When a bore travels shoreward into water at rest on a beach, then according to the first-order non-linear long-wave theory, the bore accelerates and decreases in height, until it collapses at the shore. The investigation here reported concerns the question, what happens next? It is formulated as a singular characteristic boundary-value problem with somewhat unusual mathematical properties. Its asymptotic solution predicts a rather thin sheet of run-up and back-wash with some unexpected features.
Spectra of bounded and unbounded basins are calculated on an asymptotic approximation based on smallness of the seabed slopes. This leads to the geometrical optics theory of surface waves of Keller, which is used to extend the known, exact result for an idealized beach to the prediction of the spectra of a variety of more natural water bodies. Attention is directed to cylindrically symmetrical and two-dimensional topographies corresponding, respectively, to islands and to continental slopes and laboratory channels. Marked differences are found in the character of the spectra for these two types of topography.
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