The problem of transmission of wave energy in strongly inhomogeneous media is discussed with application to long water waves propagating in a basin with a quartic bottom profile. Using the linear shallow-water theory it is shown that the wave component of the flow disturbance is described by a travelling wave solution with an amplitude and phase that vary with distance. This means that the kinetic part of the wave energy propagates over large distances without reflection. Conditions for wave breaking in the nearshore are found from the asymptotic solution of the nonlinear shallow-water theory. Wave runup on a vertical wall is also studied for a quartic bottom profile.
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