Progressive Wave Solutions Research Articles

Least‐squares finite element approximation of Fisher's reaction–diffusion equation

AbstractFisher's equation is a classical diffusion–reaction type of problem describing diffusion and nonlinear reproduction of a species. In the present study we develop a least‐squares finite element formulation of Fisher's equation and carry out supporting numerical studies. Of particular interest are questions associated with the approximation of progressive wave solutions with minimum speed and the viability of the least‐squares approach for this class of problem. © 1995 John Wiley & Sons, Inc.

On permanent nonlinear waves in a rotating fluid

The governing equation for long nonlinear gravity waves in a rotating fluid changes with the value of the Coriolis parameter f. (1) When f is large, i.e. in the strong rotation case, in an infinite ocean, there are only Sverdrup waves; in a semi-infinite ocean or in a channel, there are either solitary Kelvin waves, for which the governing equation is a KdV equation, or Poincaré waves, which can be obtained by superposition of two Sverdrup waves. (2) When f is small, i.e. in the weak rotation case, in an infinite ocean there are solitary or cnoidal waves governed by the Ostrovskiy equation, and we provide an explicit solution for both solitary and cnoidal Ostrovskiy progressive waves; and in a semi-infinite ocean or a channel, there are Sverdrup waves, which are governed either by Ostrovskiy equations or by the Grimshaw-Melville equation. (3) When f is very small, i.e. in the very weak rotation case, in an infinite ocean, or in a channel, there are solitary waves with a horizontal crest, but with a velocity component or a pressure gradient, which are governed by KdV equations as in the non-rotating case. Physically, that means that the most determining factor is the ratio of the Rossby radius of deformation over a characteristic length of the wave.

Surface waves on viscous magnetic fluid flow down an inclined plane

The objective of this paper is to study weakly nonlinear waves on a viscous magnetic fluid flow down an inclined plane under the influence of gravity and a constant magnetic field parallel to the plane. An approximate evolution equation is derived and a critical Reynolds number for the stability of three-dimensional long waves obtained. It is found that the component of the applied magnetic field aligned with the flow has a stabilizing effect but that the transverse component, however, has a destabilizing effect on the long waves. Stability regions of two-dimensional long waves at the critical Reynolds number and numerical results for the solution of a nonlinear ill-posed problem are presented. Several progressive-wave solutions are also given.

Active Wave Control of a Flexible Beam. Proposition of the Active Sink Method.

This paper deals with the flexural wave control of a flexible beam with an infinite number of vibration modes. It is the purpose of this paper to present a new vibration control method, the active sink method, which makes it possible to suppress all vibration modes of the flexible beam. Unlike a conventional vibration control method which attempts to suppress several vibration modes already excited, the active sink method seeks to keep all the vibration modes inactive. First, this paper presents the principle of the active sink method and shows a means of realizing its system. Next, in order to describe the principle of the active sink method mathematically, transfer matrices of a beam using a progressive wave solution to the wave equation are obtained. Then, the optimal conditions for the active sink system are derived, and the fundamental characteristics of the system are discussed. Finally, from the vdewpoint of vibration intensity analysis, the validity of the active sink method is clarified.

Wake-free waves in one and three dimensions

A recent paper by Gottlieb [J. Math. Phys. 29, 2434 (1988)] provides examples of acoustic wave equations, in various dimensions, that have nontrivial families of solutions that are progressing waves of order 1, and relates this to whether or not these equations satisfy Huygens’ principle. A statement made in that paper related to Huygens’ principle in one space dimension is clarified, and it is shown in this connection that, in general, the relationship between the possession of progressing wave solutions and the satisfaction of Huygens’ principle is more complex than the situation described by Gottlieb. In addition, the attractive properties of progressing waves of order 1 are retained by progressing waves of any finite order, and we use this to generalize in several ways Gottlieb’s results on ‘‘wake-free’’ solutions of the acoustic equation in three dimensions.

Self-adjoint acoustic equations with progressing wave solutions

The simplest wave equations are those whose general solutions comprise progressing waves. The author constructs a comparatively large, possibly exhaustive, family of self-adjoint acoustic equations in 1+1 dimensions that are simple in this sense.

New spiky solitary waves, explosive modes and periodic progressive waves in a magnetised plasma

A new class of nonlinear evolution equations presented here ,.P( d&/ d q )2 = c1 4' + (2) ~~4''~ + ())~~4~ + (4) c44' describes the one-dimensional wave propagation of ion acoustic waves in a magnetised plasma with trapped electrons. d, c1, c2, c3, c, are arbitrary constants. The remarkable feature of this equation is that a higher-order nonlinear term of 47'2 appears in this equation. It is found that this equation has a new type of spiky solitary-wave solution, an explosive (bursting) solution and periodic progressive-wave solutions. The explosive solution is associated with the wave with the negative potential. Periodic progressive-wave solutions are reduced to the spiky solitary wave and the explosive solution under a certain condition. This theory may be applicable to explaining the behaviour of higher-order nonlinear waves in physical systems.

