  Inhomogeneities in the wave field due to wave groups, currents, and shoaling
among other ocean processes can affect the mean water level. In this work, the
classical and unsolved problem of continuously computing the set-down and the
following set-up induced by wave breaking on a shoal of constant finite slope
is tackled. This is possible by using available theoretical knowledge on how to
approximate the distribution of wave random phases in finite depth. Then, the
non-homogeneous spectral analysis of the wave field allows the computation of
the ensemble average by means of the phase distribution and the inversion of
the integral of the second moment for the special case of a shoaling process
with uniform phase distribution. In doing so, I am able to obtain a direct
effect of the slope magnitude on the phases distribution. Therefore, an
analytical and slope-dependent mean water level with continuity over the entire
range of water depth is provided.
