Plane Waves Of Finite Amplitude Research Articles

1D Numerical Study of Nonlinear Propagation of Finite Amplitude Waves in Traveling Wave Tubes with Varying Cross Section

The nonlinear acoustic problem of a finite amplitude plane wave propagating along the axial direction in a traveling wave tube is studied. Based on the one-dimensional Westervelt equation, a one-dimensional nonlinear wave equation is derived in which the cross section of the traveling wave tube is considered. The two-order finite difference scheme is used to solve the nonlinear wave equation. The nonlinear propagation characteristics of a finite amplitude wave in the traveling wave tube is analyzed. In the expanding transition section, the acoustic pressure amplitude of the acoustic wave decreases with the increase of the cross-sectional area of the pipeline. The nonlinear characteristics of the acoustic wave show waveform distortion and harmonic growth. The waveform distortion becomes more serious in the rear of traveling wave tube than in the front of the tube. Considering the acoustic reflection condition at the mouth, the influence of differently shaped diffusion sections on the acoustic pressure distribution in the test section is investigated. The larger the change rate of the diffusion section in an area, the less amplitude of the sound pressure, and the nonlinear effect of the sound wave propagation is weakened. These nonlinear wave propagation characteristics in a travelling wave tube provide important guidance for both designing a uniform sound pressure distribution in the test section and determining the optimal measuring points for different sizes of structures in spacecraft.

Isolated frequencies at which nonlinear materials behave linearly.

In this Rapid Communication, we demonstrate that specific frequencies in weakly nonlinear lattices avoid the generation of higher harmonics, and thus the lattices behave linearly. Using a multiple scales analysis, we present plane-wave solutions that persist at only a single frequency and wave number; i.e., whose spatiotemporal production of higher harmonics is remarkably small. We study monatomic and diatomic chains with quadratic and cubic stiffness nonlinearities as example systems. Direct numerical integration of the equations of motion confirms that finite amplitude plane waves assigned to these special frequencies produce negligible higher harmonics when injected into the lattices. Such findings provide new considerations for the operating frequency of nonlinear communications devices, sensors, and transducers for enhanced signal-to-noise ratios.

Initial-Boundary Value Problem for Stimulated Raman Scattering Model: Solvability of Whitham Type System of Equations Arising in Long-Time Asymptotic Analysis

An initial-boundary value problem for a model of stimulated Raman scattering was considered in [Moskovchenko E.A., Kotlyarov V.P., J. Phys. A: Math. Theor. 43 (2010), 055205, 31 pages]. The authors showed that in the long-time range $t\to+\infty$ the $x>0$, $t>0$ quarter plane is divided into 3 regions with qualitatively different asymptotic behavior of the solution: a region of a finite amplitude plane wave, a modulated elliptic wave region and a vanishing dispersive wave region. The asymptotics in the modulated elliptic region was studied under an implicit assumption of the solvability of the corresponding Whitham type equations. Here we establish the existence of these parameters, and thus justify the results by Moskovchenko and Kotlyarov.

Comparison between the Gol’dberg number and the Morfey-Howell Q/S for finite-amplitude plane waves

In this paper we compare the two most popular indicators of nonlinearity, namely, the Gol’dberg number [Sov. Phys. Acoust. 2, 346-350 (1956)] and the Morfey-Howell Q/S [AIAA J. 19, 986-992 (1981)] for the case of an initially-sinusoidal, finite-amplitude plane wave in a thermoviscous medium. The Gol’dberg number Γ, defined as the ratio of the absorption length at the fundamental frequency to the shock formation distance, is a parameter that single-handedly dictates the entire history of the wave [for example, weak (Γ < 1) and strong (Γ >> 1) waves]. However, it fails to quantify the strength of nonlinearity that each individual harmonic “sees” at any given point. In other words, not every harmonic sees the apparent strength of nonlinearity according to Γ. The Q/S, on the other hand, is defined for each harmonic as a function of propagation distance, and hence keeps track of the ever-changing strength of nonlinearity during the course of propagation. One interesting ramification of this is that using Q/S it is possible to pinpoint the distance at which the value of nonlinearity relative to absorption actually becomes Γ for the fundamental component.

