Resonant nonlinear oscillations of an ideal gas contained in a cylindrical shell are studied. Excitation is generated by a harmonically oscillating line source positioned on the cylinder axis and symmetric, harmonic displacement of the cylinder wall of the same circular frequency; also there is a constant phase shift between both excitations taken into account. The problem is nonlinear and qualitatively resembles the analogous one in a spherical shell in that there is a similiar response curve in both cases with the response amplitude being of the order of magnitude of the cubic root of the excitation amplitude in either situation, but there are quantitative differences.

The propagation of weakly nonlinear acoustic waves in a non-uniform medium is treated. It is assumed that the waves are one-dimensional. Non-uniformities arising from variable cross-section and stratification are included. The effect of non-uniformities on unidirectional waves on an infinite interval and resonant waves on a finite interval is discussed for a near-uniform reference state (geometrical acoustics limit) and for stronger non-uniformities in the finite-interval case. Nonlinearities are taken into account up to quadratic and, wherever necessary, cubic order in the wave amplitude.Unidirectional waves in the geometrical acoustics limit can formally be reduced to the behaviour in a uniform system described by a kinematic wave equation with constant coefficients. For illustration acceleration waves in a weakly non-uniform medium are treated. The resonance case in the geometrical acoustics limit is closely related to resonance in a uniform system so that the methods developed for that situation require only slight modification. For larger influence of non-uniformity the geometrical acoustics limit does not apply and the resonance problem may lead to a Duffing oscillator type of behaviour.

Summary.Resonant nonlinear oscillations of a thermoviscous gas contained in a cylindrical or spherical shell are studied. Excitation is generated by a harmonically oscillating line or point source positioned on a cylinder axis or the center of a sphere, respectively, and symmetric, harmonic displacement of the wall; phase shift between both excitations is taken into account. The case of a constant phase shift is treated. The results also cover the unforced behavior of the undamped and damped systems. One finds that the envelope of the oscillation decays like in linear theory but the temporal evolution of the wave phase exhibits nonlinear effects. For both geometries the problems are nonlinear and qualitatively resemble each other closely in that there is the typical response curve ofa Duffing oscillator with the response amplitude being of the order of magnitude of the cubic root of the excitation amplitude in either situation when damping is assumed weak enough to be accountable for by linear approximation. There are quantitative differences, however, with the damping as well as the nonlinearity effect being more pronounced for the spherical geometry. Temporal means of all perturbation fields are found to be vanishing in contrast to plane wave resonance. scaled displacement amplitude of wall, scaled source strength isentropic velocity of sound in perturbed gas, isentropic velocity of sound in quiescent gas radial coordinate, mean position of oscillating wall of cylindrical or spherical shell, specific gas constant in Appendix II time, slow time (absolute temperature in Appendix II) dimensional, non-dimensional velocity potential radial particle velocity (see Appendix II) specific entropy circular frequency of excitation ratio of displacement amplitude of wall and R dimensional, non-dimensional source strength amplitude ratio of combination of shear and bulk viscosity and mass density, heat conductivity, and thermal diffusivity in quiescent state acoustic Reynolds number acoustic Prclet number Prandtl number scaled damping (scaled sum of reciprocal acoustic Reynolds and Prclet number) mass density, pressure in quiescent state isentropic exponent of ideal gas specific isobaric heat capacity cor non-dimensional radial coordinate --, non-dimensional time cot

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