Propagation Of Nonlinear Surface Waves Research Articles

Nonlinear piezoelectric surface acoustic waves.

The theory for nonlinear surface acoustic waves in crystals developed using Hamiltonian mechanics [Hamilton, Il'inskii, and Zabolotskaya, J. Acoust. Soc. Am. 105, 639 (1999)] is modified to account for piezoelectric material properties. The derived spectral evolution equations permit analysis of nonlinear surface wave propagation along a cut surface of any orientation with respect to the crystallographic axes and for piezoelectric crystals with any symmetry. Numerical simulations of waveform distortion in the particle velocity and electric field components are presented for surface wave propagation in Y-cut lithium niobate along the X- and Z-crystallographic axes. The influence of piezoelectricity is illustrated by comparing the nonlinear evolution of waveforms along a surface bounded by a vacuum (free space) and an ideal conductor (short circuit). Contributions to the nonlinearity from elasticity, piezoelectricity, electrostriction, and dielectricity are quantified separately for the two boundary conditions.

Wave breaking on the surface of a dielectric liquid in a horizontal electric field

The weakly nonlinear dynamics of the free surface of a dielectric liquid in an electric field directed tangentially to the unperturbed boundary is investigated numerically. Within the framework of the strong field model, when the effects of capillarity and gravity are not taken into account, it is shown that nonlinear surface waves have a tendency to break. In result of the collapse of the surface waves, the curvature of the boundary and the gradient of the local electric field undergo infinite discontinuity on the surface of the liquid. The angles of the boundary inclination remain small. The characteristic collapse time of a surface wave traveling in a given direction is calculated versus the dielectric constant of the liquid. It is shown that the time of the singularity formation increases infinitely at small and high values of the dielectric constant. The first case corresponds to the transition of the system to the neutral stability regime (the jump in electrostatic pressure at the boundary turns into zero). At high values of the dielectric constant of the liquid, the collapse time also increases. This effect is associated with the realization of a special regime of fluid motion, in which the propagation of nonlinear surface waves of an arbitrary configuration occurs without distortions. For the liquids with relative permittivity close to five, the wave breaking time reaches a minimum, i.e., the collapse of the surface waves for such liquids occurs most intensively.

Propagation of Surface Waves along a Boundary of a Photorefractive Crystal with a Nonlinear Defocusing Coating

A composite waveguide structure is considered; it is a crystal with the diffusion mechanism of formation of a nonlinear photorefractive response, on the surface of which a coating with defocusing Kerr nonlinearity is deposited. It is shown that nonlinear surface extraordinary waves of two types (with different ranges of existence and field decay profiles) exist in it. The modes of nonlinear surface wave propagation, at which the field intensity can be maximum or minimum near the surface or at the crystal–coating interface, are determined.

Propagation of nonlinear surface waves along the interface between a Kerr-type crystal and a medium characterized by stepwise dielectric permittivity

The contact of a medium characterized by Kerr nonlinearity and a medium characterized by a stepwise nonlinearity, where the dielectric permittivity depends on the amplitude of the electric field strength, is considered. It is found that an optical domain of finite thickness is formed. Two new types of surface waves are described. The influence of the switching field on the intensity distribution across the interface and on total power flow is analyzed. The intensity at the interface in the case of contact with a nonlinear focusing medium can be maximal, in contrast to the case of contact with a linear medium where the intensity at the interface could not be maximal. The total power flow monotonically decreases with an increase in the nonlinearity parameter and it takes the minimum value at a specific value of the switching amplitude. The largest fraction of the power flow is concentrated on the right side of the optical domain for small values of the switching field starting from a certain value.

