Nonlinear Resonant Interaction Research Articles

On the instability of one-dimensional acoustic standing waves

The paper theoretically considers the instability of acoustical standing waves in a nondispersive medium in a plane-parallel layer excited at the border of the layer or by a distributed force. The instability is caused by the nonlinear resonance interaction that sets in motion the parametric temporal amplification of the spontaneously formed standing waves. Discussion will focus on the results of the deviation of the frequencies of the parametrically excited subharmonic waves from the natural frequencies of the layer. This phenomenon was noted by a number of investigators [L. Adler and M. A. Breazeale, J. Acoust. Soc. Am. 48, 1077–1083 (1970); N.-C. Yen, J. Acoust. Soc. Am. 57, 1357–1362 (1975); Z. A. Goldberg, in ProblemsofNonlinearAcoustics, XIth Int. Symp. on Nonlinear Acoustics (in Russian), Part I (SO AN USSR, Novosibirsk, 1987), pp. 154–158]. It is shown that this phenomenon is not connected with the borders of the layer but it is peculiar to waves running in opposite directions that replaced the standing wave. The cause of these results is the instability of the standing waves to waves with deviated frequencies.

Nonlinear wave-wave interactions related to gravity wave reflection in the auroral upper F-region observed with the EISCAT radar

Some observational evidences of nonlinear wave-wave interactions associated with TIDs in the quiet daytime auroral F-region are presented. The dynamic frequency spectra in the upper F-region and the one-dimensional wavenumber spectra of TIDs show energy cascading from longer periods (larger scales) to shorter ones (smaller scales). The bispectra and the rates of change of the action density indicate that a chain of nonlinear resonant triad interactions take place. The TID-inducing AGW's wavenumbers along the magnetic field k ξ = k r ξ + jk j ξ are calculated with an inversion method developed in this paper by using simultaneous measurements of N e and V iξ . The k ξ r obtained as a function of height and period show a nearly non-dispersive character at greater heights. The observed nonlinear wave-wave interactions in the higher F-region are accompanied by negative values of − k ξ 3 and wave amplitude growth ( k ξ j> 1 2 H ), which probably imply an over-reflection. This reflection is supported by sharp meridional neutral wind shear and large incident wavelength. The weakness of the dispersion together with the reflection may enable the resonant conditions to be satisfied.

Internal wave eigenmodes and their weak nonlinear resonant interaction in a nonlinearly stratified fluid—A preliminary theoretical and experimental study

A preliminary theoretical and experimental study was conducted on internal wave modes and their weak nonlinear resonant interaction in a nonlinearly stratified fluid. An asymptotical solution of the modes and a dispersion relation of internal waves in a stratified fluid with density profile\(\rho = \rho _m - \Delta \rho \pi ^{ - 1} tg^{ - 1} \left( {b^{ - 1} y} \right)\) similar to that in our experiment were obtained theoretically. The resonant-interaction mechanism to 2nd order approximation is also discussed. The resonant interaction of the 3rd and 4th mode internal waves excited by the unstable 1st mode wave is analyzed on the basis of data obtained by conductivity probes. The resonant-interaction condition, i.e.,\(\omega _1 - \omega _3 - \omega _4 = 0\),k1+k3+k4=0, is examined. It is shown that the resonant instability increases with pycnocline thickness and wave maker driving frequency.

Observed and modeled resonantly interacting progressive water-waves

This study deals with the transformations of long, small waves of finite amplitude propagating in shallow water. The mechanisms of the nonlinear energy transfer between wave components and damping is first investigated by performing a set of laboratory experiments. The results of this experimental investigation are then compared to different models for the propagation of water waves. The models to be tested are the nonlinear potential flow model based on the Mixed Eulerian Lagrangian Method, the Boussinesq model, and lastly the harmonic model based on the concept of nonlinear resonant interaction. All three models are modifield to allow dissipation. The accuracy of the predictions of the modeling approximations is analysed.

A Third-Generation Model for Wind Waves on Slowly Varying, Unsteady, and Inhomogeneous Depths and Currents

Abstract A full discrete spectral model for propagation generation and dissipation of wind waves for arbitrary depth, current and wind fields is presented (WAVEWATCH). This model incorporates all relevant wave-current interaction mechanisms including changes of absolute frequencies due to unsteadiness of depth and currents. The model furthermore explicitly accounts for growth and decay of wave energy and for nonlinear resonant wave-wave interactions. The numerical schemes for propagation are basically second-order accurate. Effect of refraction and frequency shifts (due to unsteadiness of depth and current) are calculated on a fixed grid, also using second-order schemes. This paper focuses on the governing equations and the numerical algorithms. Furthermore some results for academic and realistic cases are presented to illustrate some features and merits of the model.

