  We put forward a solution to the initial boundary value (IBV) problem for the
nonlinear shallow water system in inclined channels of arbitrary cross-section
by means of the generalized Carrier-Greenspan hodograph transform (Rybkin et
al., 2014). Since the Carrier-Greenspan transform, while linearizing the
shallow water system, seriously entangles the IBV in the hodograph plane, all
previous solutions required some restrictive assumptions on the IBV conditions,
e.g., zero initial velocity, smallness of boundary conditions. For arbitrary
non-breaking initial conditions in the physical space, we present an explicit
formula for equivalent IBV conditions in the hodograph plane, which can readily
be treated by conventional methods. Our procedure, which we call the method of
data projection, is based on the Taylor formula and allows us to reduce the
transformed IBV data given on curves in the hodograph plane to the equivalent
data on lines. Our method works equally well for any inclined bathymetry (not
only plane beaches) and, moreover, is fully analytical for U-shaped bays.
Numerical simulations show that our method is very robust and can be used to
give express forecasting of tsunami wave inundation in narrow bays and fjords
