Water waves in the time domain

The initial-value problem for a structure floating on the surface of the sea is investigated under the assumptions of linear theory. Fourier transforms are used to connect the time- and frequency-domain representations of the coupled motion of the fluid and body. This allows the large-time asymptotics of the motion to be obtained from the singularity structure of the frequency-domain potential in the complex plane. Under certain initial conditions, the free motion of a body about a fixed, equilibrium position is shown not to exist for all time, and in this case the assumptions behind the linear theory are violated. For suitably moored structures, motion is found which is purely exponentially decaying in time and does not involve any oscillations.     

Algebraic integrability: a survey

We give a concise introduction to the notion of algebraic integrability. Our exposition is based on examples and phenomena, rather than on detailed proofs of abstract theorems. We mainly focus on algebraic integrability in the sense of Adler-van Moerbeke, where the fibres of the momentum map are affine parts of Abelian varieties; as it turns out, most examples from classical mechanics are of this form. Two criteria are given for such systems (Kowalevski-Painlevé and Lyapunov) and each is illustrated in one example. We show in the case of a relatively simple example how one proves algebraic integrability, starting from the differential equations for the integrable vector field. For Hamiltonian systems that are algebraically integrable in the generalized sense, two examples are given, which illustrate the non-compact analogues of Abelian varieties which typically appear in such systems.     

The integrability criterion in finite elastoplasticity and its constitutive implications

Summary In traditional Eulerian rate formulation of finite elastoplasticity, an Eulerian rate equation of hypoelastic type is used as one of the basic constituents to relate the elastic part of the stretching to a stress rate. On account of the fact that the elastic-like behavior should be expected prior to yielding, the foregoing basic elastic rate equation should be exactly integrable to deliver a conventional hyperelastic stress-strain relation. Physically, it requires a consistent combination of hypoelasticity and hyperelasticity into a single Eulerian rate equation. Since this criterion is purely a physical consistency requirement and since the basic elastic rate equation involves no strain concept and allows for any stress rate in its own right, from a physical standpoint it may be of both interest and significance to know what consequences it will imply concerning the stress rate and the finite strain measure. By a simple, straightforward procedure we demonstrate in a general sense that both Hencky strain and the logarithmic rate emerge naturally as direct consequences of the foregoing criterion. This result may be regarded to reveal the physical essence behind Hencky strain and the logarithmic rate in connection with a basic physical consistency requirement in finite elastoplasticity. Constitutive implications are discussed in a few relevant respects concerning representative formulations of finite elastoplasticity.     

Stochastic averaging in parametric regions near separatrices of integrability

Stochastic averaging in Hamiltonian framework, as a powerful tool for the dimensional reduction of strongly-nonlinear stochastic dynamical systems, achieves its enormous efficacy by distinguishing the integrability and resonance of associated conservative systems. Its implementation and mathematical expressions of drift and diffusion coefficients are completely different for different categories. For multi-degree-of-freedom systems, a slight changing of stiffness coefficients may induce an immediate conversion of the integrability, and correspondingly, notably different mathematical expressions, which means a sudden changing of dynamical behaviors. This work is devoted to this anti-intuitive phenomenon through a two-degree-of-freedom nonlinear stochastic system with adjustable parameters, by introducing the concept of the degree of integrability, applying stochastic averaging for quasi-integrable/quasi-nonintegrable systems to the parametric region near the separatrix of integrability, and comparing their accuracy of prediction. Numerical results illustrate that there exists a specific band around the separatrix of integrability, as system parameters fall in which, stochastic averaging for quasi-integrable systems achieves higher accuracy than that for quasi-nonintegrable systems even though the system itself is non-integrable. This work uncovers the existence of such specific band, and constitutes a necessary supplement to stochastic averaging in Hamiltonian framework.     

Hidden integrability in subsonic gasdynamics. Reciprocal vortex street generation

A class of multi-parameter reciprocal transformations which leaves invariant, up to the equation of state, the governing equations of steady, plane, non-homentropic gasdynamics is used to reveal hidden integrability in subsonic motion. Integrability is obtained provided that the gas law is of a generalised Kármán–Tsien type and subject to certain total enthalpy distributions. The result is applied to construct periodic vortex street motions in subsonic gasdynamics which are reciprocally linked to hydrodynamic Stuart vortices.     

