Dissipative Wave Equation Research Articles

Factorization of the dissipative wave equation and inverse scattering

A decomposition of the solution of the dissipative wave equation into incoming and outgoing components across a smooth surface in a homogeneous region is presented. (The proof of the decomposition is given only for the plane surface.) This is then applied to the factorization of the dissipative wave equation into incoming and outgoing components in a planar-stratified medium. The Ricatti integral–differential equation for the reflection operator that relates the two components is obtained. It is shown how the zeroth and second moments (with respect to the transverse variable in a planar-stratified medium) of the reflection kernel can be used in the inverse problem to recover the velocity and dissipation coefficient from knowledge of the scattered field.

Addendum to: Existence and uniqueness of solutions to nonlinear dissipative wave equations

Addendum to: Existence and uniqueness of solutions to nonlinear dissipative wave equations

Existence and uniqueness of solutions to nonlinear dissipative wave equations

On etudie l'existence et l'unicite des solutions d'equations d'ondes dissipatives non lineaires avec donnees de Cauchy dans l'espace de mesure. Les donnees de Cauchy ont une energie infinie, ce qui ne permet pas d'utiliser la methode d'energie classique

On the decay of solutions of some nonlinear dissipative wave equations in higher dimensions

On etudie la propriete de decroissance des solutions des equations d'onde dissipatives non lineaires de la forme: ∂ 2 u/∂t 2 -Δu+g(∂u/∂t)=f(x,t), (x,t)∈Ω×R + avec les conditions u(x,o)=u 0 (x), u t (x,o)=u 1 (x), x∈Ω et u(x,t)=0 sur ∂Ω×R + , ou Ω est un domaine borne de R n avec une frontiere lisse ∂Ω et g(v) est une fonction du genre k/v/ α v,k>o,α≥o

Statistical properties and correlation functions for drift waves

The dissipative one-field drift wave equation is solved using the pseudospectral method to generate steady-state fluctuations. The fluctuations are analyzed in terms of space-time correlation functions and modal probability distributions. Nearly Gaussian statistics and exponential decay of the two-time correlation functions occur in the presence of electron dissipation, while in the absence of electron dissipation long-lived vortical structures occur. Formulas from renormalized, Markovianized statistical turbulence theory are given in a local approximation to interpret the dissipative turbulence.

Non-resonance for a strongly dissipative wave equation in higher dimensions

Under some regularity conditions, a non-resonance property is established for a semi-linear forced wave equation with a strong local damping term and Dirichlet boundary conditions in a bounded open domain. In dimension less than or equal to six, the damping term can grow at infinity like an arbitrarily large power of the velocity. If a viscosity term is added, in dimension less or equal to four a stronger result is obtained, and this property allows to construct almost periodic solutions for an arbitrary forcing term in a suitable regularity class.

Forced oscillations for some nonlinear, weakly dissipative wave equations

Periodic solutions and asymptotic behavior are studied for the equation u tt − Δ u + β( u t ) ε f(t, x) on ℝ + × Ω; u = 0on ℝ + × ∂Ω. In Section I, we takeΩ = |0,π|. For some maximal monotone β such that D (β) and β(R) ≠R, we study the existence of 2π-time periodic solutions. In Section II, we take β = sgn. Then for some f such that| f(t, x) | ⩽ 1 − ɛon ℝ + × Ω (ɛ > 0), we prove that all solutions u(t, x) converge when t → ∞to functions independent of t satisfying stability conditions.

A new towed array shape estimation technique using directional sensor outputs

Based on Kennedy's (NUSC 93231‐TMU‐8‐79, July 1979) analysis of Paidoussis's [J. Fluid Mech. 26, 717–751 (1966)] equation for the dynamics of a towed cylinder, a computer model has been developed, which estimates array shape using a Kalman filter, with directional sensor information as inputs. Kennedy and Rispin [J. Underwater Acoust. 30, 111–123 (1980)] have validated the concept that the propagation down the array of disturbances at the forward end of an array is governed by a dissipative and dispersive wave equation whose coefficients are functions of the ratio of the longitudinal to the transverse drag coefficient, array length, and tow speed. We formulate the partial differential equation describing array motion as a linear, discrete, state‐variable, matrix difference equation with the state vector elements being the transverse displacements of specified array grid points. Using directional sensor outputs as the observations, the system has been shown to be observable. The model has been used to pred...

