On Wave Interactions II. Explosive Resonant Triads in Fluid Mechanics

This paper considers the occurrence of explosive resonant triads in fluid mechanics. These are weakly nonlinear waves whose amplitudes become unbounded in finite time. Previous work is expanded to include interfacial flow systems with continuously varying basic velocities and densities. The first paper in this series [10] discussed the surprisingly strong singular nature of explosive triads. Many of the problems to be studied here will be found to have additional singularities, and the techniques for analyzing these difficulties will be developed. This will involve the concept of a critical layer in a fluid, a level at which a wave phase speed equals the unperturbed fluid velocity in the direction of propagation. Examples of such waves in this context are presented.     

Resonant Triads Involving a Nondispersive Wave

A simple system is studied, involving a single nondispersive breaking wave and its interaction with two dispersive modes through a resonant triad. The dynamics of this system are shown to be quite rich, through a combined theoretical and numerical analysis. A sharply defined traveling wave with a corner seems to attract almost all initial data with enough energy, provided the nondispersive wave is unstable to the other two when standing alone. In other cases, the solution converges to quasiperiodic final states, unless extra symmetries force the solution to converge to simpler configurations.     

Permanent Wave Structures and Resonant Triads in a Layered Fluid

A closed three layer fluid with small density differences between the layers has two closely related modes of gravity wave propagation. The nonlinear interactions between the wave modes are investigated, particularly the nearly resonant or significant interactions. Permanent wave solutions are calculated, and it is shown that a permanent wave of the slower mode can generate resonantly a wave harmonic of the faster mode. The equations governing resonant triads of the two modes are derived, and solutions having a permanent structure are calculated from them. It is found that some resonant triad solutions vanish when the triad is embedded in the set of all harmonics with wavenumbers in its neighborhood     

Resonant Long–Short Wave Interactions in an Unbounded Rotating Stratified Fluid

A theoretical study is made of finite-amplitude modulated internal wavetrains and the attendant nonlinear interaction with the mean flow induced by the modulations, in an unbounded uniformly stratified Boussinesq fluid rotating around the vertical axis. When the rotation is relatively weak, in particular, 'flat' wavetrains, that feature stronger vertical than horizontal modulations, are resonantly coupled with the mean flow in a manner analogous to the resonant long–short wave interaction between gravity and capillary waves on the surface of deep water. A coupled set of evolution equations for the vertical wavenumber, the wave amplitude, and the mean flow is derived under resonant conditions, and is used to examine the propagation of locally confined wavetrains with initially uniform wavenumber and no pre-existing mean flow. The resonant interaction causes radiation of energy away from a flat wavetrain by means of the induced mean flow which forms a trailing wake; this furnishes a possible mechanism for generating low-frequency inertial–gravity waves in the atmosphere, as suggested by field observations. Moreover, owing to refraction by the mean flow, a finite-amplitude wavetrain may experience rapid wavenumber variations in certain locations, consistent with prior numerical simulations. Eventually, in these regions, the wavenumber tends to become multi-valued, suggesting the formation of caustics.     

Resonant Interactions between Vortical Flows and Water Waves. Part I: Deep Water

Any weak, steady vortical flow is a solution to leading order of the inviscid fluid equations with a free surface, so long as this flow has horizontal streamlines coinciding with the undisturbed free surface. This work considers the propagation of irrotational surface gravity waves when such a vortical flow is present. In particular, when the vortical flow and the irrotational surface waves are both periodic, resonant interactions can occur between the various components of the flow. The periodic vortical component of the flow is proposed as a model for more complicated vortical flows that would affect surface waves in the ocean, such as the turbulence in the wake of a ship. These resonant interactions are studied in two dimensions, both in the limit of deep water (Part I) and shallow water (Part II). For deep water, the resonant set of surface waves is governed by “triad-like” ordinary differential equations for the wave amplitudes, whose coefficients depend on the underlying rotational flow. These coefficients are calculated explicitly and the stability of various configurations of waves is discussed. The effect of three dimensionality is also briefly mentioned.     

On Wave Interactions III. Miscellaneous Results

In this paper, a collection of results pertaining to resonant triads in fluid flow systems is presented. The main topic is the extension of the analysis by Mahoney for wind-driven shear flows to general, layered flows, that is to say, flows which have piecewise continuous velocity and density profiles. It is seen how the results for calculating the triad interaction coefficients for two-layer flows generalize and carry over for multilayer flows.     

