The radiation and scattering of water waves
by a horizontal thin circular disc in three dimensions is treated.
The linear water wave theory is
adopted and integral equation methods are used for the
solution of the problems. The circular dock problem is formulated,
analysed and solved by using a boundary element method. Numerical
results, including the scattering cross section, are provided.
The interaction of waves with a submerged disc is also considered.
The problem is formulated as a hypersingular integral equation
and a new expansion-collocation method, allowing the analytical
evaluation of this double integral, is developed and applied
to these problems, which exhibits critical aspects when
both the depth of submergence and frequency are small.
Several relationships between the physical parameters
in this critical region are deduced.
A detailed study of a class of axisymmetric motions is carried out
by employing one-dimensional integral equation formulations.
In particular a small-submergence approximation is considered
and its connections with the dock problem show surprising
results. In addition, a relation between peaks in several
physical quantities and resonance poles is found. These
resonances, which constitute simple poles in the complex plane,
are located by using Newton's method, and estimates,
for small values of submergence, are given.