An inhomogeneous, nonlinear elastic panel, fixed at one end, is subject to a resonant harmonic forcing at the other. The vibrations in an equivalent homogeneous panel would contain shocks. We exhibit a class of finite inhomogeneities that produces a panel response that is harmonic but does not contain shocks. The finite-strength inhomogeneity is capable of inhibiting the nonlinear distortion sufficiently to prevent shocks from forming. This is not the case for a “slow” inhomogeneity that can be described using a geometrical acoustics approximation.
Mortell, M. P. and Varley, E.Finite amplitude waves in bounded media: nonlinear free vibrations of an elastic panel . Proceedings of the Royal Society of London A, 318, 169-196 (1970).
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Seymour, B. R. and Mortell, M. P.Resonant acoustic oscillations with damping: small rate theory . Journal of Fluid Mechanics58, 353-374 (1973).
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Mortell, M. P. and Seymour, B. R.Standing waves in an open pipe: a nonlinear initial-boundary value . Journal of Applied Mathematics and Physics (ZAMP), 24, 473-487 , (1973).
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Seymour, B. R. and Mortell, M. P.Nonlinear resonant oscillations in open tubes . Journal of Fluid Mechanics, 60, 733-749 (1973).
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Mortell, M. P. and Seymour, B. R.Pulse propagation in a nonlinear viscoelastic rod of finite length . SIAM Journal of Applied Mathematics, 22, 209-224 (1972).
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Lawrenson, C. C. , Lipkins, B. , Lucas, T. S. , Perkins, D. K. and Van Doren, T. W.Measurements of macrosonic standing waves in oscillating closed cavities . Journal of the Acoustical Society of America, 104, 623-636 (1998).
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Engin, H. , Askar, A. and Cakamak, A. S.An analytical method for strong motion studies in layered media . International Journal of Nonlinear Mechanics, 16, 165-186 (1981).
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Chester, W.Resonant oscillations in a closed tube . Journal of Fluid Mechanics, 18, 44-64 (1964).
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Mortell, M. P. and Seymour, B. R.Nonlinear Forced Oscillations in a Closed Tube: Continuous Solutions of a Functional Equation . Proceedings of the Royal Society of London A, 367, 253-270 (1979).
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Seymour, B. R. and Mortell, M. P.A finite rate theory of resonance in a closed tube: discontinuous solutions of a functional equation . Journal of Fluid Mechanics, 99, 365-382 (1980).
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Mortell, M. P. and Seymour, B. R.The calculation of resonance separatrices for the near-integrable case of the standard mapping . Journal of Applied Mathematics and Physics (ZAMP), 39, 861-873 (1988).
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Ellermeier, W.Resonant wave motion in a nonhomogeneous system . Journal of Applied Mathematics and Physics (ZAMP), 45, 275-286 (1994).
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Sturtevant, B.Nonlinear gas oscillations in pipes: Part 2. Experiment . Journal of Fluid Mechanics, 63, 97-120 (1974).
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Can, M. and Askar, A.Nonlinear elastic surface gravity waves: continuous and discontinuous solutions . Wave Motion, 12, 485-495 (1990).
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Varley, E. and Seymour, B. R.A method for obtaining exact solutions to partial differential equations with variable coefficients . Studies in Applied Mathematics, 78, 183-225 (1988).
