Subharmonic resonance of a trapped wave near a vertical cylinder by narrow-banded random incident waves

It is well-known that near an infinite linear array of periodically spaced cylinders trapped waves of certain eigenfrequencies can exist. If there are only a finite number of cylinders in an infinite sea, trapping is imperfect. Simple harmonic incident waves can excite a nearly trapped wave at one of the eigen frequencies through a linear mechanism. However, the maximum amplification ratio increases monotonically with the number of the cylinders; hence the solution is singular in the limit of infinitely many cylinders. Recently, a nonlinear theory of subharmonic resonance of perfectly trapped waves has been completed. In this article the theory is further extended to random incident waves with a narrow spectrum centered near twice the natural frequency of the trapped wave. The effects of detuning and bandwidth of the spectrum are examined. Dedicated to Professor J. N. Newman on his 70th birthday. We wish to express our profound admiration for Professor Newman’s scientific contributions and leadership in the ship-hydrodynamics discipline. The relation between this article and an early work of his reflects in part his impact on us.