Resonant Wave Interaction with Random Forcing and Dissipation

A new model for studying energy transfer is introduced. It consists of a "resonant duo"—a resonant quartet where extra symmetries support a reduced subsystem with only two degrees of freedom—where one mode is forced by white noise and the other is damped. This system has a single free parameter: the quotient of the damping coefficient to the amplitude of the forcing times the square root of the strength of the nonlinearity. As this parameter varies, a transition takes place from a Gaussian, high-temperature, near equilibrium regime, to one highly intermittent and non-Gaussian. Both regimes can be understood in terms of appropriate Fokker–Planck equations.     

Quasi-Periodic Solutions of Completely Resonant Forced Wave Equations

We prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases when the forcing frequency is: (Case A) a rational number and (Case B) an irrational number.     

Nonlinear Wave Interactions in Nonlinear Nonintegrable Systems

The main goal of this article is to understand the qualitative appearances of regular arrays of pulses that come up in nonintegrable systems in a variety of contexts, particularly in fluid dynamics. It is shown that even nonintegrable systems have a kind of particle dynamics made up of solitary waves. But the interaction of these solitary waves is not absolutely “clean” as in the case of the KdV and other integrable equations.     

On the Nonlinear Initial Value Problem for Vortex–Wave Interactions in Shear Flows

Small-amplitude wave systems interacting nonlinearly can produce 0(1)amplitude streamwise vortex structures through the vortex–wave interaction mechanism described, for example, by [1–3]. The key feature of the interaction is that the spanwise velocity component of a vortex is small as compared to the streamwise component so that a nonlinear wave system driving the spanwise velocity component through Reynolds stresses can provoke a 0(1) response of the vortex. The wave system can correspond to either a Rayleigh or Tollmien–Schlichting wave disturbance, but previous work on the initiation of the process has been confined to Rayleigh waves (see, for example, [5, 6]). Here, we address the nonlinear initial value problem for Tollmien–Schlichting wave–vortex interactions in channel flows. The evolution of the disturbances is accounted for using the phase equation approach of [7]. We determine the circumstances, if any, under which the finite amplitude vortex–wave equilibrium states of [4] are generated. Our discussion of the nonlinear evolution of a wave system points toward a possible mechanism for the experimentally observed breakup of three-dimensional instabilities into shorter streamwise scales.     

Resonant Interactions between Vortical Flows and Water Waves. Part II: Shallow Water

Any weak, steady vortical flow is a solution, to leading order, of the inviscid fluid equations with a free surface, so long as this flow has horizontal streamlines coinciding with the undisturbed free surface. This work considers the propagation of long irrotational surface gravity waves when such a vortical flow is present. In particular, when the vortical flow and the irrotational surface waves are both periodic and have comparable length scales, resonant interactions can occur between the various components of the flow. The interaction is described by two coupled Korteweg-de Vries equations and a two-dimensional streamfunction equation.     

Blow-up Surfaces for Nonlinear Wave Equations,I

We introduce a systematic procedure for reducing nonlinear wave equations to characteristic problems of Fuchsian type. This reduction is combined with an existence theorem to produce solutions blowing up on a prescribed hypersurface. This first part develops the procedure on the example □u = exp(u); we find necessary and sufficient conditions for the existence of a solution of the form ln(2/?²) + v, where {? = 0} is the blow-up surface, and v is analytic. This gives a natural way of continuing solutions after blow-up.     

Quasi-periodic Solutions of Completely Resonant Wave Equations with Quasi-periodically forced Vibrations

In this paper, we provide the existence of quasi-periodic solutions with two frequencies for a class of completely resonant nonlinear wave equations with quasi-periodically forced vibrations under the spatial periodic boundary conditions. We consider the frequencies vector (ω ₁,ω ₂) near the linear system. The proofs are based on the Variational Lyapunov-Schmidt reduction and Linking Theorem.     

