# Nonlinear Standing Waves, Resonance Phenomena and Frequency Characteristics of Distributed Systems

## Abstract

Resonance is one of the most interesting and fundamental phenomena in the physics of oscillations and waves. Resonance manifests itself clearly when the dependence of the amplitude of induced oscillations on frequency (frequency response of the system) has a sharp maximum. In these cases, the ratio of the central frequency ω_{0} of the spectral line, representing a response, to the characteristic width of this line is a large value. This ratio, called the quality- or *Q*-faetor, is used as a “quality” measure of the resonance system. At large values of *Q*, the system may contain a high energy density, since the ratio of the amplitude of induced oscillations to the amplitude of oscillations of the external source providing an influx of energy to the system is also equal to *Q.* In high-*Q* systems, approaching a state of equilibrium is slow process with a characteristic relaxation time on the order of Q/ω_{0}. The buildup time of oscillations (or their attenuation after the source is switched off) occurs over the course of many periods, the number of which is ~ *Q.* Excitation of strong oscillations during resonance may lead to the appearance of nonlinear effects, the most well-known of which is destruction of the system. On the other hand, high-*Q* systems are used for taking high-precision physical measurements.

## Keywords

Mach Number Standing Wave Shock Front Nonlinear Medium Resonance Phenomenon## Preview