The Quantum and Stochastic Manifestations of Chaos

The transition to chaos manifests itself in quantum systems through a change in spectral statistics due to level repulsion as a result of the interaction of nonlinear resonances. In quantum systems the transition leads to the extension of the wave function in Hilbert space (although it may be limited by dynamic Anderson localization). We have found that a similar transition occurs in the Fokker-Planck equation of a system whose Langevin equation exhibits a transition to a type noise induced deterministic chaos.

Book reviews

Abstract The Art of Electronics. Student Manual. By T. C. Hayes and P. Horowitz. (Cambridge, New York, Port Chester: Cambridge University Press, 1989.) [Pp. ix + 611 + subject index.] Price £12.50, soft cover. Resonant Nonlinear Interactions of Light with Matter. By V. S. Buytlkin, A. E. Kaplan, Yu. G. Khronopulo and E. I. Yakubovich. Translated by O. A. Germocenova (Berlin, Heidelberg, New York, London: Springer-Verlag, 1989.) [Pp. xiv + 337 + subject index.] Price DM 198.00, hard cover. Progress in Low Temperature Physics. Volume XII. Edited by D. F. Brewer. (Amsterdam, Oxford, New York: North-Holland, 1989.) [Pp. xvi + 339 + author and subject index.] Price Dfl. 225.00/US $118.50, hard cover. Magnetic Phenomena. Edited by A. P. Maclin, T. L. Gill and W. W. Zachary. (Berlin, Heidelberg, New York: Springer-Verlag, 1989.) [Pp. vi + 139.] Price DM 55.00, hard cover. Nanostructure Physics and Fabrication. Edited by M. A. Reed and W. P. Kirk. (Boston, San Diego, New York: Academic Press, 1989.) [Pp. xiv + 520 + author index.] Price US $64.50, hard cover. Principles of Magnetic Resonance. Third Enlarged and Updated Version. By C. P. Slichter. (Berlin, Heidelberg, New York: Springer-Verlag, 1990.) [Pp. xi + 646 + author and subject index.] Price DM 89.00, hard cover. Electronic Properties of Conjugated Polymers III. Basic Models and Applications. Edited by H. Kuzmany, M. Mehring and S. Roth. (Berlin, Heidelberg, New York: Springer-Verlag, 1989.) [Pp. xiv + 483 + author index.] Price DM 110.00, hard cover. (Proceedings of an International Winter School, held in Kirchberg, Tirol, March 11-18, 1989. Eighty-six contributions.) The Penrose Transform. Its Interaction with Representation Theory. By R. J. Baston and M. G. Eastwood. (Oxford, Clarendon Press, 1989.) [Pp. xv + 204 + subject index.] Price £25.00, hard cover. Optimal Structures in Heterogeneous Reaction Systems. Edited by P. J. Plath. (Berlin, Heidelberg, New York: Springer-Verlag, 1989.) [Pp. vii + 195.] Price DM89.00, hard cover. Electrostatic Atomization. By D. Michelson. (Bristol and New York: Adam Hilger, 1990.) [Pp. x +145 +subject index.] Price £30.00, hard cover. Plasma Physics and Plasma Electronics. By L. M. Kovrizhnykh. (Commack, Nova Science, 1989.) [Pp. x + 306 + subject index.] Price US $98.00, hard cover. Crystal Growth and Characterization of Advanced Materials. Edited by A. N. Christensen, F. Leccabue, C. Paorici and O. Vigil. (Singapore, New Jersey, London: World Scientific, 1988.) [Pp. xi + 541.] Price £51.00, hard cover. (Proceedings of an International School held on November 30-December 10, 1987 in La Habana, Cuba. Twenty-five contributions.) Atlas of Polymer and Plastics Analysis. Volume 2, Parts a and b. By D. O. Hummel. (Weinheim, Basel, Cambridge: VCH Verlagsgesellschaft 1988.) [Pp. xvii + x; 470 + 587 +subject index.] Price Part a £267.00; Part b £209.00, hard cover. High Temperature Superconductivity. Edited by R. Nicolsky, R. A. Barrio, O. F. De Lima and R. Escudero. (Singapore, New Jersey, London: World Scientific, 1988.) [Pp. xiv +480 +author index.] Price US $68.00, hard cover; US $41.00, soft cover. (Proceedings of the First Latin-American Conference held 4-6 May, 1988, Rio de Janeiro, Brazil. Seventy-five contributions.) Theories of Immune Networks. Edited by H. Atlan and I. R. Cohen. (Berlin, Heidelberg, New York: Springer-Verlag, 1989.) [Pp. viii + 117.] Price DM85.00, hard cover. Structure and Properties of Dislocations in Semiconductors, 1989. Edited by S. G. Roberts, D. B. Holt and P. R. Wilshaw. (Bristol and New York: Institute of Physics, 1989.) [Pp. xii + 466 + author and subject index.] Price £55.00, hard cover. (Proceedings of the Sixth International Symposium held at the University of Oxford, 5-8 April, 1989. Fifty-six contributions.) Isotropy Subgroups of the 230 Crystallographic Space Groups. By H. T. Stokes and D. M. Hatch. (Singapore and London: World Scientific, 1988.) [Pp. vii + 623.] Price £57.00/US $90.00, hard cover.