Uniform integrability of the OLS estimators,and the convergence of their moments

The problem of convergence of moments of a sequence of random variables to the moments of its asymptotic distribution is important in many applications. These include the determination of the optimal training sample size in the cross-validation estimation of the generalization error of computer algorithms, and in the construction of graphical methods for studying dependence patterns between two biomarkers. In this paper, we prove the uniform integrability of the ordinary least squares estimators of a linear regression model, under suitable assumptions on the design matrix and the moments of the errors. Further, we prove the convergence of the moments of the estimators to the corresponding moments of their asymptotic distribution, and study the rate of the moment convergence. The canonical central limit theorem corresponds to the simplest linear regression model. We investigate the rate of the moment convergence in canonical central limit theorem proving a sharp improvement of von Bahr’s (Ann Math Stat 36:808–818, 1965) theorem.     

Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves

The arbitrary control of electromagnetic waves is a key aim of photonic research. Although, for example, the control of freely propagating waves (PWs) and surface waves (SWs) has separately become possible using transformation optics and metamaterials, a bridge linking both propagation types has not yet been found. Such a device has particular relevance given the many schemes of controlling electromagnetic waves at surfaces and interfaces, leading to trapped rainbows, lensing, beam bending, deflection, and even anomalous reflection/refraction. Here, we demonstrate theoretically and experimentally that a specific gradient-index meta-surface can convert a PW to a SW with nearly 100% efficiency. Distinct from conventional devices such as prism or grating couplers, the momentum mismatch between PW and SW is compensated by the reflection-phase gradient of the meta-surface, and a nearly perfect PW-SW conversion can happen for any incidence angle larger than a critical value. Experiments in the microwave region, including both far-field and near-field characterizations, are in excellent agreement with full-wave simulations. Our findings may pave the way for many applications, including high-efficiency surface plasmon couplers, anti-reflection surfaces, light absorbers, and so on.     

Electromagnetic waves generated by opposite acoustic waves

An acoustic wave of a combined frequency (formed upon the superposition of the opposite acoustic waves of close frequencies) from a moving source generates electromagnetic waves of the same frequency with the amplitude increasing in the longitudinal direction. The problem is solved for the first time, assuming the absence of electric charges and neglecting the frequency dispersion. It is shown that the running acoustic wave is accompanied by weak electromagnetic waves. This effect may find new applications, in particular, in the space energetic.     

Interactions between spin waves and elastic waves in one-domain ferromagnetic insulators

The interaction between spin waves and the elastic waves is theoretically investigated in the ferromagnetic insulators which are made to behave like one-domain bodies by means of an applied magnetic field. The free energy of the material, that is, the potential of stresses, is derived by continuum approximations of a microscopic model, and the physical meaning of various contributions in the free energy is discussed. The dipole-dipole interaction which may be considered to be a long-range interaction is approximated by a nearest neighbor interaction. As an application of the theory, interactions of spin waves and elastic waves are studied. A correspondence between the microscopic and the continuum theories of ferromagnetic resonance is presented.     

Two-dimensional generalized solitary waves and periodic waves under an ice sheet

Two-dimensional gravity waves travelling under an ice sheet are studied. The flow is assumed to be potential. Weakly nonlinear solutions are derived and fully nonlinear solutions are calculated numerically. Periodic waves and generalized solitary waves are studied.     

Tkachenko waves

The existence of Tkachenko waves in rotating superfluid⁴He has been confirmed experimentally. Tkachenko waves are displacement waves in the vortex line array that exists in a rotating superfluid. The waves were excited and detected in resonant cavities formed by a stack of closely spaced parallel plates. The dispersion relation, studied as a function of rotation speed, disk spacing (which fixed one component of the wave vector), and temperature, is compared with existing theories and found to be in good agreement at low temperatures. The theories predict peaks in the density of states associated with a lattice of vortices; the resonant responses observed are found to correspond to these peaks. The need for a more complete theory is presented in the light of the behavior of the vortex resonances at elevated temperatures.This work was supported by NSF grant DMR-78-25409.     

Self-confining waves

We present a free space wave structure which under propagation forms a cone shaped intensity distribution along the optical axis. This means that the intensity distribution in the transverse plane is contracting in the course of free space propagation in much the same way as the field propagates in a conical hollow waveguide. The contracting feature of the wave propagation originates from the interference of waves which emerge from a cylindrically symmetric, modulated field structure located initially on a closed spherical shell. This closed wave structure can be called a self-confining wave.     

Vacancy waves

Various properties of vacancy-wave excitations are discussed in detail. Included are (a) the ground-state properties, excitation energy and band width; (b) interaction processes, vacancy-wave-phonon and vacancy-wave-vacancy-wave interactions; (c) the contribution of vacancy waves to the specific heat of ³ He; and (d) light scattering from the hydrodynamic modes of a system of vacancy-wave excitations.Supported in part by the National Science Foundation and the A. P. Sloan Foundation.     

