Water wave diffraction by an array of bottom-mounted circular
cylinders is analysed
under the assumptions of linear theory. The cylinders are identical, and equally
spaced along the array. When the number of cylinders is large, but finite,
near-resonant modes occur between adjacent cylinders at critical wavenumbers,
and cause
unusually large loads on each element of the array. These modes are associated
with the existence of homogeneous solutions for the diffraction by an array which
extends to infinity in both directions. This phenomenon is related to
the existence of
trapped waves in a channel. A second trapped wave is established, corresponding to
Dirichlet boundary conditions on the channel walls, as well as a sequence of higher
wavenumbers where ‘nearly trapped’ modes exist.
The diffraction of water waves by a vertical circular cylinder is considered in the regime where the wave amplitude A and cylinder radius a are of the same order, and both are small compared to the wavelength. The wave slope is small, and a conventional linear analysis applies in the outer domain far from the cylinder. Significant nonlinear effects exist in the complementary inner domain close to the cylinder, associated with the free-surface boundary condition. Using inner coordinates scaled with respect to a, it is shown that the leading-order nonlinear contribution to the velocity potential includes terms proportional to both A2a and A3. The wave load which acts on the cylinder near the free surface includes second- and third-harmonic components which are proportional respectively to A2a2 and A3a. In a conventional perturbation analysis, where A [Lt ] a, these components would be ordered in magnitude corresponding to the different powers of A, but here they are of the same order. The second- and third-order components of the total force are of comparable magnitude for practical values of the wave slope.
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