Showing papers on "Cnoidal wave published in 1971"
105 citations
94 citations
TL;DR: Singular perturbation method is applied to a class of nonlinear partial differential equations to reduce it into a single nonlinear equation presenting the wave propagation in an inhomogeneous medium.
Abstract: Singular perturbation method is applied to a class of nonlinear partial differential equations to reduce it into a single nonlinear equation presenting the wave propagation in an inhomogeneous medium. The coupling between nonlinear and dispersive or dissipative effects is shown to determine the order of the coordinate-stretching uniquely. The theory is applied to weak shock wave, shallow water wave and oblique magneto-acoustic wave.
40 citations
TL;DR: In this paper, a class of long planetary waves in a zonal channel analogous to the solitary and cnoidal waves of surface and internal gravity wave theory is discussed, and the wave profile along the channel is found to satisfy the Korteweg-de Vries equation.
Abstract: A class of long planetary waves in a zonal channel analogous to the solitary and cnoidal waves of surface and internal gravity wave theory is discussed. On a mid-latitude β-plane, such waves exist as the result of divergence, non-uniform zonal velocity fields or bottom topography. In all cases studied the wave profile along the channel was found to satisfy the Korteweg-de Vries equation.
38 citations
17 citations
TL;DR: In this paper, necessary conditions on dissipative operators for the existence of shock-like solutions of the Kortewegde Vries equation, which limit the damping tolerable at long wavelengths are derived.
Abstract: Necessary conditions on dissipative operators for the existence of shock‐like solutions of the Korteweg‐de Vries equation, which limit the damping tolerable at long wavelengths are derived.
10 citations
TL;DR: In this article, a general form of the nonlinear wave equation with dispersive and dissipative terms involving small coefficients has been treated for the transition period when the time is around the breakdown time.
Abstract: A general form of the nonlinear wave equation with dispersive and dissipative terms involving small coefficients has been treated for the transition period when the time is around the breakdown time. In the first part of this paper, some relations have been established in order to see how the classical overtaking phenomenon is eventually prevented when the curve has steepened sufficiently. In the second part of this paper, the growth and damping of solitary waves due to the effects of dispersion and dissipation have been analyzed.
01 Nov 1971
TL;DR: In this article, Courant discussed the solution of problems involving the propagation of discontinuities and other singularities for hyperbolic partial differential equations by means of progressing wave expansions and referred to the work of Hadamard, Friedlander, Ludwig and others on this subject.
Abstract: The solution of problems involving the propagation of discontinuities and other singularities for hyperbolic partial differential equations by means of progressing wave expansions is discussed in the book by Courant(l). He also refers to the work of Hadamard, Friedlander, Ludwig and others on this subject. More recently, Ludwig (2), Lewis(3) and others have considered 'uniform' progressing wave expansions for various problems. These expansions are valid in regions where the standard expansions are not suitable and they can be re-expanded in the standard form outside these regions. Examples of such regions are given by envelopes of bicharacteristic curves or, equivalently, caustics and by shadow boundaries such as occur in diffraction problems. In each of these regions, which we term 'transition regions' different types of uniform expansions are required.