnoidal waves. A classical observation of Korteweg and de Vries was that the solitary wave is obtained as a certain limit of cnoidal wavetrains.

Solitary waves in a compressible fluid

Abstract: Evolution equations for long nonlinear internal waves in a compressible fluid are derived, with the aim of comparing these equations with their counterparts in an incompressible fluid Both the Korteweg-de Vries equation, and the deep fluid equation are discussed, for both dry and moist atmospheres It is shown that the effects of compressibility, or non-Boussinesq terms, are generally small, but measurable, and are manifested mainly in the nonlinear term of the evolution equation For the case of a moist atmosphere the effect of a gain in energy by latent heat release is compared with the energy lost by radiation damping

Langmuir solitons in a two-temperature plasma

Abstract: The existence of finite-amplitude Langnwir solitary waves in a two-electron-temperature plastia is investigated. A now type of soliton, in which the density depression and the electric field amplitude scale in the same manner, and which travels at the effective sound speed, has been found.

Horizontal particle velocities in long waves

Abstract: 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.

High Solitary Waves in Water: Results of Calculations,

Abstract: : This report presents results of calculations of certain properties of solitary waves in water as a function of wave strength, with emphasis on the highest and other high waves. A preceding report outlines the method used to arrive at the results presented here. The method involves a Fourier series solution of the flow field at various (numerical) resolutions to arrive at an estimate of limits as the resolution becomes infinitesimal. Values of the properties of intermediate and high waves possess 4 to 6 significant figure accuracies. For some solitary wave parameters, the calculations agree well with those found using independent methods and verify that the total energy and other integral properties reach a maximum at strengths less than that of the highest wave. They also verify that the maximum angle of surface inclination can exceed 30 deg. for the not-quite-highest waves. In addition, this report gives solitary wave properties useful in constructing periodic wave solutions by the superimposition of solitary waves. The highest wave in water is higher than previously reported; its amplitude to depth ratio is 0.8332. (Author)

Simple solution method for the Korteweg–de Vries equation

Abstract: A simple method is presented to solve the Korteweg–de Vries equation. If one assumes that the solution can be written as a product of two functions, it is shown that each of them can be related to the same Schrodinger equation. The well‐known multisoliton solution then appears in a convenient way, if one deals with discrete eigenvalues.

Nonlinear Dispersive Waves on the Surface of a Self Gravitating Fluid Layer

Abstract: The evolution of two dimensional wave packets on the surface of a self-gravitating fluid layer is investigated and shown to be governed by a nonlinear Schrodinger equation. The wave train of finite amplitude is modulationally unstable. Obtained also are the dynamical equations for the second harmonic resonance. The analysis reveals that the general motion consists of both amplitude and phase modulated waves of which the pure phase and amplitude modulated waves, solitary waves, and phase jump are just the special cases.

Wave motion, oscillations and free fall through a medium

Abstract: It is shown that three different types of physical phenomena, i.e., free fall, oscillations and wave motion, can be considered identically on a common base of non-linearity. The present treatments of non-linear differential equations for these phenomena provide good educational materials in the course of physics.

Second-order solution of a nonlinear wave equation

Abstract: The method of multiple scales is used to investigate the nonlinear wave equation u„ — uxx = e(2ux uxx — 2Am,). The purpose of the investigation is to find the second-order solution, and in particular to find an approximate solution which may be used for values of t which are of order E~2.

Modulation Solitons and Internal Waves

Abstract: The propagation of a modulation soliton in a medium with inhomogenities is considered. It is shown that the center of the wave packet obeys a Newtonian force equation with the inhomogenity acting as the force. The model equation is compared with numerical solutions of the inhomogeneous nonlinear Schrodinger equation.