Linear and nonlinear resonance of water waves near periodic structures

摘要

In the first part of this thesis, we present a nonlinear theory for the excitation of trapped wave around a circular cylinder mounted at the center of a channel. It is well-known that near an infinite linear array of periodically spaced cylinders trapped waves of certain eigen-frequencies 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. A nonlinear theory is developed for the trapped waves excited subharmonically by an incident wave of twice the eigen-frequency. The effects of geometrical parameters on the initial growth of resonance and the final amplification are studied in detail. The nonlinear theory is further extend 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. In the second part of the thesis, we study the Bragg resonance of surface water waves by (i) a line of periodic circular cylinders in a long channel, and (ii) a two-dimensional periodic array of cylinders.