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## References

- 1.O.V. Rudenko, S.I. Soluyan,
*Theoretical Foundations of Nonlinear Acoustics*(Plenum, New York, 1977)zbMATHGoogle Scholar - 2.V.B. Braginsky, V.P. Mitrofanov, V.Il. Panov,
*Systems with Small Dissipation*(University of Chicago Press, Chicago, 1986)Google Scholar - 3.V.B. Braginsky, V.P. Mitrofanov, K.V. Tokmakov, Energy dissipation in the pendulum mode of the test mass suspension of the gravitational wave antenna, Phys. Lett. A
**218**, 164–166 (1996)ADSCrossRefGoogle Scholar - 4.L.K. Zarembo, O.Y. Serdobolskaya, I.P. Chernobai, Effect of phase shifts accompanying the boundary reflection on the nonlinear interaction of longitudinal waves in solids, Sov. Phys. Acousl.
**18**, 333–338 (1972)Google Scholar - 5.Y.A. Ilinskii, B. Lipkens, T.S. Lucas, T.W.V. Doren, H.A. Zabololskaya, A theorelical model of nonlinear standing waves in an oscillating closed cavity, J. Acoust. Soc. Am.
**104**, 623–636 (1998)ADSCrossRefGoogle Scholar - 6.R. Landbury, Ultrahigh-energy sound wave promise new technologies, Physics Today
**51**, 23–24 (1998)CrossRefGoogle Scholar - 7.C.C. Lawrenson, B. Lipkens, T.S. Lucas, D.K. Perkins, T.W.V. Doren, Measurements of macrosonic standing waves in oscillating closed cavities, J. Acoust. Soc. Am.
**104**, 623–636 (1998)ADSCrossRefGoogle Scholar - 8.O.V. Rudenko, Artificial nonlinear media with a resonant absorber, Sov. Phys. Acoust.
**29**, 234–237(1983)Google Scholar - 9.O.V. Rudenko, Nonlinear acoustics — achievements, prospects, problems, Priroda (Nature) (7), 16–26 (1986). In RussianGoogle Scholar
- 10.V.G. Andreev, V.E. Gusev, A.A. Karabutov, O.V. Rudenko, O.A. Sapozhnikov, Enhancement of the q-factor of a nonlinear acoustic resonator by means of a selectively absorbing mirror, Sov. Phys. Acoust.
**31**, 162–163 (1985)Google Scholar - 11.V.V. Kaner, O.V. Rudenko, R.V. Khokhlov, Theory of nonlinear oscillations in acoustic resonators, Sov. Phys. Acoust.
**23**, 432–437 (1977)Google Scholar - 12.V.V. Kaner, O.V. Rudenko, Propagation of waves of finite amplitude in acoustic waveguides. Mosc. Univ. Phys. Bulletin
**33**, 63–70 (1978)Google Scholar - 13.M.B. Vinogradova, O.V. Rudenko, A.P. Sukhorukov,
*Theory of Waves*, 2nd edn. (Nauka, Moscow, 1990). In RussianGoogle Scholar - 14.O.V. Rudenko, A.V Shanin, Nonlinear phenomena accompanying the development of oscillations excited in a layer of a linear dissipative medium by finite displacements of its boundary. Acoust. Phys.
**46**, 334–341 (2000)ADSCrossRefGoogle Scholar - 15.A.A. Karabutov, O.V. Rudenko, Nonlinear plane waves excited by volume sources in a medium moving with transonic velocity. Sov. Phys. Acoust.
**25**, 306–309 (1979)Google Scholar - 16.A.A. Karabutov, E.A. Lapshin, O.V. Rudenko, Interaction between light waves and sound under acoustic nonlinearity conditions, Sov. Phys. JETP
**44**, 58–68 (1976)ADSGoogle Scholar - 17.O.V. Rudenko, Feasibility of generation of high-power hypersound with the aid of laser radiation, JETP Lett.
**20**, 203–206 (1974)ADSGoogle Scholar - 18.M.J.O. Strutt,
*Lame, Mathieu and Related Functions in Physics and Technology*(Edwards Brothers, New York, 1944)Google Scholar - 19.O.V. Rudenko, C.M. Hedberg, B.O. Enflo, Nonlinear standing waves in a layer excited by the periodic motion of its boundary, Acoust. Phys.
**47**, 452–460 (2001)ADSCrossRefGoogle Scholar - 20.B.O. Enflo, C.M. Hedberg, O.V. Rudenko, Resonant properties of a nonlinear dissipative layer excited by a vibrating boundary: Q-factor and frequency response, J. Acoust. Soc. Am.
**117**, 601–612 (2005)ADSCrossRefGoogle Scholar - 21.M. Abramovitz, I.A. Stegun,
*Handbook of Mathematical Functions*(Dover, N.Y., 1970)Google Scholar - 22.M.V. Dyke,
*Perturbation Methods in Fluid Mechanics*(Parabolic, Stanford, 1975)zbMATHGoogle Scholar - 23.W. Chester, J. Resonant oscillations in closed tubes. Fluid Mech.
**18**, 44–66 (1964)ADSzbMATHCrossRefGoogle Scholar - 24.O.V. Rudenko, A.L. Sobisevich, L.E. Sobisevich, C.M. Hedberg, Enhancement of energy and q-factor of a nonlinear resonator with an increase in its losses. Dokl. Phys.
**47**, 188–191 (2002)ADSCrossRefGoogle Scholar - 25.O.V. Rudenko, Nonlinear distortion of waves excited in a linear medium by finite chaotic piston vibrations, Dokl. Phys.
**43**, 346–348 (1998)MathSciNetADSGoogle Scholar - 26.O.V Rudenko, Nonlinear interactions of regular and noise spectra in intense radiation from a piston in a linear medium, Acoust. Phys.
**44**, 717–721 (1998)ADSGoogle Scholar - 27.O.V. Rudenko, Nonlinear oscillations of linearly deformed medium in a closed resonator excited by finite displacements of its boundary, Acoust. Phys.
**45**, 351–356 (1999)ADSGoogle Scholar - 28.I.P. Lee-Bapty, D.G. Crighton, Nonlinear wave motion governed by the modified burgers equation, Philos. Trans. Roy.Soc. London
**323**, 173–209 (1987)MathSciNetADSzbMATHCrossRefGoogle Scholar - 29.O.V. Rudenko, O.A. Sapozhnikov, Wave beams in cubically nonlinear nondispersive media, JETP
**79**, 220–229 (1994)ADSGoogle Scholar - 30.V.G. Andreev, TA. Burlakova, Measurement of shear elasticity and viscosity of rubberlike materials, Acoust. Phys.
**53**, 44–47 (2007)ADSCrossRefGoogle Scholar - 31.A.P. Sarvazyan, O.V. Rudenko, S.D. Swanson, J.B. Folwkes, S.Y. Emelianov, Shear wave elasticity imaging — a new ultrasonic technology of medical diagnostics, Ultrasound in Medicine and Biology
**24**, 1419–1436 (1998)CrossRefGoogle Scholar - 32.L.A. Ostrovsky, P.A. Johnson, Dynamic nonlinear elasticity in geomaterials, La Rivista del Nuovo Cimcnto
**24**, 1–47 (2001)Google Scholar - 33.O.V. Rudenko, S.N. Gurbatov, CM. Iledberg,
*Nonlinear Acoustics through Problems and Examples*(Trafford, 2010)Google Scholar - 34.O.V. Rudenko, Nonlinear sawtooth-shaped waves, Phys. Usp.
**38**, 965–989 (1995)MathSciNetADSCrossRefGoogle Scholar - 35.O.V. Rudenko, O.A. Sapozhnikov, Self-action effects for wave beams containing shock fronts, Phys. Usp.
**47**, 907–922 (2004)ADSCrossRefGoogle Scholar - 36.O.V. Rudenko, C.M. Hedberg, B.O. Enflo, Finite-amplitude standing acoustic waves in a cubically nonlinear medium, Acoust. Phys.
**53**, 455–464 (2007)ADSCrossRefGoogle Scholar - 37.A.P. Kuznetsov, S.P. Kuznetsov, N.M. Ryskin,
*Nonlinear Oscillations*(Fizmatlit, Moscow, 2002). In RussianGoogle Scholar