Peculiarities of the nonlinear surface wave propagation in the composite waveguide with different nonlinearity types of the layers

In this paper, we explore the photorefractive crystal with diffusion nonlinearity mechanism covered by self-focusing Kerr-type crystal contacting with dielectric medium. Four types of nonlinear surface TM electromagnetic waves can propagate along the interfaces of such three-layered composite waveguide. The dispersion equations and conditions of existence of the waves depending on waveguide characteristics are analyzed. This allows to find exact and approximate solutions of dispersion equations in different cases and to obtain the dependence of propagation constant of the surface wave in explicit form. Conditions to control the field distribution across the interface planes by varying the waveguide temperature or waveguide optical and structure characteristics are presented.

Study on nonlinear surface waves along the boundary of LiNbO3 crystals

The propagation of nonlinear surface waves along the boundary of LiNbO3 crystals with diffusion and photovoltaic nonlinearities is investigated theoretically and experimentally. Surface waves with different oscillating period are obtained by changing the values of the propagation constant, and the energy of surface waves would increase monotonically with the propagation constant. Our experimental results are in good agreement with the theoretical analysis. Experimental results show also that the generation of surface waves can be shortened by increasing the power of the input beam; and the excitation efficiency of the surface waves can be enhanced by increasing the angle (less than 90 degree) between the input beam and the crystal c-axis.

Surface waves at the interface of superconductor media and nonlinear metamaterials

We theoretically study the nonlinear surface wave propagation at the interface between superconductor media and nonlinear metamaterials. The dispersion equation is analytically derived and solved numerically. Moreover, we present the power for the propagating waves at the interface. The results display different behaviors of the propagating waves as the nonlinear term or temperature is tuned. These results indicate that this structure can have potential applications in superconductor waveguide devices and integrated optics.

Analytical solution for nonlinear wave propagation in shallow media using the variational iteration method

An analytical solution is presented for nonlinear surface wave propagation. A variational iteration method (VIM) was employed and time-dependent profiles of the surface elevation level and velocity obtained analytically for different initial conditions. It is shown that the VIM used here is a flexible and accurate approach and that it can rapidly converge to the same results obtained by the Adomian decomposition method.

A model for the propagation of nonlinear surface waves over viscous muds

The effect of a thin viscous fluid–mud layer on nearshore nonlinear wave–wave interactions is studied using a parabolic frequency-domain nonlinear wave model, modified to incorporate a bottom dissipation mechanism based on a viscous boundary layer approach. The boundary-layer formulation allows for explicit calculation of the mud-induced wave damping rate. The model performed well in tests based on laboratory data. Numerical tests show that damping of high frequency waves occurs, mediated by “difference” nonlinear interactions. Simulations of 2-dimensional wave propagation over a mud “patch” of finite extent show that the wave dissipation causes significant downwave diffraction effects.

Propagation Characteristics of a Nonlinear TM Surface Wave in a Parallel Plate Superconductor/Antiferromagnet Waveguide

Nonlinear surface wave propagation properties for a symmetric planar antiferromagnet/superconductor/antiferromagnet waveguide are theoretically investigated. The attenuation constant and the phase constant, in the infrared frequency region, are numerically analyzed as a function of the thickness of superconductor. It is seen that both the attenuation constant and the phase constant decrease with increasing the thickness of superconducting film. In addition, numerical result in the total power flow reveals the existence of a threshold value in the propagation constant. Electromagnetic wave can propagate in the waveguide when propagation constant is greater than this threshold value.

Tracking of Cracks in Airplane Components Using Nonlinear Surface Wave Propagation Techniques

Ultrasonic surface waves provide a sensitive means to detect cracks in airplane structures. Until now several obstacles remained to use ultrasonic surface waves for on-line damage detection (i.e. in-flight). In this article a method will be proposed to detect a growing fatigue crack while the aircraft is operating. In contrast to classical ultrasonic measurement methods, that use a high voltage pulse, we applied an optimized multi-sine excitation signal with an amplitude of a few volts only (this agrees better with the applicable safety regulations for aircraft). Furthermore, an indicator quantifying the nonlinearity of the ultrasonic surface wave propagation is used. By using a nonlinearity index the influence of changing operation conditions that can be observed with most linear methods is eliminated. The proposed method is validated on a steel beam that is fatigue loaded with a force signal obtained from in-flight data.

