Showing papers on "Cnoidal wave published in 1972"

Nonlinear Modulation of Gravity Waves

Abstract: Slow modulation of gravity waves on water layer with uniform depth is investigated by using singular perturbation methods. It is found, to the lowest order of perturbation, that the complicated system of equations governing such modulation can be reduced to a simple nonlinear Schrodinger equation. A nonlinear plane wave solution to this equation is found to correspond to the so-called Stokes wave. The linear stability of this plane wave solution is essentially determined by the sign of the product of two coefficients in this equation, yielding Benjamin and Whitham's criterion. The same equation is found to give a weak cnoidal wave derived from the Korteweg-de Vries equation in the shallow-water limit.

Weak Nonlinear Dispersive Waves: A Discussion Centered Around the Korteweg–De Vries Equation

Abstract: This article is concerned with wave propagation processes where some balance occurs in the competition between a nonlinear effect and a higher order derivative effect which might be of a dispersive or a dissipative nature. The Burgers equation and the Korteweg–de Vries equation, which are prototype scalar nonlinear dissipative and dispersive equations, are shown to be fundamental to this study, even when quite general systems of equations are involved. The role of the solitary wave solution is shown to be central to the study which is applied to gravity waves, plasma waves and to waves in lattices. Both steady state solutions and initial value problems are reviewed together with questions of stability, existence and uniqueness.

Airy wave theory and breaker height prediction

Abstract: Using a critical value for Yt> = H./ h. as a wave breaking criterion, where Hb and hb are respectively the wave breaker height and depth, applying Airy wave theory, and assuming conservation of the wave energy flux, one obtains 1/5 2 2/5 Hb = k g (TH. ) relating Hb to the wave period T and to the deep-water wave height H^ . Three sets of laboratory data and one set of field data yield k = 0.39 for the dimensionless coefficient. The relationship, based on Airy wave theory and empirically fitted to the data, is much more successful in predicting wave breaker heights than is the commonly used equation of Munk, based on solitary wave theory. In addition, the relationship is applicable over the entire practical range of wave steepness values.

The Exact N-Soliton Solution of the Korteweg-de Vries Equation

Abstract: The exact N -soliton solution of the Korteweg-de Vries equation is obtained through the procedure suggested by Gardner, Greene, Kruskal and Miura. From this solution, it is shown that solutions are stable and behave like particles. The collisions are well described by the phase shifts. Explicit calculation of the phase shifts assures the conservation of the total phase shift. This fact turns out to be a special expression for the constant motion of he center of mass.

IMPROVED KORTEWEG--deVRIES EQUATION FOR ION-ACOUSTIC WAVES.

Abstract: The Korteweg‐deVries equation describing nonlinear ion‐acoustic waves in a plasma with finite ion temperature is derived and the temperature dependences of soliton width and speed are obtained.

Some numerical solutions of a variable-coefficient Korteweg-de Vries equation (with applications to solitary wave development on a shelf)

Abstract: Some numerical solutions of a variable-coefficient Korteweg-de Vries equation are presented. This particular equation was derived by the author recently (Johnson 1972) in an attempt to describe the development of a single solitary wave moving onto a shelf. Soliton production on the shelf was predicted and this is confirmed here. Results for two and three solitons are reproduced and two intermediate shelf depths are also considered. In these latter two cases both solitons and an oscillatory wave occur. One of the profiles corresponds to the integrations performed by Madsen & Mei (1969) and a comparison is made.

Shoaling of cnoidal waves

Abstract: 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.

On the N-tuple Wave Solutions of the Korteweg-de Vries Equation

Abstract: Each of such solutions is called a soliton or solitary wave solution. Recently it was discovered that there exist solutions of the KdV equation which behave like superposition of two solitons as t -> ± oo (Kruskal and Zabusky [JT]). The existence and properties of such solutions (called double wave solutions) were studied by Lax |Jf]. The structure of the solution of the KdV equation for the rapidly decreasing initial value was clarified by Gardner, Greene, Kruskal and Miura Q2]. They related the solution u(x, t) to the Schrodinger equation with the potential u (for each t) and found that discrete eigenvalues remain invariant. The reflection coefficient and the normalization coefficients

Solitary waves in concentrated vortices.

Abstract: A nonlinear integrodifferential equation governing finite amplitude wave propagation on concentrated vortices is solved numerically. The solution to the Cauchy problem shows a solitary wave development qualitatively similar to solutions of the Korteweg-de Vries equation. A perturbation solution of the stationary form of the evolution equation confirms the unsteady calculation.

The Interaction of Large Amplitude Shallow Water Waves with an Ambient Shear Flow: Non-Critical Flows.

Abstract: : An analysis is given of the behavior of a gravity wave moving over a horizontal bed into a region where the flow is sheared in a vertical direction. No restriction is placed on the amplitude of the wave. The only assumptions made are that the fluid is inviscid and that the hydrostatic pressure law holds. The results can be used to analyze the behavior of tsunamis and weather fronts. It is shown that even with an arbitrary ambient shear, the variation in height of the free surface in a long progressing wave can be similar to that in a simple wave except that the local wave speed depends on the shear profile. The interaction of such a wave with a uniform shear is described in detail. (Author)

Perturbation Theory of the Nonlinear Plasma Wave with Landau Damping

Abstract: A generalized Korteweg-de Vries type equation is obtained for small but finite amplitude waves in plasma by treating the Vlasov equation perturbation theoretically. The influence of ion Landau damping on the ion sound soliton is discussed. Even if the ion acoustic wave is stable, it is shown that under a certain condition the amplitude of soliton grows in time. In the case of the unstable ion acoustic wave, the amplitude of soliton saturates at the finite value.

Cnoidal wave theory for application to offshore structural design

Abstract: The theory presented in this paper was programmed specifically for engineering calculations. It can easily be combined with any general purpose structural mechanics program for simplified application to offshore structure design. This version of cnoidal theory was obtained by differentiating Keulegan-Patterson's velocity potential. It predicts velocities close to laboratory measurements for large amplitude, shallow water waves. The predicted vertical velocity and both components of acceleration differ from previously published results.

On waves generated by a moving pressure field, II: Cnoidal wave effects

Abstract: In a recent paperHunt andTanner [3]2) investigated the waves generated by a steadily moving two-dimensional pressure distribution, which was zero ahead of the disturbance and a constantp0, tehind it, these regions being joined smoothly by a cubic function. Only those solutions with supercritical flow in both regions were considered, these were found to lead to an asymmetric solitary wave.

Mathematical modeling of large objects in shallow water waves and uniform current

Abstract: A mathematical model is presented which portrays the physical system of a large axially symmetric structure in a flow field of finite water depth, large amplitude wave and strong current. The flow field, which enters as the input, is derived from a velocity potential similar to that of the cnoidal wave of Keulegan and Patterson. The inclusion of a uniform velocity in the derivation of velocity potential results in a cross interference term in addition to the well known Doppler shift effect. The numerical results are compared with experiments on a bridge pier (Ref. 6) which is partially cylindrical with base diameter equivalent to 100 feet in prototype; close to the surface, where the wave action is greatest it is conical. These results are also compared with theoretical calculations based on linear wave theory and fifth-order wave theory. It is concluded that the results based on the modified cnoidal wave theory come closest to the experimental value.