Abstract
This chapter discusses interactions of strongly distorted waves containing shock fronts. Such sawtooth-shaped perturbations are formed in the course of wave propagation through media where nonlinearity predominates over competitive factors like dispersion, diffraction and absorption. Specific character of nonlinear processes in the fields of sawtooth waves is particularly emphasized. We describe experimentally observed phenomena, as well as present-day applied trends.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
O.V. Rudenko, S.I. Soluyan, Theoretical Foundations of Nonlinear Acoustics (Plenum, New York, 1977)
V.A. Burov, V.A. Krasilnikov, Direct observation of the deformation of intense ultrasonic waves in liquids, Sov. Phys. Dokl. 3, 173–176 (1958)
L.K. Zarembo, V.A. Krasil’nikov, Introduction to Nonlinear Acoustics (Nauka, Moscow, 1966). In Russian
L.K. Zarembo, V.A. Krasil’nikov, V.V. Shklovskaya-Kordi, On the absorption of ultrasonic waves of a finite amplitude in liquids, Sov. Phys. Dokl. 109(4), 731–734 (1956)
K.A. Naugolnykh. in High-Intensity Ultrasonic Fields, ed. by L.D. Rosenberg (Plenum, New-York, 1974)
R.T. Beyer, Nonlinear Acoustics, 2nd edn. (Acoustical Society of America, New York, 1997)
A.A. Karabutov, O.V Rudenko, O.A. Sapozhnikov, Theory of thermal self-focusing with allowance for the generation of shock waves and acoustic streaming, Sov. Phys. Acoust. 34, 371–374(1988)
A.A. Karabutov, O.V. Rudenko, O.A. Sapozhnikov, Thermal self-focusing of weak sock waves, Sov. Phys. Acoust. 35, 40–43 (1989)
O.V. Rudenko, M.M. Sagatov, O.A. Sapozhnikov, Thermal self-focusing of sawtooth waves, Sov. Phys. JETP 71, 449–557 (1990)
O.V. Rudenko, O.A. Sapozhnikov, Inertialess self-focusing of nondispcrsive waves with a broadband spectrum, Quantum Electron. 23, 896 (1993)
O.V. Rudenko, O.A. Sapozhnikov, Wave beams in cubically nonlinear nondispersive media., JETP 79, 220–229 (1994)
O.V. Rudenko, O.A. Sapozhnikov, Theoretical description of thermal self-focusing of saw-tooth sound waves. Moscow Univ. Phys. Bull. 43(6), 106–107 (1989)
O.V. Rudenko, in Advances in Nonlinear Acoustics, ed. by H. Hobæk (World Scientific, Singapore, 1993), pp. 3–6
A.G. Musatov, O.V. Rudenko, O.A. Sapozhnikov, Accounting for nonlinear refraclion and nonlinear absorption in focusing high-power pulses, Sov. Phys. Acoust. 38, 274–279 (1992)
A.G. Musatov, O.A. Sapozhnikov, Nonlinear effects in process of focusing of acoustic pulses with shock front, Acoust. Phys. 39, 266–269 (1993)
O.V. Rudenko, Nonlinear sawtooth-shaped waves, Phys. Usp. 38, 965–989 (1995)
O.V. Rudenko, Nonlinear methods in acoustic diagnostics, Russ. J. Nondestructive Testing 29(8), 583–588 (1993)
V.G. Andreev, V.Y. Veroman, G.A. Denisov, O.V. Rudenko, O.A. Sapozhnikov, Nonlinear acoustical aspects of extracorporeal lithotripsy, Sov. Phys. Acoust. 38(4). 325–328 (1992)
O.V. Rudenko, O.A. Sapozhnikov, Intense acoustic beams: self-action of discontinuous waves, focusing of pulses and extracorporeal lithotripsy, Moscow Univ. Phys. Bull. 46, 3–17 (1991)
