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Waveform evolution for nonlinear surface acoustic waves. (English) Zbl 0627.73022

A new nonlinear analysis is given for the propagation of surface waves on a homogeneous, elastic half-space of general anisotropy. Avoiding the need for a multiple scale analysis, it yields the evolution equation for the Fourier transform of the surface elevation as a criterion ensuring that corrections to the displacements within linear theory are everywhere sufficiently small. For second-order elasticity, this equation is similar to that of R. W. Lardner [J. Elasticity 16, 63-73 (1986; Zbl 0575.73032)]. For periodic waveforms it reduces to an infinite set of ordinary differential equations governing the Fourier coefficients. The behaviour of special solutions is illustrated by numerical integration for a particular isotropic material. This confirms that initially sinusoidal waveforms steepen until the vertical velocities develop a singularity within each wavelength. However, waveforms which are initially close to be ‘nondistorting waveforms’ of Parker and Talbot [see P. Chadwick, ibid. 6, 73-80 (1976; Zbl 0332.73034)] may travel large distances without significant alteration to their dominant feature, which is one sharp peak per wavelength.

MSC:

74J15 Surface waves in solid mechanics
74H45 Vibrations in dynamical problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
74E10 Anisotropy in solid mechanics
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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[1] Kalyanasundaram, N., Int. J. Engng Sci., 19, 279 (1981)
[2] Kalyanasundaram, N.; Ravindran, R.; Prasad, P., J. Acoust. Soc. Am., 72, 488 (1982)
[3] Kalyanasundaram, N., J. Sound Vib., 96, 411 (1984)
[4] Lardner, R. W., Int. J. Engng Sci., 21, 1331 (1983)
[5] Lardner, R. W., Int. J. Engng Sci., 23, 113 (1985)
[6] Lardner, R. W., J. Elast., 16, 63 (1986)
[7] Planat, M., J. Appl. Phys., 57, 4911 (1985)
[8] Parker, D. F.; Talbot, F. M., (Nigul, U.; Engelbrecht, J., Nonlinear Deformation Waves (1983), Springer: Springer Berlin)
[9] Parker, D. F.; Talbot, F. M., J. Elast., 15, 389 (1985)
[10] Chadwick, P., J. Elast., 6, 73 (1976)
[11] A. SOKOLOV, private communication.; A. SOKOLOV, private communication.
[12] Farnell, G. W.; Adler, E. L., (Mason, W. P.; Thurston, R. N., Physical Acoustics IX (1972), Academic Press: Academic Press New York)
[13] David, E. A., Int. J. Engng Sci., 23, 699 (1985)
[14] Murnaghan, F., Finite Deformation of an Elastic Solid (1951), Dover: Dover New York · Zbl 0045.26504
[15] Parker, D. F., (Maiellaro, M., Waves and Stability in Continuous Media III (1987)), Bari
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