Linear and nonlinear resonance of water waves near periodic structures

Abstract

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.

(cont.) For case (i), strong reflection takes place in a channel when the cylinder spacing is one-half that of the incident waves. Solutions for a large but finite number of cylinders in a channel are examined and compared with finite element results. For case (ii) we study an array of cylinders extending in both horizontal directions toward infinity, the Bragg resonance condition is found to be the same as that in the physics of solid state and photonic crystals, and can be determined by Ewald construction. Envelope equations of Klein-Gordon type for resonated waves are derived for multiple resonated waves. For a wide strip of cylinders, analytical solutions of both two-wave and three-wave resonance are discussed in detail. We also extend the theory to include second-order nonlinear effects of the free surface. For a train of periodically modulated incident waves scattered by an one-dimensional line of cylinders, free long waves are found to exist and propagate faster than the set-down long wave bound to the short wave envelopes. At Bragg resonance, the short waves are reflected by the array but the induced free long wave can pass through it. For a train of periodically modulated waves scattered by a finite strip of cylinders, the free long waves can propagate away from the strip or be trapped near the strip depending on the angle of incidence.