O.V. Rudenko, Interactions of intense noise waves, Phys. Usp. 29, 620–641 (1986)
O.V. Rudenko, A.K. Sukhorukova, A.P. Sukhorukov, Equations of high-frequency nonlinear acoustics of inhomogeneous media, Acoust. Phys. 40, 264–268 (1994)
D.J. Maglieri, Overview of current knowledge and activity in sonic boom research, J. Acoust. Soc. Am. 92, 2328–2337 (1992)
A.D. Pierce, in Advances in Nonlinear Acoustics, ed. by H. Hobæk (World Scientific, Singapore, 1993), pp. 7–20
S.A. Akhmanov, R.V. Khokhlov, Problems of Nonlinear Optics (VINITI, Moscow, 1964). In Russian
S.A. Akhmanov, B.A. Vysloukh, A.S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, New York, 1992)
F.V. Bunkin, Y.A. Kravtsov, G.A. Lyakhov, Acoustic analogues of nonlinear-optics phenomena, Sov. Phys. Usp 29(7), 607–619 (1986)
D.T. Blackstock, Connection between the fay and fubini solutions for plane sound waves of finite amplitude, J. Acoust. Soc. Am. 39, 1019–1026 (1966)
G.A. Liakhov, O.V. Rudenko, The effect of parametric amplification of weak signals in non-linear acoustics, Sov. Acoust. Phys. 20(5), 738–744. (1974)
B.K. Novikov, O.V. Rudenko, Degenerate parametric amplification of sound, Sov. Phys. Acoust. 22(3), 258–259 (1976)
R. Landauer, Parametric amplification along nonlinear transmission lines, J. Appl. Phys. 31, 479–484 (1960)
O.A. Vasil’eva, A.A. Karabutov, E.A. Lapshin, O.V. Rudenko., Interaction of One-Dimensional Waves in Nondispersive Media (MGU, Moscow, 1983). In Russian
S.N. Gurbatov, A.N. Malakhov, On the possibility of using parametric interaction of waves to highlight the weak acoustic signals, Sov. Phys. Acoust. 25(1), 53–59 (1979)
V.G. Andreev, A.A. Karabutov, O.V. Rudenko, Experimental study of the propagation of nonlinear sound beams in free space, Sov. Phys. Acoust. 31, 252–255 (1985)
D.A. Webster, M.A. Theobald, D.T. Blackstock. in Proc. 9th Int. Congress on Acoust., vol. 2(32) (Madrid, 1977), vol. 2(32), p. 740
D.A. Webster, D.T. Blackstock, Finite-amplitude saturation of plane sound waves in air, J. Acoust. Soc. Am. 62, 518–523 (1977)
O.V. Rudenko, S.I. Soluyan, R.V. Khokhlov, Bounded quasiplane beams of periodic perturbations in a nonlinear medium,, Sov. Phys. Acoust. 19, 871–876 (1973)
L.A. Ostrovskii, V.E. Fridman, Directivity of high-intensity acoustic radiation, Sov. Phys. Acoust. 18, 478–182(1973)
N.S. Bakhvalov, Y.M. Zhilcikin, E.A. Zabolotskaya, Nonlinear Theory of Sound Beams (AIP, New York, 1987)
R.V. Khokhlov, Theory of shock radiowaves in nonlinear lines, Sov. Phys. — Radiotech. Electron, (new title: J. of Communications Technology and Electronics) 6(6). 917–925 (1961)
E.A. Zabolotskaia, R.V. Khokhlov, Quasiplanar waves in nonlinear acoustics of bounded beams, Sov. Acoust. Phys. 15(1), 40–46 (1969)
O.V. Rudenko, The 40th anniversary of the khokhlov-zabolotskaya equation, Acoust. Phys. 56(4), 457–466 (2010)
V.P. Kuznetsov, Equations of nonlinear acoustics. Sov. Phys. Acoust. 16, 467–470 (1970)
B.B. Kadomtsev, V.I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl. 15(6), 539–542 (1970)