Bore formation, evolution and disintegration into solitons in shallow inhomogeneous channels

Abstract. The propagation of nonlinear surface waves in channels of smoothly variable in space cross section is studied theoretically and by means of numerical computations. The mathematical model describing wave evolution is based on the generalized Korteweg-de Vries equation with additional terms due to spatial inhomogeneity and energy dissipation. Specifically we consider channels of variable depth and width. The breaking of Riemann waves and the disintegration of hydraulic jumps into trains of solitons have been examined. The results obtained can be useful in particular for the understanding some peculiarities of bore (mascaret) formation, viscous evolution and disintegration into solitons in inhomogeneous channels or rivers.

Boundary Element Method, Volume 1: Applications in Thermo-Fluids and Acoustics

Boundary Element Method, Volume 1: Applications in Thermo-Fluids and Acoustics

Electromigration-induced propagation of nonlinear surface waves.

Due to the effects of surface electromigration, waves can propagate over the free surface of a current-carrying metallic or semiconducting film of thickness h(0). In this paper, waves of finite amplitude, and slow modulations of these waves, are studied. Periodic wave trains of finite amplitude are found, as well as their dispersion relation. If the film material is isotropic, a wave train with wavelength lambda is unstable if lambda/h(0)<3.9027 ..., and is otherwise marginally stable. The equation of motion for slow modulations of a finite amplitude, periodic wave train is shown to be the nonlinear Schrödinger equation. As a result, envelope solitons can travel over the film's surface.

A numerical method for the determination of weakly non-hydrostatic non-linear free surface wave propagation

A numerical method is described that may be used to determine the propagation characteristics of weakly non-hydrostatic non-linear free surface waves over a general, bottom topography. In shallow water of constant undisturbed depth, such waves are equivalent to the familiar cnoidal waves characterized by sharp crests and relatively flat troughs. For a certain range of parameters, these propagate without change of form by virtue of the weakly non-hydrostatic balance in the vertical momentum equation. Effectively, this counters the tendency for the non-linearity in a purely hydrostatic theory to lead to a continuously deforming surface wave profile. The realistic representation furnished by cnoidal wave theory of free surface waves in the shallow near-shore zone has led to its utilization in evaluating their propagation characteristics. Nonetheless, the classic analytical theory is inapplicable to the case of wave propagation over a sloping beach or off-shore sand bar topography. Under these conditions, a local change in form of the surface wave profile is anticipated before the waves break and knowing this is required in order to evaluate fully the propagation process. The efficacy of the numerical method is first demonstrated by comparing the solution for water of constant depth with the evaluation of the analytical solution expressed in terms of the Jacobian elliptic function cn. The general method described in the paper is then illustrated by experiments to determine the change in profile of weakly non-hydrostatic non-linear surface waves propagating over bed forms representative of those found in shallow coastal seas. Copyright © 2001 John Wiley & Sons, Ltd.

Nonlinear Stoneley and Scholte waves

Spectral evolution equations are derived for plane, progressive, finite-amplitude Stoneley and Scholte waves that propagate along plane interfaces formed by two semi-infinite, isotropic media in contact. The evolution equations have mathematical forms identical to those obtained previously for Rayleigh waves [E. A. Zabolotskaya, J. Acoust. Soc. Am. 91, 2569–2575 (1992)], and they are expressed explicitly in terms of the second- and third-order elastic constants of the media. Calculations were performed to simulate nonlinear surface wave propagation in several pairs of real media. Harmonic generation and shock formation associated with the Stoneley and Scholte modes are compared with the corresponding processes in Rayleigh waves. Waveform distortion is shown to be very similar for the three types of surface waves when the propagation distance is normalized by an appropriate shock formation distance.