Self-similar viscous gravity currents: phase-plane formalism

A theoretical model for the spreading of viscous gravity currents over a rigid horizontal surface is derived, based on a lubrication theory approximation. The complete family of self-similar solutions of the governing equations is investigated by means of a phase-plane formalism developed in analogy to that of gas dynamics. The currents are represented by integral curves in the plane of two phase variables, Z and V, which are related to the depth and the average horizontal velocity of the fluid. Each integral curve corresponds to a certain self-similar viscous gravity current satisfying a particular set of initial and/or boundary conditions, and is obtained by solving a first-order ordinary differential equation of the form dV/dZ = f(Z, V), where f is a rational function. All conceivable self-similar currents can thus be obtained. A detailed analysis of the properties of the integral curves is presented, and asymptotic formulae describing the behaviour of the physical quantities near the singularities of the phase plane corresponding to sources, sinks, and current fronts are given. The derivation of self-similar solutions from the formalism is illustrated by several examples which include, in addition to the similarity flows studied by other authors, many other novel ones such as the extension to viscous flows of the classical problem of the breaking of a dam, the flows over plates with borders, as well as others. A self-similar solution of the second kind describing the axisymmetric collapse of a current towards the origin is obtained. The scaling laws for these flows are derived. Steady flows and progressive wave solutions are also studied and their connection to self-similar flows is discussed. The mathematical analogy between viscous gravity currents and other physical phenomena such as nonlinear heat conduction, nonlinear diffusion, and ground water motion is commented on.

Flexural wave control of a flexible beam. Proposition of the active sink method.

This paper deals with the flexural wave control of a flexible beam. It is the purpose of this paper to present a new vibration control method, the active sink method, which makes it possible to suppress all vibration modes of the flexible beam. Unlike the conventional vibration modal control method, the active sink method is designed for making all the vibration modes of the beam asleep. First, this paper presents the principle of the active sink method and shows a means of realizing it. Then, in order to describe the principle mathematically, a transfer matrix of a beam using a progressive wave solution to the wave equation is obtained. Moreover, based upon the transfer matrix method, the optimal conditions for the active sink system are derived, the fundamental characteristics of the system being discussed. Finally, from the viewpoint of vibration intensity analysis, the validity of the active sink system is clarified.

Some Invariant Solutions to Two-Phase Fluid Displacement Problems Including Capillary Effect

Summary Analytical investigation of the two-phase displacement problem that considers capillary force is one of the most important and difficult problems in the theory of fluid flow through porous media. This paper presents some invariant solutions on the subject, including a self-similar solution for the axisymmetric case, a self-similar solution in a linear, isoclinal reservoir that considers the effects of gravity and capillary forces simultaneously, and a progressive wave solution corresponding to a stabilized zone of saturation distribution. The relations and differences among (1) a model that completely includes capillary effect, (2) a model that considers capillary action partially and implicitly (the Buckley-Leverett model), and (3) a model that completely ignores capillary effect are indicated and discussed.

A class of wave equations with progressive wave solutions of finite order

We obtain a family of wave equations with progressive wave solutions of finite order that combines and generalizes a subfamily involving the nonreflecting Pöschl-Teller potentials and several subfamilies reported by Chang and Janis. The elementary equivalence of some of the known families is derived.

Notes on long-crested water waves

Fully three-dimensional surface gravity waves in deep water are investigated in the limit in which the length of the wave crests become long. We describe an analytic solution to fourth order in wave steepness, which matches onto known short-crested wave solutions on the one hand and onto the well-known two-dimensional progressivewave solution on the other. In the progressive-wave limit a particular solution in which the wave crests are semi-finite is given to sixth-order accuracy. These solutions are part of a more general set of solutions which are found from a nonlinear Schrodinger equation.

Periodic Rayleigh waves of finite amplitude on an isotropic solid

Existence of a periodic progressive wave solution to the nonlinear boundary value problem for Rayleigh surface waves of finite amplitude is demonstrated using an extension of the method of strained coordinates. The solution, obtained as a second-order perturbation of the linearized monochromatic Rayleigh wave solution, contains harmonics of all orders of the fundamental frequency. It is shown that the higher harmonic content of the wave increases with amplitude, but the slope of the waveform remains finite so long as the amplitude is less than a critical value.

Stages in the transient cooling of a slab

The classical solutions to the one dimensional transient heat flow situation are examined and criteria are set up to map stages in the cooling process in a parallel sided slab following step changes at the exposed surface. The temperature θ, at a point within the slab, undergoes a perceptible deviation from the undisturbed state, and from the state determined by cooling from the nearer side alone, at times ζ 0 (the Fourier number) found from a relation of type d 3θ dζ 0 3 = 0 in the appropriate progressive wave solution. Cooling becomes virtually exponential at a time ζ 0 when d 2θ dζ 0 2 for the first term in the standing wave solution equals the sum of the higher terms. The response times are conveniently expressed in terms of ζ 0, ζ 0 B or ζ 0 B 2 ( B is the Biot number) according to the model involved. Examples are given illustrating these results in a building context.

