Springer Online Journal Archives 1860-2000
Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
Abstract There has been substantial effort recently put into proving, for a variety of different geometries, the existence of trapped waves, that is unforced time-harmonic motions which do not radiate energy to large distances. Thus it is known that such motions can exist in a deep channel which includes a cylinder spanning the channel, for various shapes of cylinder. The converse problem of proving the absence of such trapped waves has received much less consideration, and the only relevant uniqueness proof for a channel spanned by a cylinder is that of McIver (1991). In an appendix to that paper, McIver demonstrates that no trapped-wave motions can exist for the case in which the cylinder is surface piercing and is entirely contained between vertical planes through the free-surface intersections. This is exactly the same geometrical condition which John (1950) found would ensure uniqueness in water-wave radiation and scattering problems, in finite or infinite depth. Both John and McIver achieved their uniqueness results by consideration of integrals of the potential along vertical lines down from the free surface. John's work was extended by Simon and Ursell (1984) who established uniqueness for a wider class of two-dimensional radiation and scattering problems by consideration of integrals along nonvertical lines. The work presented in this paper is the corresponding extension of McIver's work; although this extension does not rule out trapped waves at all frequencies for any geometry except that already considered by McIver, it does yield an easy lower bound for the ratio of the trapped-mode frequency to the cut-off frequency, in finite or infinite depth.
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