Measurements of surface-wave harmonic generation in nonpiezoelectric materials

Methods for exciting surface waves suitable for harmonic generation experiments in nonpiezoelectric materials have been investigated. Such methods are needed for improved characterization of engineered surfaces. Unlike conventional ultrasonic techniques, nonlinear experiments require the ability to produce finite-amplitude, spectrally pure surface waves. Comb structures were selected from the methods evaluated, because they appear to offer the greatest potential for harmonic generation experiments on arbitrary substrates. The out-of-plane component of the surface waves was measured using a Michelson interferometer. The apparatus enabled direct measurement of absolute displacements with a spatial resolution of approximately 50 μm. Using this combination of excitation and detection techniques, the evolution of narrowband surface waves in the 1–10 MHz range was studied. Displacement amplitudes as large as 35 nm at a fundamental frequency of 10 MHz were observed. The spatial behavior of the fundamental and second-harmonic displacements indicated that, for correct interpretation, it was necessary to account for diffraction and other geometrical effects. Quantitative results will be presented for a variety of experimental configurations, including different substrates and comb shapes. Experimental results will also be compared to predictions of available theories for nonlinear surface wave propagation.

Temporal modulation of nonlinear surface waves and soliton emission at nonlinear slab waveguides

The propagation of pulsed Gaussian beams end-firing the fundamental mode of a linear thin film covered with a nonlinear medium has been analysed by means of a variational method. Fundamental parameters for soliton emission and nonlinear surface wave propagation, such as critical power and film thickness, have been calculated. The role of the pulse shape has been investigated by comparing the behaviour of Gaussian and hyper-Gaussian pulses. The variational expressions obtained for the evolution of the beam width and trajectory are shown to be in good agreement with numerical simulations.

Nonlinear surface wave propagation in a piezoelectric material

Model equations were derived from first principles for nonlinear surface wave propagation in a piezoelectric material. There is no dependence on ad hoc or empirical parameters. The present work extends an earlier analysis of surface waves in crystals, described by the authors at the previous meeting [J. Acoust. Soc. Am. 99, 2538(A) (1996)]. As in the earlier work, the model equations account for crystals having arbitrary symmetry, and for surface wave propagation in arbitrary directions in planes having arbitrary orientations with respect to the crystalographic axes. Here, elastic, piezoelectric, dielectric, and electrostrictive properties of the material are taken into account. The model is formulated as spectral evolution equations that are integrated numerically to illustrate the distortion of a finite amplitude surface wave that is sinusoidal at the source. The material we considered is LiNbO3, for which measured values of all required second- and third-order elastic constants are available in the literature [e.g., the latter are reported by Cho and Yamanouchi, J. Appl. Phys. 61, 875 (1987)]. Analysis of the nonlinearity matrix permits identification of which physical effects contribute most to the distortion of nonlinear surface waves. [Work supported by ONR, NSF, and Schlumberger Foundation.]

Nonlinear surface wave propagation in crystals.

Elastic wave propagation along the free surface of a crystal differs from Rayleigh wave propagation in an isotropic solid. In particular, the phase speed and depth dependence vary with the direction of propagation with respect to the crystallographic axes, and the wave-vector component perpendicular to the surface is complex. Results from a theoretical investigation of nonlinear propagation along a free surface of a crystal are presented. The analysis is based on spectral equations that are derived with Hamiltonian formalism. The equations were integrated numerically to illustrate finite-amplitude distortion of a surface wave that is sinusoidal at the source. In one example, calculations based on published material properties are presented for surface wave propagation in KCl out to distances beyond where shock formation occurs. Waveform distortion associated with propagation along the [111] axis in KCl was found to be similar to that of Rayleigh waves in isotropic solids. However, for propagation along the [001] axis in KCl, energy transfer from the fundamental to the second harmonic component is very inefficient, and the waveform distortion was found to differ considerably from that of Rayleigh waves. [Work supported by NSF, ONR, and the Schlumberger Foundation.]