A general theory of harmonic wave propagation in linear periodic systems with multiple coupling

A general theory is presented of harmonic wave propagation in one-dimensional periodic systems with multiple coupling between adjacent periodic elements. The motion of each element is expressed in terms of a finite number of displacement coordinates. The nature and number of different wave propagation constants at any frequency are discussed, and the energy flow associated with waves having real, complex or imaginary propagation constants is investigated. Kinetic and potential energy functions are derived for the propagating waves and a generalized Rayleigh's Quotient and Rayleigh's Principle for the complex wave motion have been found. This is extended to yield a generalized Rayleigh-Ritz method of finding approximate, yet accurate, relationships between the frequencies and propagation constants of the propagating waves. The effect of damping is also considered, and a special class of “damped forced waves” is postulated for hysteretically damped periodic systems. An energy definition for the loss factor of these waves is found. Briefly considered is the two-dimensional multi-coupled periodic system in which a simple wave motion analogous to a plane wave propagates across the whole system.     

Local run-up amplification by resonant wave interactions

Until now, the analysis of long wave run-up on a plane beach has been focused on finding its maximum value, failing to capture the existence of resonant regimes. One-dimensional numerical simulations in the framework of the nonlinear shallow water equations are used to investigate the boundary value problem for plane and nontrivial beaches. Monochromatic waves, as well as virtual wave-gage recordings from real tsunami simulations, are used as forcing conditions to the boundary value problem. Resonant phenomena between the incident wavelength and the beach slope are found to occur, which result in enhanced run-up of nonleading waves. The evolution of energy reveals the existence of a quasiperiodic state for the case of sinusoidal waves. Dispersion is found to slightly reduce the value of maximum run-up but not to change the overall picture. Run-up amplification occurs for both leading elevation and depression waves.     

Reflexion und Brechung an optisch einachsigen Medien in Richtung fehlender Doppelbrechung

Reflection and Refraction at Uniaxial Media in Directions of Vanishing Double Refraction The amplitude relations for the reflection and refraction of plane monochromatic waves by uniaxial media in the case of vanishing double refraction of the refracted respectively reflected waves in the uniaxial medium are obtained by a limiting process from generalized Fresnel formulae earlier derived. The incidence both from the isotropic and from the uniaxial medium is considered. The boundary conditions may be fulfilled in the case considered only by addition of refracted respectively reflected waves with amplitudes varying linearly with the distance from the boundary plane in the uniaxial medium. The requirement of vanishing double refraction restricts the possible choice of the common tangential components of all refraction vectors and consequently also of the refraction vectors of incident waves from a primarily twodimensional complex manifold to onedimensional complex manifolds, which are determined in this paper.     

A feasibility study of the use of bounded beams resembling the shape of evanescent and inhomogeneous waves

Plane waves are solutions of the visco-elastic wave equation. Their wave vector can be real for homogeneous plane waves or complex for inhomogeneous and evanescent plane waves. Although interesting from a theoretical point of view, complex wave vectors normally only emerge naturally when propagation or scattering is studied of sound under the appearance of damping effects. Because of the particular behavior of inhomogeneous and evanescent waves and their estimated efficiency for surface wave generation, bounded beams, experimentally mimicking their infinite counterparts similar to (wide) Gaussian beams imitating infinite harmonic plane waves, are of special interest in this report. The study describes the behavior of bounded inhomogeneous and bounded evanescent waves in terms of amplitude and phase distribution as well as energy flow direction. The outcome is of importance to the applicability of bounded inhomogeneous ultrasonic waves for nondestructive testing.     

Harmonic Green's functions for flexural waves in semi-infinite plates with arbitrary boundary conditions and high-frequency approximation for convex polygonal plates

This paper provides a method for obtaining the harmonic Green's function for flexural waves in semi-infinite plates with arbitrary boundary conditions and a high frequency approximation of the Green's function in the case of convex polygonal plates, by using a generalised image source method. The classical image source method consists in describing the response of a point-driven polygonal plate as a superposition of contributions from the original source and virtual sources located outside of the plate, which represent successive reflections on the boundaries. The proposed approach extends the image source method to plates including boundaries that induce coupling between propagating and evanescent components of the field and on which reflection depends on the angle of incidence. This is achieved by writing the original source as a Fourier transform representing a continuous sum of propagating and evanescent plane waves incident on the boundaries. Thus, the image source contributions arise as continuous sums of reflected plane waves. For semi-infinite plates, the exact Green's function is obtained for an arbitrary set of boundary conditions. For polygonal plates, a high-frequency approximation of the Green's function is obtained by neglecting evanescent waves for the second and subsequent reflections on the edges. The method is compared to exact and finite element solutions and evaluated in terms of its frequency range of applicability.     

Non-equilibrium thermodynamics of interfacial systems

The theory of non-equilibrium thermodynamics is applied to a multi-component system containing an interface using a method developed by Bedeaux, Albano, and Mazur. A singular mass density is allowed at the interface as well as singular densities of energy and entropy. All currents are also allowed to be singular at the interface. The conservation laws and the Gibbs relation for the entropy are used to derive the entropy production at the surface. Linear laws relating the fluxes and thermodynamic forces are presented. The theory is then applied to a two component system where one of the phases is a liquid and the other a low density gas and the boundary conditions at the free surface of the liquid derived. The boundary conditions include the conditions used by Levich in his theory of the damping of waves by surface-active substances, but include other effects as well.     

