Showing papers on "Cnoidal wave published in 1989"
154 citations
TL;DR: In this paper, the authors treated travelling waves in a heterogeneous, inviscid, non-diffusive fluid bounded between two horizontal boundaries and analyzed the small amplitude solutions using a dynamical systems approach.
Abstract: This paper treats travelling waves in a heterogeneous, inviscid, non-diffusive fluid bounded between two horizontal boundaries. The fluid has two incompressible components of different, but constant density and is acted on by gravity. The flow is steady when viewed in a moving reference frame and gives rise to a quasilinear elliptic problem with an eigenvalue parameter related to the wave speed. The small amplitude solutions are analyzed using a dynamical systems approach. A center manifold reduction in combination with a conserved quantity for the flow is used to parametrise all ‘small’ solutions of the full elliptic system in terms of solutions of an autonomous first order ordinary differential equation for a principal component of the wave amplitude. The result is a characterization of all small waves, irrotational in each fluid, near the critical speed for the system. They are: solitary waves; surges connecting distinct conjugate flows at extreme ends of the channel; conjugate flows; and periodic waves.
45 citations
TL;DR: In this paper, the amplitude equations for two-dimensional long waves on the free surface of a thin film flowing down an inclined plane were considered and parallel N -soliton solutions were shown to exist in each case.
32 citations
TL;DR: The integration procedure based on the inverse scattering method was developed for a 2D elliptic sine-Gordon equation with a periodic wave type asymptotic behavior on one space variable as mentioned in this paper.
Abstract: The integration procedure based on the inverse scattering method is developed for a 2D elliptic sine-Gordon equation with a periodic-wave type asymptotic behaviour on one space variable The soliton solutions are found in explicit form
27 citations
TL;DR: Based on Boussinesq equations and parabolic approximation, the forward diffraction of cnoidal waves by a straight breakwater is studied numerically in this article, where the formation of stem waves along the breakwater and the relation between the stem waves and the incident wave characteristics are discussed.
Abstract: Based on Boussinesq equations and parabolic approximation, the forward diffraction of cnoidal waves by a straight breakwater is studied numerically. The formation of stem waves along the breakwater and the relation between the stem waves and the incident wave characteristics are discussed. A numerical scheme for computing uniform cnoidal waves also is presented.
16 citations
01 Jan 1989
TL;DR: In this article, the authors describe geometrical optics theories for nonlinear waves and derive a theory for hyperbolic waves with large-amplitude, rapidly varying initial data.
Abstract: We describe geometrical optics theories for nonlinear waves and derive a theory for hyperbolic waves with large-amplitude, rapidly varying initial data. We consider initial data which is either compactly supported or periodic in a phase variable. We also analyze the decay of periodic solutions of hyperbolic conservation laws and the resonant interaction of weakly nonlinear sawtooth waves.
15 citations
TL;DR: In this paper, the effect of two-temperature electrons on the formation and propagation of cnoidal waves in a weak relativistic plasma is considered. And the phase velocity of the modulated periodic wave is plotted as a function of the amplitude, and the form of the wave for small values of ellipticity of the Cnoidal wave shows up as a series of harmonics.
Abstract: We have considered the effect due to the presence of two-temperature electrons on the formation and propagation of cnoidal waves in a weak relativistic plasma. Since the cnoidal wave is equivalent to an infinite series of solitary waves, it is worthy of study for the situation where more than one solitary wave is important. The phase velocity of the modulated periodic wave is plotted as a function of the amplitude. The form of the wave for small values of ellipticity of the cnoidal wave shows up as a series of harmonics. Such cnoidal excitations are important in determining the anomalous transport coefficients of a plasma.
10 citations
TL;DR: In this article, a coupled Korteweg de Vries (KdV)-modified KdV equation is presented, some explicit solutions of which are exhibited in terms of Jacobi elliptic functions.
Abstract: A coupled Korteweg de Vries (KdV)-modified KdV equation is presented, some explicit solutions of which are exhibited in terms of Jacobi elliptic functions.
7 citations
TL;DR: In this paper, weakly nonlinear longwaves in a horizontally sheared current flowing along a longitudinal boundary in a two-layer ocean are investigated by using a quasi-geostrophic β-plane model under the assumptions that the depth ratio of two layers is small, the β effect is weak and the waves are almost stationary.