Photo-elastic waves

The basic equations of the dynamical theory of a hyperelastic dielectric are presented in spatial form and include Gibbs equation and the conservative form of energy equation here for the first time. The study of weak waves and shock waves, based on singular surface theory, is used to bring out the structure of linear and nonlinear theories of photo-elastic waves. An equation governing the lowest order nonlinear wave is derived for a mode and its solution exhibited to bring out amplitude dependence of frequency; further, though it can be considered isentropic, it is necessarily accompanied with elastic strain.     

Helmholtz-Gauss waves

A detailed study of the propagation of an arbitrary nondiffracting beam whose disturbance in the plane z = 0 is modulated by a Gaussian envelope is presented. We call such a field a Helmholtz-Gauss (HzG) beam. A simple closed-form expression for the paraxial propagation of the HzG beams is written as the product of three factors: a complex amplitude depending on the z coordinate only, a Gaussian beam, and a complex scaled version of the transverse shape of the nondiffracting beam. The general expression for the angular spectrum of the HzG beams is also derived. We introduce for the first time closed-form expressions for the Mathieu-Gauss beams in elliptic coordinates and for the parabolic Gauss beams in parabolic coordinates. The properties of the considered beams are studied both analytically and numerically.     

Unsteady pressure waves and shock waves in elastic tubes containing bubbly air-water mixtures

Summary The flow of a two-phase fluid through elastic tubes is more complex than that of a single phase fluid. The mathematical model is based on an one-dimensional approach to the flow of a liquid-gas mixture. The one-dimensional equations for transient two-phase flow through elastic tubes are a system of nonlinear hyperbolic partial differential equations if the bubbles and the liquid particles move with the same velocity. Included in the model are the effects of wall elasticity, compressibility of the gas and the liquid, the surface tension and the variable area change. The propagation of finite pressure waves and shock waves in a liquid containing gas bubbles has been investigated. The results show a differently strong influence of the parameters on the wave propagation speed and on the shock wave relations.

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Zur Ausbreitung von Druck- und Stoßwellen in instationären Blasen-Flüssigkeitsströmungen durch elastische Leitungen

Zusammenfassung Es ist bekannt, daß bei der mathematischen Beschreibung einer Zweiphasenströmung insofern Schwierigkeiten auftreten können, als unter bestimmten Voraussetzungen sowohl reelle als auch komplexe charakteristische Richtungen auftreten können. Für den Fall gleicher Geschwindigkeiten von Blasen und Flüssigkeit erhält man aus den instationären Gleichungen ein nichtlineares hyperbolisches Differentialgleichungssystem. Berücksichtigt werden die Elastizität der Wandungen, die Kompressibilität des Gases und der Flüssigkeit sowie die Oberflächenspannung. Wellenausbreitungsgeschwindigkeiten und Stoßrelationen werden angegeben. Die Resultate zeigen einen unterschiedlich starken Einfluß der verschiedenen Parameter auf die Wellenaus-breitungsgeschwindigkeit und die Stoßrelationen.

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With 13 Figures     

Parametric generation of acoustic waves and mode-locking of spin waves through magnetoelastic coupling

A YIG disk with parallel and polished surfaces was bonded onto a quartz rod with a transducer attached to it. The YIG sample was placed in a microwave cavity in a parallel-pumping configuration. For microwave power several decibels above the threshold for spin-wave instability, parametric generation of low-frequency acoustic waves was observed, and was accompanied by the simultaneous appearance of low-frequency modulation of the microwave reflected power. The effects were attributed to the beating of spin-wave side bands with the central band. The existence of side bands was confirmed by measuring the spin-wave spectrum. In the mode-locking experiment, a low-frequency acoustic wave was sent into the YIG sample through the transducer. Experimental evidences show that the acoustic wave, through magnetoelastic coupling, not only locks the phase but also enhances the amplitude of the sidebands relative to the central band.     

Droplet-shaped waves: causal finite-support analogs of X-shaped waves

A model of steady-state X-shaped wave generation by a superluminal (supersonic) pointlike source infinitely moving along a straight line is extended to a more realistic causal scenario of a source pulse launched at time zero and propagating rectilinearly at a constant superluminal speed. In the case of an infinitely short (delta) pulse, the new model yields an analytical solution, corresponding to the propagation-invariant X-shaped wave clipped by a droplet-shaped support, which perpetually expands along the propagation and transversal directions, thus tending the droplet-shaped wave to the X-shaped one.