Plane harmonic waves in orthorhombic thermoelastic materials

Keeping in view the increased usage of orthorhombic materials in the development of advanced engineering materials such as fibers and composites and other multilayered media, the aim of the present paper is to give a detailed account of the plane harmonic generalized thermoelastic waves in orthorhombic materials. According to the frequency equation, the four waves, a quasi-longitudinal, two quasi-transverse, and a quasi-thermal wave, can propagate in an orthorhombic crystal. When plane waves propagate along the axis of an orthorhombic solid, then only the longitudinal and thermal waves are coupled, whereas the transverse waves get decoupled from the rest of the motion. For plane waves propagating in one plane of the solid, only the SH wave in that plane remains purely transverse and gets decoupled from the rest of the motion and vice versa. The other three waves are coupled and get modified due to thermal variations and relaxation time. The particle paths and stability of the waves have been discussed and the results verified numerically. These have been represented graphically for single crystals of solid helium and cobalt material.     

Propagation of acoustic waves in a medium with the Rayleigh energy release mechanism

This research continues theoretical studies of propagation of acoustic waves in a plasma considering it in the context of a Rayleigh medium. For the first time, the solution to the problem with the boundary and not the initial conditions is examined. It is shown that for small values of the parameter characterizing the energy input in the plasma, the amplification coefficients of a harmonic acoustic wave in the problem of propagation of the initial perturbation and in the problem with the boundary conditions are close. However, if the energy input increases, the amplification of the wave propagating from the source is larger than in the problem of the initial perturbation propagation. The same concerns the amplification of waves with different frequencies for fixed parameters of the plasma; i.e., the difference between the amplification coefficients is larger, the lower the wave frequency. The resultant analytic dependences make it possible to determine exactly which of the problems (with the initial or boundary conditions) should be solved to compute the amplification coefficient of acoustic waves under specific experimental conditions.     

The investigation of a two-layer fluid soliton pair using phase plane analysis

Nonlinear long waves theory in a two-layer fluid system has been studied. The dynamical equations according to the normalized heights in first order are obtained using the reductive perturbation, method and the equations of shallow water in each fluid and taking boundary conditions appropriate into account Conserve energy form by definition a independent variable is found. By definition a Ljapunov function, the condition for stability are shown. A newtechnique was used to prove stability as well as existence of soliton pair using phase plane analysis.     

A vertical eigenfunction expansion for the propagation of sound in a downward-refracting atmosphere over a complex impedance plane

The propagation of sound in a stratified downward-refracting atmosphere over a complex impedance plane is studied. The problem is solved by separating the wave equation into vertical and horizontal parts. The vertical part has non-self-adjoint boundary conditions, so that the well-known expansion in orthonormal eigenfunctions cannot be used. Instead, a less widely known eigenfunction expansion for non-self-adjoint ordinary differential operators is employed. As in the self-adjoint case, this expansion separates the acoustic field into a ducted part, expressed as a sum over modes which decrease exponentially with height, and an upwardly propagating part, expressed as an integral over modes which are asymptotically (with height) plane waves. The eigenvalues associated with the modes in this eigenfunction expansion are, in general, complex valued. A technique is introduced which expresses the non-self-adjoint problem as a perturbation of a self-adjoint one, allowing one to efficiently find the complex eigenvalues without having to resort to searches in the complex plane. Finally, an application is made to a model for the nighttime boundary layer.     

Phase velocities of plane waves in a pipe with a flow whose velocity varies along the radius

The phase velocities of plane waves in a pipe filled with a moving acoustic medium are studied for different laws of flow velocity variation along the pipe radius. The wave equation is solved by the discretization method, which breaks the entire pipe volume into individual cylinders under the assumption that, within each of the cylinders, the flow velocity of the medium is constant. This approach makes it possible to reduce the solution to the wave problem to solving Helmholtz equations for individual cylinders. Based on boundary conditions satisfied at the boundaries between neighboring cylinders, a homogeneous system of linear algebraic equations is obtained. From this system, with the use of the scattering matrices, a simple dispersion equation is derived for determining the phase velocities of plane waves. The stability of the numerical solution to the dispersion equation with respect to the number of cylinders is investigated. The phase velocities of quasi-homogeneous and inhomogeneous waves in a pipe are numerically calculated and analyzed for different velocities of a moving medium and different laws of flow velocity variation along the radius. It is shown that the variation that occurs in the phase velocity of a homogeneous plane wave in a pipe due to the motion of the medium is identical to the mean flow velocity for different laws of flow velocity variation along the radius. For inhomogeneous plane waves, the phase velocity increment exceeds the mean flow velocity several times and depends on both the law of wave amplitude distribution along the radius and the law of the flow velocity variation along the radius.     

