Lev A. Ostrovsky

Nonlinear Wave Processes in Acoustics

Preface 1. Nonlinearity, dissipation and dispersion in acoustics 2. Simple waves and shocks in acoustics 3. Nonlinear geometrical acoustics 4. Nonlinear sound beams 5. Sound-sound interaction (nondispersive medium) 6. Nonlinear acoustic waves in dispersive media 7. Self-action and stimulated scattering of sound Conclusion Subject index.

Long Nonlinear Surface and Internal Gravity Waves in a Rotating Ocean

Nonlinear dynamics of surface and internal waves in a stratified ocean under the influence of the Earths rotation is discussed. Attention is focussed upon guided waves long compared to the ocean depth. The effect of rotation on linear processes is reviewed in detail as well as the existing nonlinear models describing weakly and strongly nonlinear dynamics of long waves. The influence of rotation on small-scale waves and two-dimensional effects are also briefly considered. Some estimates of the influence of the Earths rotation on the parameters of real oceanic waves are presented and related to observational and numerical data.

Internal solitons in the ocean and their effect on underwater sound

Nonlinear internal waves in the ocean are discussed (a) from the standpoint of soliton theory and (b) from the viewpoint of experimental measurements. First, theoretical models for internal solitary waves in the ocean are briefly described. Various nonlinear analytical solutions are treated, commencing with the well-known Boussinesq and Korteweg-de Vries equations. Then certain generalizations are considered, including effects of cubic nonlinearity, Earths rotation, cylindrical divergence, dissipation, shear flows, and others. Recent theoretical models for strongly nonlinear internal waves are outlined. Second, examples of experimental evidence for the existence of solitons in the upper ocean are presented; the data include radar and optical images and in situ measurements of wave forms, propagation speeds, and dispersion characteristics. Third, and finally, action of internal solitons on sound wave propagation is discussed. This review paper is intended for researchers from diverse backgrounds, including acousticians, who may not be familiar in detail with soliton theory. Thus, it includes an outline of the basics of soliton theory. At the same time, recent theoretical and observational results are described which can also make this review useful for mainstream oceanographers and theoreticians.

Nonlinear acoustics of micro-inhomogeneous media

Abstract Acoustic waves can interact in micro-inhomogeneous media much more intensively than in homogeneous media. This has been repeatedly observed in experiments with ground species, marine sediments, porous materials and metals. This paper considers two models of such media which seem to be applicable to the description of these results. One of them is based on the consideration of nonlinear sound scattering by separate spherical cavities in liquids and solids. The second model is based on the phenomenological stress-deformation relation in solids with microplasticity which often has hysteresis (heritage) properties associated with the micro-inhomogeneities. In metals, for example, it is caused by the movement of dislocations. Different nonlinear effects in such media (harmonic and combination frequency generation, nonlinear, variations of resonance frequency amplitude-dependent losses) are considered. Some results of experiments with metallic resonators supporting the theory developed here are also presented. These mechanisms may determine the nonlinear properties of real soils and rocks summarized in a table given in the paper.

Internal solitons in the ocean

Internal waves (IW) are among the important factors affecting sound propagation in the ocean. A special role may be played by solitary IWs because of their spatial localization and high magnitudes. Here, nonlinear IWs are discussed (a) from the standpoint of soliton theory and (b) from the viewpoint of experimental measurements. First, basic theoretical models for solitary IWs in the ocean are described, and various analytical solutions are treated, commencing with the well‐known Korteweg–de Vries equation and its important generalizations including effects of rotation, cylindrical divergence, eddy viscosity, shear flows and instabilities, and turbulence. Experimental evidence for the existence of solitons in the upper ocean is presented both for shallow and deep sea regions. The data include radar and optical images and in situ measurements of waveforms, propagation speeds, and dispersion characteristics. It is suggested that internal solitons in the ocean are ubiquitous and are generated primarily by ti...

Evolution equations for strongly nonlinear internal waves

This paper is concerned with shallow-water equations for strongly nonlinear internal waves in a two-layer fluid, and comparison of their solitary solutions with the results of fully nonlinear computations and with experimental data. This comparison is necessary due to a contradictory nature of these equations which combine strong nonlinearity and weak dispersion. First, the Lagrangian (Whitham’s) method for dispersive shallow-water waves is applied to derivation of equations equivalent to the Choi–Camassa (CC) equations. Then, using the Riemann invariants for strongly nonlinear, nondispersive waves, we obtain unidirectional, evolution equations with nonlinear dispersive terms. The latter are first derived from the CC equations and then introduced semiphenomenologically as quasistationary generalizations of weakly nonlinear Korteweg–de Vries and Benjamin–Ono models. Solitary solutions for these equations are obtained and verified against fully nonlinear computations. Comparisons are also made with available observational data for extremely strong solitons in coastal zones with well expressed pycnoclines.

Internal solitons in laboratory experiments: Comparison with theoretical models

Nonlinear internal solitary waves observed in laboratory experiments are discussed from the standpoint of their relation to different soliton theories, from the classical integrable models such as the Korteweg-de Vries, Gardner, Benjamin-Ono, and Joseph-Kubota-Ko-Dobbs equations and their modifications, through the nonintegrable models describing higher-order nonlinear effects, viscosity, rotation, and cylindrical spreading, to the strongly nonlinear models. First, these theoretical models are briefly described and, then, laboratory data and their comparison with the theory are presented.

Terminal Damping of a Solitary Wave Due to Radiation in Rotational Systems

The evolution of a solitary wave under the action of rotation is considered within the framework of the rotation-modified Korteweg–de Vries equation. Using an asymptotic procedure, the solitary wave is shown to be damped due to radiation of a dispersive wave train propagating with the same phase velocity as the solitary wave. Such a synchronism is possible because of the presence of rotational dispersion. The law of damping is found to be “terminal” in the sense that the solitary wave disappears in a finite time. The radiated wave amplitude and the structure of the radiated “tail” in space–time are also found. Some numerical results, which confirm the approximate theory developed here, are given.

Nonlinear Surface and Internal Waves in Rotating Fluids

A class of nonintegrable equations related to a wide range of physical problems including surface and internal waves in rotating ocean is considered. A characteristic feature of these equations is the presence of a broad “dispersionless” band in the frequency spectrum that separates the regions of low- and high-frequency dispersion. The structures of plane and two-dimensional steady-state solutions are studied analytically and numerically. Results of the numerical calculations of non-stationary perturbation dynamics under different initial conditions are presented.

Nonlinear wave interactions in bubble layers.

Due to the large compressibility of gas bubbles, layers of a bubbly liquid surrounded by pure liquid exhibit many resonances that can give rise to a strongly nonlinear behavior even for relatively low-level excitation. In an earlier paper [Druzhinin et al., J. Acoust. Soc. Am. 100, 3570 (1996)] it was pointed out that, by exciting the bubbly layer in correspondence of two resonant modes, so chosen that the difference frequency also corresponds to a resonant mode, it might be possible to achieve an efficient parametric generation of a low-frequency signal. The earlier work made use of a simplified model for the bubbly liquid that ignored the dissipation and dispersion introduced by the bubbles. Here a more realistic description of the bubble behavior is used to study the nonlinear oscillations of a bubble layer under both single- and dual-frequency excitation. It is found that a difference-frequency power of the order of 1% can be generated with incident pressure amplitudes of the order of 50 kPa or so. It appears that similar phenomena would occur in other systems, such as porous waterlike or rubberlike media.