The Radiation Of Sound Waves From A Lightly Loaded Finite Elastic Shell, II: Non-Linear Shell Resonances

This paper describes the second half of an investigation into the scattering of sound waves by a circular cylindrical elastic shell of finite length. The flexible body is bound to an infinite rigid cylindrical baffle, and is completely immersed in a compressible inviscid fluid. Here the system is forced by plane acoustic waves, and we seek to obtain an asymptotic approximation of the scattered field by expanding in powers of a suitable small parameter. This is the ratio of fluid to shell density, and is conveniently small in aeroacoustic situations. The first part of this work revealed the possibility of a large scattered field when the incident radiation oscillated at a frequency close to an infinite discrete set of eigenfrequencies, corresponding to either shell resonances or interior duct resonances. In Part I attention was confined to linear motions, which was satisfactory for small amplitude radiation, but for larger magnitude incident waves, non-linear terms in the structural equation become important. The objective in the work described in this paper, Part II, was to investigate the way in which the non-linearities modify the results of Part I, and also to see if new resonances occur. We show that the scattered field is not only distorted near resonance but can attain different long-time states depending on the initial conditions of the model. We determine new secondary resonant states, and prove that limit-cycle solutions are also possible. The sensitivity of the system to changes in initial conditions is also discussed.