The Benjamin–Feir instability and propagation of swell across the Pacific

About 40 years ago, Snodgrass and other oceanographers (1966) tracked ocean swell propagating across the entire Pacific Ocean. At about the same time, several investigators (including Benjamin and Feir) showed that a uniform train of plane waves of finite amplitude on deep water is unstable. Comparing these two results, each of which is highly cited, leads to the following question: in light of this instability, how did the waves tracked by oceanographers travel coherently more than 10,000km across the Pacific Ocean? A possible explanation is provided in recent work that re-examined the Benjamin–Feir instability in the presence of linear damping. The conclusion was that even small amounts of damping can stabilize the instability before nonlinear effects become important. In addition, the theoretical predictions agree well with results from laboratory experiments. In this paper we re-examine ocean data from 1966 to estimate whether the oceanic damping that was measured could have controlled the Benjamin–Feir instability for the swell that was tracked. We find that for one set of ocean swell, dissipation controls the Benjamin–Feir instability enough to allow coherent wave propagation across the Pacific. For a second set of ocean swell, it does not. For a third set of ocean swell, an integral that the theory predicts to be constant is not constant in the data; it decreases and correspondingly the spectral peak shifts to a lower frequency—this is frequency downshifting. For this case the theory is not an adequate model, so the corresponding Benjamin–Feir analysis can be misleading. Thus, our results from the historical records are inconclusive: we can assert neither that dissipation of ocean swell is always negligible, nor that it is always important. But our results show that dissipation can control the Benjamin–Feir instability for small-amplitude waves and that downshifting occurs in ocean swell with relatively small wave slopes.

Multipole expansion for nonlinear acoustic scattering at the difference-frequency.

In this work, the scattering at the difference-frequency (DF) of two finite-amplitude plane waves by a rigid sphere in a fluid is studied. The DF-scattering is present in some acoustic imaging methods such as vibro-acoustography. This has motivated us to perform this study. We obtain for the first time the multipole expansion for the DF-scattered pressure in the farfield. Theory is based on the successive approximation and the Green’s function methods. From the multipole expansion, we derive some quantities such as scattering cross-section and the total DF-scattered power. Furthermore, we show that the DF-scattered pressure increases with the observation distance r like r ln r, while it varies almost linearly with the difference-frequency. Despite the amplification, we demonstrate that the DF-scattered pressure does not grow indefinitely because of absorption in the fluid. The theory is applied to the scattering in the Rayleigh limit (scatterer much smaller than the incident wavelengths). Only monopole and dipole moments are relevant in this limit. In conclusion, we believe that this theory can help to understand better the nonlinear scattering in acoustics.

Linearly polarized finite-amplitude plane waves in pre-strained incompressible elastic materials

In this paper, general incompressible non-linear elastic materials are considered and also a sub-class of these called “generalized neo-Hookean materials”. We deal with the propagation of finite-amplitude linearly polarized transverse waves in such materials subjected to an arbitrary static homogeneous deformation. Contrary to the case of waves of infinitesimal amplitude (incremental theory), linearly polarized waves are in general not possible for every propagation direction. In this paper, we present two cases when such finite-amplitude waves may propagate. The first case is that of generalized neo-Hookean materials, for which it is shown that a finite-amplitude linearly polarized wave may propagate in any direction. In general, the polarization direction is uniquely determined. The second case is that of general incompressible elastic materials, but the propagation and polarization directions are both assumed to be in the same principal plane of the static deformation. In both cases, the wave motion is governed by a single scalar non-linear wave equation. For simple wave solutions of this equation, a property relating the energy density and the projection of the energy flux onto the propagation direction is derived.

Determination of the nonlinear parameter by propagating and modeling finite amplitude plane waves

The acoustic nonlinear parameter, B∕A, is an important piece of data whenever high intensity pressure fields are under consideration. In this work, an alternative method is proposed to measure this parameter. First, the method involves measuring the sound velocity and nonlinear waveform distortion of a finite amplitude plane wave propagating through a medium, Butanediol, whose density and attenuation law have been preliminarily determined. Measurements were performed in the nearfield of a piston where plane wave propagation regime exists. Impulse response of the hydrophone was determined and pressure waveforms were obtained by a convolution process. Then, the method involves modeling, in time domain and under experimental conditions, the theoretical nonlinear waveform distortion and fitting it to the experimental results by adjusting the B∕A parameter. Comparative measurements were performed using the technique of parametric interaction. The respective results for the two methods (B∕A=11.0±10% and 10.9±5%) are in a good agreement despite a smaller degree of accuracy for the proposed method.