O.V. Rudenko, S.I. Soluyan, R.V. Khokhlov, Problems of the theory of nonlinear acoustics, Sov. Phys. Acoust. 20(3), 356–359 (1974)
V.G. Andreev, O.V. Rudenko, O.A. Sapozhnikov, VA. Khokhlova, Suppression of nonlinear damping of a sound wave in a medium which contains a resonant absorber with a finite line width. Moscow Univ. Physics Bulletin 40(3), 3–17 (1985)
M.B. Vinogradova, O.V. Rudenko, A.P. Sukhorukov, Theory of Waves, 2nd edn. (Nauka, Moscow, 1990). In Russian
S.I. Aanonsen, T. Barkve, J.N. T’jotta, S. T’jotta, Distortion and harmonic generation in the near-field of a finite amplitude sound beam. J. Acoust. Soc. Am. 75, 749–768 (1984)
M.F. Hamilton, J.N. Tjotta, S. Tjotta, Nonlinear effects in the far-field of a directive sound source, J. Acoust. Soc. Am. 78, 202–216 (1985)
H. Hobaek, Parametric acoustic transmitting arrays — a survey of theories and experiment. Sci. Techn. Report 99 (1977)
B.K. Novikov, O.V. Rudenko, V.I. Timoshenko, Nonlinear Underwater Acoustics (AIP, New York, 1987)
O.V. Rudenko, S.I. Soluyan, R.V. Khokhlov, To the nonlinear theory of paraxial sound beams, Sov. Phys. Dokl. 225(5) (1975)
S.A. Akhmanov, A.P. Sukhorukov, R.V. Khokhlov, Self-focusing and diffraction of light in a nonlinear medium. Sov. Phys. Usp. 10. 609–636 (1968)
R. Courant, D. Hilbert, Methodes of Mathematical Physics (Interscience, New York, 1953)
N.S. Bakhvalov, Y.M. Zhileikin, H.A. Zabolotskaia, R.V. Khokholov, Nonlinear propagation of a sound beam in a nondissipative medium, Sov. Acoust. Phys. 22(4), 487–491 (1976)
M.F. Hamilton, V.A. Khokhlova, O.V. Rudenko, Analytical method for describing the paraxial region of finite amplitude sound beams, J. Acoust. Soc. Am. 96, 3321–3332 (1994)
Y.R. Lapidus, O.V. Rudenko, New approximations and findings of theory of nonlinear acoustic beams, Sov. Phys. Acoust. 30(6), 473–476 (1984)
Y.R. Lapidus, S.I. Soluyan, Evolution of nonlinear waves along the axis of a weakly diffracting and focused beams, Sov. Phys. Acoust 31, 615–619 (1985)
J.H. Ginsberg, Nonlinear king integral for arbitrary axisymmetric sound beams at finite amplitude, J. Acoust. Soc. Am. 76, 1201–1214 (1984)
F.Y. Coulouvrat, An analytical approximation of strong nonlinear effects in bounded sound beams, J. Acoust. Soc. Am. 90, 1592–1600 (1991)
D.R. Bacon, A.C. Baker, Comparison of two theoretical models for predicting nonlinear propagation in medical ultrasonic fields, Phys. Med. Biol. 34, 1633–1643 (1989)
A.C. Baker, K. Anastasiadis. V.F. Humphrey, The nonlinear pressure field of a planar circular piston: Theory and experiment, J. Acoust. Soc. Am. 84, 1483–1487 (1988)
L.D. Landau, K.M. Lifshilz, Fluid Mechanics, 2nd edn. (Butterworth-Heinemann, Oxford, 1987)
C.C. Lin, E. Reissner, H.S. Tsien, On two-dimensional non steady motion of the slenderbody in a compressible fluid. J. Math. Phys. 27, 3–15 (1948)
A.A. Karabutov, O.V. Rudenko, The modified Khokhlov method for investigation of non-steady transonic flows of a compressed gas, Sov. Phys. Dokl. 24, 835–838 (1979)
M.S. Cramer, A.R. Seebass, Focusing of weak shock waves at an arěte, J. Fluid Mech. 88, 209–222 (1978)
