Showing papers on "Cnoidal wave published in 1988"

The anomalous behavior of the runup of cnoidal waves

Abstract: A new solution to the linearized shallow‐water wave equations is introduced for the case of cnoidal waves climbing up a plane beach. The solution is used to calculate the maximum runup. It is shown that the maximum relative runup of cnoidal waves is significantly larger than the runup of monochromatic waves with the same wave height and wavelength far from the shore. It is also shown that the maximum relative runup of cnoidal waves is not a monotonically varying function of the normalized wavelength.

Instability and nonlinear evolution of a density-driven coastal current with a surface front in a two-layer ocean

Abstract: Coupled equations describing the nonlinear evolution of a disturbance in a two-layer coastal current with a surface front are derived by assuming that the alongshore scale of the disturbance is much longer than the current width, the lower layer is much thicker than the upper layer and the potential vorticity in the upper layer is zero. Linear stability and the characteristics of the cnoidal wave and soliton solutions governed by these equations are examined analytically, and the nonlinear evolution of disturbance is discussed using numerical methods. The main results are as follows. (1) The current is linearly stable when the current speed is greater than a certain critical value. (2) Even in such a situation, there is no stable nonlinear solution whose amplitude lies between certain critical values, a c and a′c, which are functions of the wavelength and the basic flow speed. (3) A solitary disturbance whose amplitude lies between a c and a′c grows with an eddy-pair-like structure in the lower l...

Generalized exponential, circular, and hyperbolic functions for nonlinear wave equations

Abstract: Wave functions are presented in the form of generalized exponentials that are solutions of some of the most usual linear and nonlinear wave equations. The solutions are given in terms of the elliptic functions of Jacobi and presented in a form as similar as possible to the usual circular functions. Some simple theorems are demonstrated to present the solutions as the simplest possible extension of the usual exponentials.

Nonlinear fourier analysis of laboratory generated, broad-banded surface waves

Abstract: This paper describes the application of a new data analysis procedure, which the authors refer to as nonlinear Fourier analysis, to broad-banded surface waves measured in the laboratory. Using this approach a nonlinear surface wave train or wave field may be described as a linear superposition of nonlinearly interacting sine waves, Stokes waves, cnoidal weaves and solitons. The theory is based upon the spectral/scattering transform solution to the Korteweg-de Vries equation with periodic boundary conditions. It is shown how the method can be used to analyse broad-banded surface wave data described by the Pierson-Moskowitz spectrum. Certain nonlinear spectral components are interpreted in terms of radiation stress, a low frequency contribution to the spectrum induced by second order nonlinearities in the wave motion.

Coupling stokes and cnoidal wave theories in a nonlinear refraction model

Abstract: 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.

Gauge and Bäcklund transformations for the variable coefficient higher‐order modified Korteweg–de Vries equation

Abstract: A family of higher‐order modified Korteweg–de Vries equations with variable coefficients (t‐ho‐mKdV) is introduced. A one‐to‐one correspondence between a real solution of these equations and a complex solution of the variable coefficient higher‐order Korteweg–de Vries (t‐ho‐KdV) equations is established through a complex Miura transformation. An auto‐Backlund transformation for these t‐ho‐mKdV equations is derived from that of the t‐ho‐KdV equations. The associated gauge transformations of the corresponding AKNS systems are presented. They enable one to construct a hierarchy of solutions of the t‐ho‐mKdV equations from a known hierarchy without solving the differential equations for the wave functions except the first one. A new family of higher‐order evolution equations with an auto‐Backlund transformation is also derived in connection with the gauge transformation of the t‐ho‐mKdV equations.

Theory of wave-trapping in space plasmas

Abstract: 39 A new method of perturbation for the study of nonlinea r wave propagation in a dielectric medium such as a magnetoplasma is proposed. By this method the wave equations which include the effects due to a nonlinearity or an inhomogeneity of the transverse waves propagating along magnetic field lines are rederived and summarized. Among many kinds of wave modulations described by these equations, the condition for wave-trapping in space plasmas is especially discussed in detail.

Nonlinear Wave Equations for Low-Frequency Acoustic Gravity Waves

Abstract: It is shown that low-frequency acoustic gravity waves propagating parallel to the earth's surface satisfy the Korteweg-De Vries equation or the Kadomtsev-Petviashvili equation and have a discrete spectrum of group velocities. The atmosphere is considered to be incompressible, homogeneous in composition and isothermal and the gravitational acceleration depends upon the height. The hydrostatic approximation is used and adapted in such a way that dispersion is not neglected. The nonlinear wave equations are obtained using the reductive perturbation technique. Finally, changes for a compressible atmosphere are discussed.

Integrability of Nonlinear Wave Equations

Abstract: It is now well known that nonlinear evolution equations like the Kortewegde Vries equation (KdV), U, = 61411, + uxXx, (1) is completely integrable.'-' Gardner et al.' first showed that equation 1 is the compatibility condition of two linear equations Lf#J = (a: + u ) 4 = A4 @a) and A4 = (48: + 6ua, + 3 ~ , ) 4 = 4,. The so-called Lax pair,' L and A, satisfies the operator equation L, = [A , L ] = AL LA.