Classic Papers in Shock Compression Science

A Paper on the Theory of Sound.- On a Difficulty in the Theory of Sound.- On the Mathematical Theory of Sound.- The Propagation of Planar Air Waves of Finite Amplitude.- On the Thermodynamic Theory of Waves of Finite Longitudinal Disturbance.- The Life and Work of Pierre Henri Hugoniot.- On the Propagation of Motion in Bodies and in Perfect Gases in Particular -I.- On the Propagation of Motion in Bodies and in Perfect Gases in Particular - II.- Aerial Plane Waves of Finite Amplitude.- The Conditions Necessary for Discontinuous Motion in Gases.- On the Theory of Shock Waves for an Arbitrary Equation of State.- Shock Waves in Arbitrary Fluids.

On self-consistent stationary propagation of relativistically coupled electromagnetic and electrostatic waves. II. Effects of electron–positron pair creation

Effects of electron–positron pair creation on the stationary one-dimensional propagation of relativistically coupled electromagnetic and electrostatic waves are studied based on the self-consistent model presented in the previous paper for an electron–ion plasma [L. N. Tsintsadze and K. Nishikawa, Phys. Plasmas 3, 511 (1996)]. The pair-created particles are treated as a cold electron–positron plasma at their creation point. In addition to the results similar to those obtained for the electron–ion plasma, a new type of instability of the finite amplitude plane wave is found which is purely growing in the wave frame near threshold. A novel solution describing an envelope shock which represents a wake-field excitation by a solitary electromagnetic pulse is obtained by taking into account the trapping of the pair-created particles in the upstream region.

A study of finite amplitude plane wave propagation in a rubber-like solid

Finite amplitude propagation of plane longitudinal waves in an isotropic compressible hyperelastic solid, with properties typical of solid rubbers, is considered. The surface of a half space is subjected to a spatially uniform application of compressive stress which results in wave propagation with small finite deformation. A nonheat conducting solid is assumed and results of the adiabatic and isentropic approximations are compared. The magnitude of the jump in the Riemann invariant across a shock is investigated and, for certain boundary-initial value problems, numerical solutions which neglect this jump are obtained and compared with other numerical solutions.

On self-consistent stationary propagation of relativistically coupled electromagnetic and electrostatic waves in cold electron-ion plasma

A general self-consistent theory is presented for one-dimensional stationary propagation of relativistically coupled electromagnetic and electrostatic waves. A finite amplitude plane wave solution is found to be always unstable against modulation. A modulated finite amplitude plane wave solution is found in a narrow region of the frequency, and the possibility for a solitary wave solution and its properties are discussed.

Finite-amplitude plane waves in deformed Hadamard elastic materials : P. Boulanger, M. Hayes & C. Trimarco, Geophysical Journal International, 118(2), 1994, pp 447–458

Finite-amplitude plane waves in deformed Hadamard elastic materials : P. Boulanger, M. Hayes & C. Trimarco, Geophysical Journal International, 118(2), 1994, pp 447–458

Finite-amplitude plane waves in deformed Hadamard elastic materials

SUMMARY It has been shown previously (John 1966; Currie & Hayes 1969) that Hadamard compressible elastic materials are the only ones for which three linearly polarized finite-amplitude plane waves, one longitudinal and two transverse, may propagate in any direction when the material is maintained in a state of arbitrary static finite homogeneous deformation. Here we first obtain simple explicit expressions for the three wave speeds and a simple characterization of the polarization directions of the transverse waves. Also, although the theory is non-linear, it is seen that the three waves may be superposed: they do not interact. Special directions, called acoustic axes, are introduced. These are the only directions along which circularly polarized transverse waves may propagate. They are determined only by the basic static deformation of the material. Results are expressed in terms of the angles that the propagation direction makes with the acoustic axes. Then, the energy flux and energy density of the waves are considered. Relations between the projection on to the propagation direction of the energy flux vector and the energy density are derived for each individual wave. Finally, periodic transverse waves are considered.