Abstract: Weakly nonlinear longwaves in a horizontally sheared current flowing along a longitudinal boundary in a two-layer ocean are investigated by using a quasi-geostrophic β-plane model. Under the assumptions that the depth ratio of two layers is small, the β effect is weak and the waves are almost stationary, we obtain a set of coupled equations similar to that derived perviously by Kubokawa for a coastal current with a surface density front on an f-plane. This set of equations contains soliton and cnoidal wave solutions and allows baroclinic instability to occur. Considering a perturbation around the marginally stable condition, we obtain an analytic solution of a growing solitary disturbance with an amplitude larger than a certain critical value in a linearly stable eastward current. This disturbance propagates eastward, and grows by a baroclinic energy conversion. A numerical computation on its further evolution shows that after the amplitude exceeds another certain critical value, the disturbance ...
7 citations
TL;DR: In this paper, the higher-order KP equation for water wave dynamics was proposed and the exact and explicit two-dimensional solitary wave solutions were also given, where the dimensionless surface-tension coefficient τ of fluid is close to 1/3.
Abstract: Using the method of multiple scales and considering the nearly two-dimensional modulation of the waves when the dimensionless surface-tension coefficient τ of fluid is close to 1/3, we proposed a new model equation which we call the higher-order KP equation for water wave dynamics. The exact and explicit two dimensional solitary wave solutions are also given.
TL;DR: In this paper, the theoretical expressions for near-bed, shorenormal flows are re-examined and the predictions are tested with morphodynamic and laboratory data, and the results suggest that Stokes wave theory over-predicts the observed peak flows.
Abstract: Although the near-bed, wave induced currents predicted by the various wave theories are responsible for seabed sediment transport, it is presently unclear which theory should be included in coastal geomorphic models. The theoretical expressions for near-bed, shorenormal flows are re-examined and the predictions are tested with morphodynamic and laboratory data. Five wave theories (Airy, Stokes, Cnoidal, Solitary, and Gerstner) are summarized here and arranged to predict the maximum onshore and offshore near-bed flows and the flow asymmetry as functions of the wave period, the wave height, and the water depth. Morphodynamic argument suggests that only Stokes and Cnoidal wave theories predict the asymmetry required to generate prototype seabed profiles in intermediate water depths. Laboratory measurements of wave induced near-bed flows are then reported and analysed. The results suggest that Stokes wave theory over-predicts the observed peak flows. A correction is derived which is shown to be analogous to the pressure attenuation corrections which are routinely applied to wave recording, seabed mounted transducer measurements.
TL;DR: In this paper, the authors studied a family of equations of the Korteweg-de Vries type and characterized their linear dispersion relation w(k) by a polynomial conserved quantity.
Abstract: The authors study a family of equations of the Korteweg-de Vries type. Different elements of this family are characterised by their linear dispersion relation w(k). They prove that there are only three members of this family which possess a non-trivial polynomial conserved quantity and have a linear dispersion relation analytic in a neighbourhood of the origin. These correspond to the Korteweg-de Vries equation itself, the Benjamin-Ono equation and the intermediate long wave equation. As is well known, these equations have indeed infinitely many conserved quantities.
TL;DR: In this article, the wave processes in a system described by a fourth-order partial differential equation with Burgers-Korteweg-de Vries nonlinearity are considered, and the initial equation is reduced to a dynamical system of three equations, which is analyzed by means of a numerical method.
Abstract: The wave processes in a system described by a fourth-order partial differential equation with Burgers-Korteweg-de Vries nonlinearity are considered. The initial equation is reduced to a dynamical system of three equations, which is analyzed by means of a numerical method. It is shown that the equation for the waves in dissipative-dispersive systems with instability has solutions in the form of solitary waves and wave fronts.
TL;DR: In this paper, the Cauchy problems of the propagation of a single wave and the interaction of two solitary waves of different amplitude are solved numerically for the case of slow symmetric surface waves in a magnetic tube.
Abstract: The Cauchy problems of the propagation of a single wave and the interaction of two solitary waves of different amplitude are solved numerically for the case of slow symmetric surface waves in a magnetic tube. It is found that the solitary waves interact in the same way as the solitons of the known soliton equations such as the Korteweg-de Vries and Benjamin-Ono equations, i.e., preserve their shape after interacting. The way in which the solitons decrease at infinity is discussed.