A mathematical analysis of nonlinear waves in a fluid filled visco-elastic tube

Our investigations in this paper are centred around the mathematical analysis of a “modal wave” problem. We have considered the axisymmetric flow of an inviscid liquid in a thinwalled viscoelastic tube under certain simplifying assumptions. We have first derived the propagation space equations in the long wave limit and also given a general procedure to derive these equations for arbitrary wave length, when the flow is irrotational. We have used the method of operators of multiple scales to derive the nonlinear Schrodinger equation governing the modulation of periodic waves and we have elaborated on the “long modulated waves” and the “modulated long waves”. We have also examined the existence and stability of Stokes waves in this system. This is followed by a discussion of the progressive wave solutions of the long wave equations. One of the most important results of our paper is that the propagation space equations are no longer partial differential equations but they are in terms of pseudo-differential operators.

Viscous sheets advancing over dry beds

The unsteady creeping motion of a thin sheet of viscous liquid as it advances over a gently sloping dry bed is examined. Attention is focused on the motion of the leading edge under various influences and four problems are discussed. In the first problem the fluid is travelling down an open channel formed by two straight parallel retaining walls placed perpendicular to an inclined plane. When the channel axis is parallel to the fall line there is a progressive-wave solution with a straight leading edge, but inclination of the axis generates distortions and these are calculated. In the second problem a sheet with a straight leading edge travelling over an inclined plane penetrates a region where the bed is uneven, and the subsequent deformation of the leading edge is followed. The third problem considers the flow down an open channel of circular cross-section (a partially filled pipe) and the time-dependent shape of the leading edge is calculated. The fourth problem is that of flow down an inclined plane with a single curved retaining wall. These problems are all analysed by assuming that a length characteristic of the geometry is large compared with the fluid depth divided by the bed slope, and all the solutions display extreme sensitivity to the data.

The relative diffusion of strong magnetic fields and tenuous gases

General equations are given for treating the diffusion of a magnetic field in a fluid with electrical conductivity and a resistive diffusion coefficient. Consequences of nonnegligible gas motion are investigated by considering the one-dimensional problem where a tenuous isothermal gas with pressure expressed as a function of x-coordinate and time is enmeshed in a magnetic field in the y-direction. Approximate equations are obtained for the case of weak fluctuations in gas or field pressure, and the properties of a nonlinear diffusion equation are illustrated with the example of an initial thin slab of gas confined by a uniform field on either side. The advance of a gas front into a nonvanishing gas density is illustrated by using the progressive wave solution. Application of the results to the evolution of streamers and filaments observed in the solar corona is briefly discussed

Nonlinear mode coupling of elastic waves

The theory of nonlinear elasticity is applied to a study of mode coupling of elastic waves at a particular frequency, called the critical frequency, in a long circular wire. It is assumed that the wire is of homogeneous isotropic elastic material and that the nonlinearity of medium (the effects of higher-order elasticity) is primary rather than that involved in the Lagrangian stress and strain tensors. The latter is suggested from the fact that the third-order elastic constants are of larger order of magnitude than the Lamé constants. The method of multiple scales is employed to obtain a system of equations which describes the behavior of the amplitudes involved in the mode coupling. The analysis of the equations shows that nonlinear mode coupling can occur at the critical frequency and that, except at this frequency, the wave undergoes only a phase shift. Further, progressive wave solutions show that the two wave amplitudes can be expressed in terms of Jacobian elliptic functions, and energy exchange between two modes takes place. Under special conditions, these periodic solutions degenerate to the solitary or shocklike solutions.

The entropy condition and the admissibility of shocks

The author proposed ( Trans. Amer. Math. Soc. 199 (1974), 89–112) the extended entropy condition (E) and solved the Riemann problem for general 2 × 2 conservation laws. The Riemann problem for 3 × 3 gas dynamics equations was treated by the author ( J. Differential Equations 18 (1975), 218–231). In this paper we justify condition (E) by the viscosity method in the spirit of Gelfand [ Uspehi Mat. Nauk 14 (1959), 87–158]. We show that a shock satisfies condition (E) if and only if the shock is admissible, that is, it is the limit of progressive wave solutions of the associated viscosity equations. For the “genuinely nonlinear” 2 × 2 conservation laws, Conley and Smoller [ Comm. Pure Appl. Math. 23 (1970), 867–884] proved that a shock satisfies Lax's shock inequalities [cf. Comm. Pure Appl. Math. 14 (1957), 537–566] if and only if it is admissible. In this paper, we consider systems that are not necessarily genuinely nonlinear.

A new shock locus for similarity solutions in one-dimensional unsteady gas dynamics

This paper deals with shock conditions for the progressing wave (or similarity) solutions of one-dimensional, unsteady gas dynamics. These solutions have hitherto been used to deal with the flow behind shocks moving into stationary atmospheres. By generalising the shock conditions to the case of moving atmospheres, it is shown that the progressing wave solutions can be used to describe a certain class of flows, and a new shock locus can be constructed in the phase plane of the solutions. It is hoped that such solutions will be of use in describing the unsteady flow behind shocks propagating into the ambient solar wind.