  In this paper, we derive the exact solutions of the long-wave equation over
nonlinear depth and breadth profiles having power-law forms given by $h(x)=c_1
x^a$ and $b(x)=c_2 x^c$, where the parameters $c_1, c_2, a, c$ are some
constants. We show that for these types of power-law forms of depth and breadth
profiles, the long-wave equation admits solutions in terms of Bessel functions
and Cauchy-Euler series. We also derive the seiching periods and resonance
conditions for these forms of depth and breadth variations. Our results can be
used to investigate the long-wave dynamics and their envelope characteristics
over equilibrium beach profiles, the effects of nonlinear harbor entrances and
angled nonlinear seawall breadth variations in the power-law forms on these
dynamics, and the effects of reconstruction, geomorphological changes,
sedimentation, and dredging to harbor resonance, to the shift in resonance
periods and to the seiching characteristics in lakes and barrages.