TL;DR: In this paper, the authors proposed a cross-hole seismic surveying approach using wave equation inverse scattering methods, which can provide a useful model of the gross velocity structure of interwell regions.
Abstract: Although the potential of cross-hole seismic surveying is widely appreciated, in practice results have sometimes failed to realize this. Travel time inversions rely on the zero wavelength approximation of ray theory, and parameterize the data as a set of arrival times. This can provide a useful model of the gross velocity structure of interwell regions, but structural imaging can be dramatically improved using wave equation inverse scattering methods.
TL;DR: In this paper, the nonlinear equation governing the nonpropagating solitary surface wave and its single solitary wave solution were derived by the method of multiple scales, and the results obtained were compared with that in rectangular trough.
Abstract: The nonlinear equation governing the nonpropagating solitary surface wave and its single solitary wave solution were derived by the method of multiple scales. The results obtained were compared with that in rectangular trough.
TL;DR: In this paper, the Korteweg-De Vries equation is derived by means of simple physical arguments, and the approximations involved are discussed, and solutions that propagate without changing their shape are studied using an analogy with a mechanical oscillator.
Abstract: Nonlinear phenomena are a very important topic of modern physics that is usually taught only at an advanced level. An elementary introduction of this subject is presented, considering nonlinear waves in shallow water. The Korteweg–De Vries equation is derived by means of simple physical arguments, and the approximations involved are discussed. Solutions that propagate without changing their shape are studied and their properties are described using an analogy with a mechanical oscillator. The solitary and cnoidal waves are obtained as well as harmonic waves as a limiting case. The mathematics employed is simple and straightforward.
TL;DR: In this article, the problem of a nonlinear gravity inertial wave of two dimensions and the possibility of solitary wave's existence was discussed, and the existing condition and analytic solution expression of shallow water waves were obtained by the qualitative method of O. D. Es.
Abstract: In this paper we discuss the problem of a nonlinear gravity inertial wave of two
dimensions and the possibility of solitary wave's existence. First of all, the existing
condition and analytic solution expression of shallow water waves are obtained by the
application of the qualitative method of O. D. Es. We find that when the problem is de-
generated, some physical values produce the nonlinear solitary wave, while other physi-
cal values will be unbounded, so we consider that the nonlinear solitary wave for the
system does not exist. Then we introduce concepts of the generalized energy (i. e. pseu-
do-energy): when the pseudo-energy produces the tiny change at acting on a special ex-
ternal effect, there will be solitary waves in this system. Finally, we obtain the repre-
sentative of the nonlinear solitary wave which is different from KdV equation.
TL;DR: In this article, an evolution equation generalizing the usual Burgers-Korteweg-de Vries (BKdV) equation is derived for describing the nonlinear waves in a gas-containing liquid.
Abstract: An equation describing the kinetics of the mass transfer accompanying the process of gas bubble growth in an incompressible liquid is proposed. The equation of state of the two-phase system, whose derivation is based on the mechanism of interphase mass transfer due to the solubility of the gas atoms in the liquid, is obtained. An evolution equation generalizing the usual Burgers-Korteweg-de Vries (BKdV) equation is derived for describing the nonlinear waves in a gas-containing liquid. The solution of this equation is obtained in the form of a solitary wave.
TL;DR: In this paper, a boundary element method using nonlinear potential theory has been developed for analyzing wave diffraction around large structure, and the numerical results for solitary wave propagating through a three-dimensional wave channel are compared with analytical solutions in terms of wave profile and hydrodynamic pressure.
Abstract: A boundary element method using nonlinear potential theory has been developed for analyzing wave diffraction around large structure. To confirm the validity of the proposed method, the numerical results for solitary wave propagating through a three-dimensional wave channel are compared with analytical solutions in terms of wave profile and hydrodynamic pressure. The comparison shows excellent agreement regardless of H0/h (H0: wave height, h: water depth). Furthermore, nonlinear effects on wave loads exerted by solitary wave on a vertical cylinder are investigated by comparing the numerical results with first approximations given by Isaacson. The first approximations for horizontal force, overturning moment and wave run-up are appropriate with H0/h being 0.1. It is, however, found that the first approximations for overturning moment are underestimated remarkably with H0/h being 0.4.