M.D. Spektor, Stability of weak shock waves with a finite-width front, JETP Lett. 35, 221–223 (1982)
A.G. Munin, V.E. Kvitka, Aviation Acoustics (Mashinostroenie, Moscow. 1973). In Russian
A.D. Pierce, D.J. Maglieri, Effects of atmospheric irregularities on sonic boom propagation, J. Acoust. Soc. Am. 51, 702–721 (1972)
V.E. Fridman, Ray theory of acoustic waves of finite amplitude. Dr. Sci. Thesis (1985). In Russian
K.A. Naugolnykh, L.A. Ostrovsky, Nonlinear Wave Processes in Acoustics (Cambridge University Press, 1998)
E.N. Pelinovskii, V.E. Fridman, Y.K. Engelbrecht, Nonlinear Evolution Equations (Valgus, Tallin, 1984). In Russian
G.. Whitham, Linear and Nonlinear Waves (Wiley Interscience, New York, 1999)
D.I. Blokhintsev, Acoustics of Nonhomogeneous Moving Medium. Tech. Memo. 1399. (National Advisory Committee for Aeronautics, Washington, 1956)
Y A. Kravtsov, Y.I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer. 1990)
V.E. Ostashcv, Sound Propagation in Moving Media (Nauka, Moscow, 1992). In Russian
K.E. Gubkin, Propagation of discontinuities in the sound waves, J. Appl. Math. Mech. 22(4), 561–564 (1958)
O.S. Ryzhov, Attenuation of shock waves in inhomogeneous media, J. Appl. Mech. Techn. Phys. 2, 15–24(1961)
Y.L. Zhilin, in Trudy TsAGI (1973), 1489, pp. 3–24. In Russian
E. Reiso, J.N. Tjotta, S. Tjotta, in Frontiers of Nonlinear Acoustics, ed. by M.F. Hamilton, D.T. Blackstock (Elsevier, London, 1990), pp. 177–182
L.K. Zarembo, I.P. Chunchuzov, Sound beam in an inhomogeneous medium with varying sound speed. Sov. Phys. Acoust. 23, 78–79 (1977)
O.V. Rudenko, A.K. Sukhorukova, A.P. Sukhorukov, Two-dimensional nonlinear waves with discontinuities in stratified media, Acoust. Phys. 41, 251–255 (1995)
A.G. Musatov, O.A. Sapozhnikov, Focusing of powerful acoustic pulses under different convergence angles of wave front, Acoust. Phys. 39, 166–169 (1993)
K.A. Naugolnykh, S.I. Soluyan, R.V. Khokhlov, Spherical waves of finite amplitude in a viscous thermally conducting medium, Sov. Phys. Acoust. 9, 42–48 (1963)
A.M. Sutin, in Nonlinear Acoustics (Appl. Phys. Inst., Gorky, 1980), pp. 45–67. In Russian
L.A. Ostrovskii, A.M. Sutin, Focusing of acoustic waves of a finite amplitude, [Sov, Phys. Dokl. 20, 275–277 (1975)
Y.Y. Borisov, N.M. Gynkina, Investigation of elliptic sound concentrators, Sov. Phys. Acoust. 19, 389–391 (1974)
C.W. Smith, R.T. Beyer, Ultrasonic radiation field of a focusing source at finite amplitudes, J. Acoust. Soc. Am. 46, 806–813 (1969)
L. Agarval, V.R. Singh, S.P. Sud, in Proc. 14th Int. Congress on Acoust. (Beijing, 1992), pp. 1–10
D. Cathignol, J.Y. Chapelon, in Advances in Nonlinear Acoustics, ed. by H. Hobaek (World Scientific, Singapore, 1993), pp. 21–35
C.C. Church, L.A. Crum, in Proc. 13th Int. Congress on Acoust. (Belgrade, 1989), pp. 205–208
M. Delius. in Frontiers of Nonlinear Acoustics, ed. by M.F. Hamilton, D.T. Blackstock (Elsevier, London. 1990). pp. 31–46
K. Takayama, in Proc. of 15th Int. Workshop Shock Wave Focusing (Sendai, 1989), pp. 217–226