On the kinetics of spinodal decomposition: a numerical solution of a dissipative wave equation

A numerical solution of a recently derived dissipative wave equation governing the kinetics of Spinodal decomposition of a Lennard-Jones fluid is presented. In addition, the results are compared with those of Cahn'S and Abraham's generalized diffusion theories for the case of the early stages of the coarsening process.

Asymptotic behavior for an almost periodic, strongly dissipative wave equation

This paper deals with asymptotic behavior for (weak) solutions of the equation u tt − Δu + β(u t) ∋ ƒ(t, x) , on R + × Ω; u( t, x) = 0, on R + × ∂Ω. If ƒ∈L∞(R +,L 2(Ω)) and β is coercive, we prove that the solutions are bounded in the energy space, under weaker assumptions than those used by G. Prouse in a previous work. If in addition ƒ t∈S 2(R +,L 2(Ω)) and ƒ is srongly almost-periodic, we prove for strongly monotone β that all solutions are asymptotically almost-periodic in the energy space. The assumptions made on β are much less restrictive than those made by G. Prouse: mainly, we allow β to be multivalued, and in the one-dimensional case β need not be defined everywhere.

Energy decay and partition for dissipative wave equations

Energy decay and partition for dissipative wave equations

Periodic solution of the dissipative wave equation in a time-dependent domain

Periodic solution of the dissipative wave equation in a time-dependent domain

Quasiperiodic solutions of a weakly dissipative nonlinear wave equation

Quasiperiodic solutions of a weakly dissipative nonlinear wave equation

Energy decay of solutions of dissipative wave equations

Energy decay of solutions of dissipative wave equations

Qualitative behavior of dissipative wave equations on bounded domains

The qualitative behavior of solutions of the mixed problem utt = Δu-a(x)ut in IR x Ω, u=0 on IR x ∂Ω, is studied in the case when a>0 and Ω⊂IRn is bounded. Roughly speaking, if a≧amin>0, then solutions decay at least as fast as exp t(ɛ −1/2amin), with the possible exception of a finite dimensional set of smooth solutions whose existence is associated with a phenomenon of overdamping. If amax is sufficiently small, depending on Ω, then no overdamping occurs.

Asymptotic method for nonlinear dispersive and weakly dissipative wave equations

We propose an asymptotic method to study nonlinear partial differential equations for dispersive waves, modified by the addition of a small term—which may be a small damping, for example. This work constitutes a generalization of that of J. C. Luke[1]. A number of new effects appear.

Amplitude Dispersion and Stability of Nonlinear Weakly Dissipative Waves

A method to solve nonlinear dissipative wave equations using ideas of Luke, Krylov, and Bogolyubov is presented. The method is compared to Whitham's theory. Dispersion relations for nonlinear dissipative waves, including amplitude dispersion, are discussed. Furthermore, stability problems of such waves are investigated.

On a simple wave approximation to a non-linear problem

The range of validity of a simple wave approximation to a non-linear set of two dissipative wave equations has been studied. The non-linear set is, when the dissipative terms are omitted, totally exceptional. It describes the propagation of longitudinal waves in an ideal elastic bar with some viscous stress. Upon a non-linear transformation, the equations become linear. These linear equations have been studied first. The results for the non-linear equations are then easily obtained by transforming backwards. It turns out that, if the non-linearity is small enough, they are similar to those obtained for the linear equations.

Finite-Amplitude Traveling Waves in Ducts

A nonlinear, dissipative, acoustic wave equation generalized from the linear, dissipative wave equation valid for one-dimensional motion in a duct (where the dominant absorptive mechanism is the energy loss in the boundary layer) has been obtained. The equation admits straightforward solution for traveling waves by a perturbation expansion in Mach number. Prominent features of the results include a first-order contribution in absorption to phase velocity. This effect generates spatially dependent phase angles among the different harmonics. Preliminary results suggest that the solution can profitably be applied to all distances when the Mach number is small enough to preclude shock. Differences in the properties of standing-wave and traveling-wave solutions may be discussed.

On the principle of limiting amplitude for dissipative wave equations

On the principle of limiting amplitude for dissipative wave equations