  Long waves bring many important challenges in the ocean and coastal
engineering, including but are not limited to harbor resonance and run-up.
Therefore, understanding and modeling their dynamics is crucially important.
Although their dynamics over various types of geometries are well-studied in
the literature, the study of the geometries with power-law variations remains
an open problem in this setting. With this motivation, in this paper, we derive
the exact analytical 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.