O.V. Rudenko, V.Y. Veromann, G.A. Denisov, et al. Optoacoustic radiator for contactless destruction of calculi in the body of a biological object. Certificate of authorship No. 1673085. In Russian
G.A. Askar’yan, L.D. Klebanov, The focusing and cumulation of pholothermoacoustic shock pulses from a concave surface heated by a flash of laser light, Sov. J. Quantum Electron. 18, 1359–1360 (1988)
S.A. Akhmanov, V.M. Gordienko, A.A. Karabutov, A.B. Reshilov, O.V. Rudenko, V.I. Shmalgauzen, in Proc. 9th Ail-Union Acoustics Conference, 4 IV-6 (Acoustics Institute, Moscow. 1977), pp. 25–28. In Russian
A.I. Bozhkov, F.V. Bunkin, A.M. Galstyan, A.A. Kolomenskii, V.G. Mikhalevich, Laser excitation of sound in liquids, Bull. Acad Sci. USSR, Phys. Ser. 46(8), 168–174 (1982)
V.E. Gusev, A.A. Karabutov, Laser Optoacoustics (AIP, New York, 1993)
A.G. Musatov, O.V. Rudenko, O.A. Sapozhnikov, in Advances in Nonlinear Acoustics, ed. by H. Hobaek (World Scientific, Singapore, 1993). pp. 321–326
A.J. Coleman, J.E. Saunders, R.C. Preston, D.R. Bacon, Pressure waveforms generated by dornier extracorporeal shock wave lithotripier, Ultrasound in Medicine and Biology 13, 651–657(1987)
A.J. Coleman, J. E.Saunders, A survey of the acoustic output of commercial extracorporeal shock wave lithotripters. Ultrasound in Medicine and Biology 15, 213–227 (1989)
M. Muller, in Proc. 13th Int. Congress on Acoust., vol. 1 (Belgrade, 1989), vol. 1, pp. 259–262
O.A. Sapozhnikov, Focusing of high-intensity acoustic pulses, Sov. Phys. Acoust. 37, 395–404 (1991)
Y.M. Zhileikin, Y.I. Osipik, in Modern Problems of Mathematical Modeling (MGU, Moscow, 1984). pp. 363–366. In Russian
Y.M. Zhileikin, V. Rudenko, Nonlinearly diffractive transformation of acoustic signals, Sov. Phys. Acoust. 27(3), 200–203 (1981)
O.V. Rudenko, Feasibility of generation of high-power hypersound with the aid of laser radiation, JETP Lett. 20, 203–206 (1974)
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)
O.V. Rudenko, Artificial nonlinear media with a resonant absorber, Sov. Phys. Acoust. 29, 234–237 (1983)
V.G. Andreev, O.V. Rudenko, inProc. 10th All-Union Acoustics Conference, Sect. B (Acoustics Institute, Moscow, 1983), p. 24. In Russian
V.G. Andreev, V.G. 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)
O.V. Rudenko, S.N. Gurbatov, C.M. Hedberg, Nonlinear Acoustics through Problems and Examples (Trafford, 2010)
V.E. Zakharov, R.Z. Sagdeev, Speclrum of acouslic turbulence, Sov. Phys. Dokl. 15, 439–442 (1970)
B.B. Kadomtsev, V.I. Petviashvili, Acoustic turbulence. Sov. Phys. Dokl. 18, 115–117 (1973)
K.A. Naugolnykh, S.A. Rybak, Spectrum of sonic turbulence, Sov. Phys. JETP 41, 39–43 (1975)
O.V. Rudenko, V.A. Khokhlova, Kinetic approach to the description of one-dimensional acoustic turbulence. Sov. Phys. Acoust. 34, 289–293 (1988)
O.V. Rudenko, V.A. Khokhlova, Kinetics of one-dimensional sawtooth waves, Sov. Phys. Acoust. 37, 90–93 (1991)
S.N. Gurbatov, A.I. Saichev, Degeneration of one-dimensional acouslic turbulence, Sov. Phys. JETP 53, 347–350 (1981)