The Stability of Two, Three, and Four Wave Interactions of a Prototype System

The stability of plane wave interactions of coupled nonlinear Schrödinger (CNLS) equations can be analyzed within a bisymplectic framework. This framework is a generalization of the Hamiltonian formulation. The current study considers a family of CNLS equations that are used as a prototype system for studying the combined interaction of unstable and stable component waves in optics. This popular family has a drawback when cast into a bisymplectic framework: the determinant controlling various types of fiber regime is zero. To solve this problem, it is proposed that a limit is taken from a more general CNLS family to the family in question. This method is then bench-marked against known stability results for the simple two plane wave interactions when amplitudes are equal and are found to agree. It is then applied to two wave interactions with unequal amplitudes as well as three and four wave interactions. The latter interactions for this particular system are not spectrally stable. By suggesting a slightly larger family of CNLS equations, it is illustrated that spectral stability can occur. This adapted prototype system may be of use in optics; in particular, to show that long-wave stability is possible given a judicious choice of parameter values.     

圆柱形容器中自由表面波的三波内共振相互作用

由Luke变分原理导出了圆柱形容器中自由毛细重力表面波的基本方程.首先对速度势和自由面波高作Galerkin展开,用多重尺度法导出了控制方程前二阶摄动方程,在此基础上讨论了三个波二阶内共振的非线性相互作用,导出了三波内共振的非线性耦合作用方程和守恒律.针对非退化的作用方程,分析了相平面上平衡点的位置,研究了参数对应共振与非共振的各种情况,在不同参数情况下求出二阶作用方程的稳态解并分析了解的稳定性态,还讨论了只在有限时间内有效的解.分析表明,在非退化情况下,由于初始条件不同,3个波之间能量传递的模式不尽相同,有可能能量在3波之间周期性传递,亦可能单波的能量有衰减或增长.     

Resonant Equations with Classical Orthogonal Polynomials. I

Ukrainian Mathematical Journal - We study some resonant equations related to the classical orthogonal polynomials and propose an algorithm for finding their particular and general solutions in the...     

相变演化Suliciu模型中波的相互作用

描述相变演化的Suliciu模型，其基本波可由行波分析得到．对于任何给定分两段常值的初始状态，相应的Riemann解是某些基本波的组合．对分三段常值的初始状态，解由上述Piemann解构成，其中相邻两状态间以基本波连接．当基本波发生碰撞时，新的Riemann问题形成．通过研究这些Riemann。问题，可以在适当的参数空间中对基本波之间复杂的相互作用加以分类．     

Lamb Wave Interactions with Non-symmetric Features at Structural Boundaries

The paper initially describes on a numerical basis how a Lamb wave would have to perform that has been initiated in a pure mode (either symmetric or anti–symmetric) and what the wave would have to anticipate in terms of mode conversion when being reflected at a surface not perpendicular to its traveling direction. The effects of changing in geometric specifications of non–symmetric artificial features like angle of sloping edge or partially sloping edges are studied. The results obtained from these studies are presented as the reflected and converted parts of the incident wave versus angle of the edge or percentage of the sloped edge. It has been further shown that Lamb waves being generated experimentally by a finite size transducer into a plate like structure thus most likely result in a combination of modes. Reflection of these combined modes at structural boundaries will therefore generate an even more complex coupling of modes. This situation is further aggravated if the structural boundary is not purely perpendicular to the traveling wave but has a slightly varying angle such as it might have to be anticipated at a countersunk rivet, a notch or even more extreme a crack in a metallic component. However from understanding the background of Lamb wave generation, mode separation and superposition, a systematic approach can be established that allows complex Lamb waves, such as they are observed when monitoring true structures, to be interpreted and understood. This approach has been explained on the basis of numerical result obtained from finite element analyses first before proving the findings by some fundamental experiments performed with variable angle beam transducers which demonstrates the difficulties in de–coupling Lamb wave modes and how to handle those coupled modes in terms of structural condition monitoring. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On N-Summations, I

This paper continues recent work on N-summations (see [8, 9, 12]). More specifically, it addresses the issue of existence of N-summations both for cone semirings and for prenormed semitopological semimodules. In the case of a cone semiring C we assume N-order completeness plus compatibility of N-joins with addition and multiplication to make the class of summarily bounded elements of C ^N into an N-summation for C. In the case of a prenormed semitopological semimodule M we use certain completeness properties of semitopologies on M to make the class of Cauchy elements of M ^N into an N-summation for M. Results on semitopologies and their connection with closure operators are contained in the Appendix.