Resonances of the Rayleigh waves in an elastic semi-infinite strip

Kirchhoff’s theory of plates is used to study forced harmonic vibrations of a semi-infinite strip when the latter is in the generalized stressed state or experiences flexural deformation. The forced vibrations are excited by a load applied to the strip end. Cross-boundary conditions are imposed on the strip’s sides, which allows one to obtain a closed solution. The presence of an infinite real frequency spectrum corresponding to the edge resonances is revealed. The relation of these resonances to the Rayleigh planar and flexural waves is established.     

Reflection and refraction of waves in oscillatory media

This paper uses the two-dimensional Brusselator model to study reflection and refraction of chemical waves. It presents some boundary conditions of chemical waves, with which occurence of observed phenomena at interface as refraction and reflection of chemical waves can be interpreted. Moreover, the angle of reflection may be calculated by using the boundary conditions. It finds that reflection and refraction of chemical waves can occur simultaneously even if plane wave goes from a medium with higher speed to a medium with lower speed, provided the incident angle is larger than the critical angle.     

Electromagnetic waves in uniaxial crystals with metallized boundaries: Mode conversion,pure reflections,and bulk polaritons

A theory is constructed for the reflection of plane electromagnetic waves in uniaxial crystals with a positive definite permittivity tensor and an arbitrarily oriented metallized boundary. The problem is solved both for general-position orientations corresponding to three-partial reflection and for special conditions allowing two-partial reflections: mode conversions when the incident and reflected waves belong to different sheets of the refraction surface and “pure” reflections when both waves belong to the same sheet. The space of pure reflections is shown to be formed by two types of optical-axis orientations: arbitrary directions in the plane of the crystal surface and in the plane of incidence. The configurations of the conversion surface for optically positive and negative crystals are investigated. A subspace of pure reflections that transform into one-partial bulk polaritons with the energy flux parallel to the surface at grazing incidence has been found. The domain of existence of such bulk eigenmodes is bounded by two “lines” of solutions. These are any directions along the boundary containing the optical axis for ordinary polaritons and the direction along the projection of the optical axis onto the surface at an arbitrary orientation of the axis with respect to the boundary for extraordinary polaritons.     

Periodic and chaotic behaviour in a reduced form of the perturbed generalized Korteweg–de Vries and Kadomtsev–Petviashvili equations

The dynamical behaviour of a reduced form of the perturbed generalized Korteweg–de Vries and Kadomtsev–Petviashvili equations (extension of the Korteweg–de Vries equation to two space variables) are studied in this paper. Harmonic solutions of non-resonance and primary resonance are obtained using the perturbation method. Chaotic motion under harmonic excitations is studied using the Melnikov method.A wide range of solutions for the reduced perturbed generalized Korteweg–de Vries equations, in which non-linear phenomena appearing within transition from regular harmonic response (periodic solutions) to chaotic motion, are obtained using the time integration Runge–Kutta method. When chaos is found, it is detected by examining the phase plane, the Poincaré map, the sensitivity solution of the solution to initial conditions, and by calculating the largest Lyapunov exponent.     

Propagation of electromagnetic waves through very thin non-Hermitian slabs

A theoretical model is proposed to study the propagation of monochromatic electromagnetic plane waves through very thin slabs having gain or loss. The analysis is based on solving the differential equations directly and applying the appropriate boundary conditions. The optical response of the slabs is considered in the linear and nonlinear regimes with a Kerr-type nonlinearity. A number of relations between the real and imaginary parts of the dielectric function are derived in order that useful effects related to the non-Hermiticity of the system be presented. The transmission of plane waves through a system consisting of two thin slabs satisfying the condition of Parity-Time symmetry is also considered.     

A note on Snell laws for electromagnetic plane waves in lossy media

Based on Maxwell's equations and Ohm's law, we rederived the Snell laws for reflection and transmission of harmonic inhomogeneous plane electromagnetic (EM) waves propagating through planar lossy interfaces. The present results are new, simple and exact and they recover the ordinary Snell laws in the case of lossless media. Besides, these results show that the wave propagation direction strongly depends on the polarization state of the EM wave and the lossy media can behave as a polarizing device. Moreover, we verify that in low frequency regime these traveling waves do not exhibit total internal reflection at interfaces between two adjacent lossy media.     

圆形活塞声源的有限振幅反射波

本文研究了圆形活塞声源的有限振幅声波的反射。应用像源法计算已产生的谐波本身在边界上的反射,将它们看成是反射边界上振动的新声源所产生的辐射场,再应用弱冲击理论来计算这部分场自己产生的谐波场。在水池中作了二阶谐波的反射实验,反射界面是水池壁。结果表明,谐波声压随距离的平均衰减规律与理论符合得很好。实验还表明,反射谐波场沿纵向距离以及与它相垂直的横断面上有强烈的干涉现象,而纵向相干长度比横向相干长度大一个数量级。通过理论估计,认为这种现象是由于反射壁的有限厚度、它的两个平面之间有一个很小的角度以及发射波束有一定的宽度所造成的。这样估算出来的两个相干长度与实验符合。 关键词：     

同频率互相垂直简谐振动的合成

讨论了质点参与两个同频率互相垂直简谐振动时的角动量、速率、能量以及同频率互相垂直简谐振动的合成推广到三维的情况。