Resonant Nonlinear Interactions of Light with Matter

Resonant Nonlinear Interactions of Light with Matter

Quantum inverse scattering method for a nonlinear N-wave resonance interaction system with both boson and fermion fields

The nonlinear N-wave resonance interaction system with both boson and fermion fields is studied in the framework of the quantum inverse scattering method. The model Hamiltonian is diagonalised and the degeneracy of the eigenstates and the existence of the quantum bound states are analysed. Moreover, the classical limit is also discussed.

Very large phase velocities of non two‐stream, meter scale irregularities in the high latitude E region ionosphere

We present experimental evidence on a new category of meter wavelength radio auroral irregularities characterized by a unique Doppler spectrum signature. The new echo type has been identified with the Scandinavian Twin Auroral Radar Experiment radars which are capable of observing simultaneously full Doppler spectra from the same scattering volume along two widely different azimuths. The new echoes were seen only by the Finland radar which observes at large angles to the electrojet flow and at times when the Norway radar observed very stable type 1 echoes with phase velocities near the ion acoustic speed. The new spectrum, named type 5, is composed of three main frequency bands, two being symmetric about zero Doppler shift near ±450 m/s and a third always at very large phase velocities near 1000 m/s, this resulting to a characteristic asymmetry in the spectrum. The Doppler polarity of the large phase velocity component is always at negative shifts in the eastward electrojet and at positive shifts in the westward electrojet. These echoes, which are not strong and last from a few minutes to tens of minutes, were observed during strong but also moderate geomagnetic conditions but, apparently, always in the presence of large and stable electric fields. The evidence suggested that the new echoes cannot be due to direct generation by the two‐stream instability and, most likely, a secondary generation mechanism is involved. Although there are some similarities with 50 MHz type 4 spectral observations, type 5 echoes have unique characteristics which we believe are sufficient to justify their classification into a different echo category. We have no satisfactory theoretical explanation for the new echoes. That the fast moving irregularities have frequencies about two times the frequency of the simultaneously observed ion acoustic waves led us to suggest that a nonlinear resonant wave‐wave interaction process might be in operation under some appropriate conditions in the plasma.

Resonant Wave-Wave Interactions of Electromagnetic and Electrostatic Waves in a Homogeneous Magnetized Plasma

General expressions of the matrix elements for nonlinear resonant wave-wave interactions of electromagnetic and electrostatic waves in a homogeneous magnetized plasma are derived from the Vlasov-Maxwell equations. The kinetic wave equations obtained for electromagnetic waves are represented by three-order tensors in the rotating and Cartesian coordinates. The waves are allowed to propagate at arbitrary angles to the uniform magnetic field, and no restrictions are imposed upon the mean Larmor radius, the frequencies, or the wave numbers. By electrostatic approximation of the dielectric tensor and the matrix elements the kinetic equations can be also applicable to the case in which the three waves are electrostatic altogether or partially electrostatic. The matrix elements in the limit of parallel or perpendicular propagation to the magnetic field are given.

Nonlinear interaction of Na Rydberg atoms (Pseries) with infrared and millimeter radiation

An experimental investigation was made of nonlinear resonant interaction of CO2 laser radiation (λ = 10.6 μm) and of tunable millimeter radiation (λ = 3.8–5.6 mm) with Rydberg atoms of Na excited to the nP states with n = 11 and 36. These Na Rydberg atoms were formed in an atomic beam as a result of three-stage excitation in accordance with the scheme 3S1/2→3P3/2→4S1/2→nP1/2,3/2. Pulsed tunable organic dye lasers and a laser utilizing the color centers in an LiF crystal were used. The saturation parameters were determined for one-photon 36P–37S (Δn = 1) and 11P–30D (Δn 1) transitions. The transition frequencies and the dipole moments were calculated, an analysis was made of the saturation mechanism, and the field broadening of the resonances was studied. The experimental results compared with theoretical calculations led to the conclusion that under experimental conditions an important role in the saturation effects was played by the Stark effect in an inhomogeneous field which reduced significantly the coherent interaction time. Saturation was observed and a study was made of the characteristics of two-photon absorption due to the 36P–37P transition.