S.N. Gurbatov, A.N. Malakhov, A.I. Saichev, Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays and Particles. (Manchester University Press, 1991)
E.A. Lapshin, O.A. Vasileva, et al. Application package NACSI (1–1.2: Plane, converging and diverging waves in dissipative and relaxing media; 2.1-2: diverging waves in dissipative and relaxing media: 2.1-4: High-intense acoustic beams in nonlinear media; etc.). Software package (1992)
V.A. Khokhlova, O.A. Sapozhnikov, Modification of the spectral method for describing non-linear acoustic waves containing shocks, J. Acoust. Soc. Am. 96, 3321–3326 (1994)
S.N. Gurbatov, A.I. Saichev, I. Yakushkin, Nonlinear waves and one-dimensional turbulence in nondispersive media, Sov. Phys. Usp. 26, 857–864 (1983)
VA. Burov, V.A. Krasilnikov, E.Y. Tagunov, Experimental investigation of collinear interaction of a weak ultrasonic signal with intense low-frequency perturbations in water, Moscow Univ. Phys. Bull. Sen 3 Phys. Astron. 33(4), 60–65 (1978)
M.B. Moffett, W.L. Konrad, L.K Carlton, Experimental demonstration of the absorption of sound by sound in water, J. Acoust. Soc. Am. 63, 1048–1051 (1978)
K.I. Volyak, Nonlinear Waves in the Ocean. Selected Works (Nauka, Moscow, 2002). In Russian
K.I. Volyak, A.S. Gorshkov, O.V. Rudenko, On the back-traveling wave appearance in homogeneous nonlinear media, Moscow Univ. Phys. Bull. 16(1), 32–36 (1975)
S.N. Makarov, Self-reflection in nonlinear acoustics, theoretical ground and possible applications, Acustica 80. 1–13 (1994)
C.L. Morfey, in Advances in Nonlinear Acoustics, ed. by H. Hobæk (World Scientific, Singapore. 1993), pp. 167–172
P.J. Westervelt, Nonscattering of sound by sound resulting from a head-on collision of two plane-wave pulses, J. Acoust. Soc. Am. 96, 3320 (1994)
V.V. Kaner, O.V. Rudenko, R.V. Khokhlov, Theory of nonlinear oscillations in acoustic resonators, Sov. Phys. Acoust. 23, 432–437 (1977)
V.V. Kaner, O.V. Rudenko, Propagation of waves of finite amplitude in acoustic waveguides, Mose. Univ. Phys. Bulletin 33, 63–70 (1978)
A.H. Nayfeh, J.E. Kaiser, D.P. Telionis, Acoustics of aircraft engine-duct systems, AIAA J. 13, 130–146 (1975)
M.F. Hamilton, J.A. Ten Gate, Finite amplitude sound near cutoff in higher-order modes of a rectangular duct, J. Acoust. Soc. Am. 84, 327–334 (1988)
D.A. Webster, D.T. Blackstock, Collinear interaction of noise with a finite-amplitude tone. J. Acoust. Soc. Am. 63, 687–691 (1978)
Y.M. Zhileikin, T.M. Zhuravleva, O.V. Rudenko, Nonlinear phenomena under propagation of high-frequency sound waves in tubes, Sov. Phys. Acoust. 26(1), 32–35 (1980)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2011 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Gurbatov, S.N., Rudenko, O.V., Saichev, A.I. (2011). Nonlinear Sawtooth Waves. In: Waves and Structures in Nonlinear Nondispersive Media. Nonlinear Physical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23617-4_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-23617-4_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-23616-7
Online ISBN: 978-3-642-23617-4
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)