The dispersion of the first two longitudinal wave modes, L(0,1) and L(0,2), was experimentally investigated for a cylindrical shell (such as a pipe or tube) that was completely filled with a liquid. It was observed that the presence of a liquid inside the cylinder dramatically alters the dispersion curve for the L(0,2) mode by dividing (or branching) the curve into approximately equally spaced regions separated by cutoff-type behavior. This branching was attributed to coupling between the unperturbed L(0,2) mode in the shell and the unperturbed longitudinal modes in a liquid cylinder with rigid boundaries, LL(0,2N), where N is an integer. The physical mechanism for the mode coupling was determined to be radial resonances in the combined liquid/pipe system. In time domain, the liquid effects on the dispersion are manifested as a long-duration signal or a series of short-duration pulses, depending on the pulse length of the transmitted signal relative to the reciprocal of the frequency interval between branching.
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Kwun et al. (1999) studied this question.