Nonlinear interactions of two compressional hydromagnetic waves

Nonlinear interactions of two azimuthally symmetric compressional hydromagnetic waves propagating in a cylindrical waveguide filled with cold magnetized plasma are investigated. Two cases are considered: the nonlinear interaction of two identical oppositely propagating compressional waves and the nonlinear interaction of two compressional waves propagating with equal group velocities. In the first case the second-order perturbation fields generated through self- and mutual interactions of the waves are calculated and their effect on the otherwise-formed simple linear standing-wave pattern is studied. The possibility of observing a resonant nonlinear interaction is shown. In the second case, in order to describe the nonlinear evolution of the wave amplitudes, two coupled nonlinear Schrödinger (NLS) equations are presented. When excited individually, both the waves are seen to be modulationally stable; but when excited simultaneously, a strong nonlinear wave-wave coupling comes into play, which makes the waves modulationally unstable. The corresponding growth rate of the instability is also calculated.

Nonlinear interaction of waves in an inhomogeneous medium

Nonlinear resonant interaction of three waves in a dispersive medium is considered within the framework of Hamiltonian formalism. The steady-state distribution function is found for random inhomogeneity and simplified diffusion equations are derived. Interaction in a periodically inhomogeneous medium is defined by the character of the inhomogeneity. For fine scale inhomogeneity the interaction slows significantly, while for a coarse scale it becomes chaotic. The dependence of the chaos level on the Manly-Row integrals is studied.

A Note on Second Harmonic Resonance of a Magneto‐Acoustic Wave in Planar Plasma‐Filled Waveguide

AbstractThe second harmonic generated by non‐linear self interaction of a magneto‐acoustic wave in a planar plasma‐filled waveguide is a freely propagating mode. Coupled‐mode equations for resonant nonlinear interaction of this second harmonic mode with the fundamental mode are derived. Using these equations the possibility of generation of the second harmonic is analysed.

Nonlinear resonant interaction between acoustic impulse and electrons in superconductors

The problem of inelastic electron interaction in a perfectly pure superconductor with a longitudinal sound impulse is considered. It is shown that the problem reduces to the equivalent one of elastic scattering by the static potential, and the sound absorption is expressed in terms of the reflection coefficient of this scattering. The classical and quantum properties of the scattering are studied and the phase region in which new excitations are created is indicated. A formula is derived that expresses the density matrix of the excitations created in terms of the exact scattering matrix of an impulse. The quasiclassical creation of excitations by a smooth-shaped impulse is investigated with regard to both overbarrier and underbarrier processes.

Nonlinear resonance interaction of ultrasonic waves under applied stress

The stress effect on the nonlinear resonance interaction T+T→L is experimentally investigated using an Al 5083 specimen. Compressive stress applied at the localized region where two noncollinear transverse waves intersect each other is shown to vary, up to ±8.5%, of the amplitude of the longitudinal wave generated through the resonance interaction. The amplitude variation with stress is explained by relating the stress induced change of elastic wave velocities (acoustoelastic effect) to the change in the resonance angle, and then to the directional characteristics of the two incoming transverse beams. The weak anisotropy of the material is taken into account.

Nonlinear resonant interaction of acoustic fields with Tollmen-Schlichting waves

Nonlinear resonant interaction of acoustic fields with Tollmen-Schlichting waves

Nonlinear Resonant Interaction of Two Longitudinal Waves and a Transverse Wave in a Hot Magnetized Plasma

AbstractBy Whitham's method of averaged Lagrangian and using Low's form of Lagrangian, coupled mode equations and coupling coefficients are derived for resonant nonlinear interaction of two longitudinal and one transverse wave in a magnetized plasma, in which the later wave propagates along the external uniform magnetic field. The limiting form of these coupling coefficients are obtained when the external magnetic field vanishes.

Interaction of quantum nonlinear resonances : G. P. Berman, G. M. Zaslivski, and A. R. Kolovskii; L. V. Kirenskii Physics Institute Siberian Division, USSR Academy of Sciences, Krasnoyarsk, 660036, USSR.

Interaction of quantum nonlinear resonances : G. P. Berman, G. M. Zaslivski, and A. R. Kolovskii; L. V. Kirenskii Physics Institute Siberian Division, USSR Academy of Sciences, Krasnoyarsk, 660036, USSR.