Fundamental Physics of Ultrasound

First Page

On the linearity of the momentum equation for progressive plane waves of finite amplitude

This Letter is primarily in response to the Letter to the Editor by G. M. Tarkenton entitled ‘‘Remarks on ‘On the coefficient of nonlinearity β in nonlinear acoustics’ ’’ [J. Acoust. Soc. Am. 88, XXXX–XXXX (1990)]. It is reaffirmed that for progressive plane waves, the momentum equation contributes nothing at second order to the expressions for the coefficient of nonlinearity and finite amplitude propagation speed.

Nearly real fronts in a Ginzburg–Landau equation

SynopsisSubcritical fronts are shown to exist in a quintic version of the well-known complex Ginzburg–Landau equation, which has a subcritical pitchfork as well as a supercritical saddle-node bifurcation. The fronts connect a finite amplitude plane wave state to a stable zero solution. The unstable manifold at finite amplitude and stable manifold of vanishing amplitude solutions are shown to intersect transversely on an invariant zero-wavenumber manifold with parameters set to be real. By the persistence of transverse intersection, frontal connections exist for a continuum of nearly real fronts parametrised by appropriate variables that exhibit some interesting changes in dimension.

Sum and difference frequency generation due to noncollinear wave interaction in a rectangular duct

Noncollinear wave interaction in a rectangular duct is investigated both theoretically and experimentally. An inhomogeneous wave equation, exact to second order in the field variables, is derived for the sum and difference frequency pressure waves generated by noncollinear interaction of two finite amplitude plane waves in a lossless fluid. This equation is extended to the interaction of waves in higher-order modes of a rectangular duct. Quasilinear solutions are obtained, and tube wall attenuation is included ad hoc. Experimental results are reported for the interaction of waves in the (0,0) and (1,0) modes of an air-filled rectangular duct. Theory and experiment are in excellent agreement with regard to oscillatory structure of the sum and difference frequency wave fields. Although overall agreement between theory and experiment is reasonable (±2 dB), it is not within estimated experimental error (±1 dB). It is shown that because local rather than cumulative nonlinear effects dominate the interaction, knowledge of the proper second-order source condition is of crucial importance. Discrepancies between the predicted and measured amplitudes are attributed to an inadequate description of the source condition.

Generalized Burgers equation for plane waves

Burgers’ equation, an equation for plane waves of finite amplitude in thermoviscous fluids, is generalized by replacing the thermoviscous term Aut′t′ (A is the thermoviscous coefficient, u particle velocity, and t′ retarded time) with an operator L(u). This operator represents the effect of attenuation and dispersion, even if known only empirically. Specific forms of L(u) are given for thermoviscous fluids, relaxing fluids, and fluids for which viscous and thermal boundary layers are important. A method for specifying L(u) when the attenuation and dispersion properties are known only empirically is described. A perturbation solution of the generalized Burgers equation is carried out to third order. An example is discussed for the case α2=2α1, where α1 and α2 are the small-signal attenuation coefficients at the fundamental and second-harmonic frequencies, respectively. The growth/decay curve of the second harmonic component is given both with and without the inclusion of dispersion. Dispersion causes a small reduction of the component. The extension of the generalized Burgers equation to cover nonplanar one-dimensional waves is given.

Plane waves in simple elastic solids and discontinuous dependence of solution on boundary conditions

Using stress as the dependent variable instead of the deformation gradient, plane waves of finite amplitude in simple elastic solids are studied. For isotropic materials there are two plane polarized simple waves as well as shock waves and one circularly polarized simple wave which can also be regarded as a shock wave. With the aid of the stress paths for simple waves and shock waves in the stress space introduced here, one can see clearly what combination of simple waves and/or shock waves is needed to satisfy the initial and boundary conditions. We use second order isotropic hyperelastic materials to illustrate the ideas. In one example we show that the solution requires as many as four simple waves. In another we show that depending on the boundary condition there are more than eight possible solutions to the problem. We also present an example in which the solution does not depend continuously on the boundary condition. This implies that in experiments if the applied load at the boundary is not properly controlled, any slight deviation in the applied load would result in a finite different response in the material.