  Dispersive shock waves (DSW) are a salient feature of long water waves often
observed in tidal bores and tsunami/meteotsunami contexts. Their interaction
with bathymetry is poorly understood. The shoreline hazard from tsunamis and
meteotsunamis critically depends on the fraction of incoming energy flux
transmitted across the shallow nearshore shelf. Here, by considering nonlinear
dynamics of waves over variable depth within the framework of the Boussinesq
equations we show that the transmitted energy flux fraction can strongly depend
on the initial amplitude of the incoming wave and the distance it travels. The
phenomenon is similar to self-induced transparency in nonlinear optics: small
amplitude waves are reflected by bathymetry inhomogeneity, while a larger
amplitude ones pass through. The mechanism of self-induced transparency of long
water waves can be explained as follows. In linear setting a bathymetry
inhomogeneity, of length comparable to that of the incident wavelength, by
transmitting high wavenumber components acts as a high-pass filter. The DSW
evolution efficiently transfers wave energy into high wavenumber band, where
reflection is negligible. By examining an idealized model of bathymetry we show
that this is an order one effect and explore its dependence on parameters in
the range relevant for meteotsunamis. The role of wave energy transfer into
high wavenumber band owing to the growth of bound harmonics unrelated to the
DSW was found to be small.
