theory and the principle of linear superposition permit derivations for 

 the hydrodynamic coefficients and forcing function Ui j , A^j and Fj (t) . 



Steady-state solutions of the form: 



a.(t) = a. sin (wt + 6.) for i = 1,2,3 (3) 



are assumed. Substitution of the assumed solution (eq, 3) into the 

 equations of motion (eq. 2) yields a set of linear algebraic equations 

 which may be solved for the unknown amplitudes and phase angles aj^ and 

 6^. Transfer functions, Hj.^, are then defined by the a^ and (S^ since 

 the incident waves are assumed to be sinusoidal. 



b. Hydrodynamic Coefficients and Waves . Potential theory is em- 

 ployed in computing the reflected and transmitted waves, hydro- 

 dynamic coefficients and the exciting forces. Under the assumptions of 

 small incident waves, small breakwater motions and an inviscid fluid, 

 the velocity potentials may be found and the problem subdivided using 

 the principle of linear superposition. The total velocity potential: 



^total ^incident Miffracted ^ motion ' ' 



is the sum of the incident wave potential, the diffracted wave poten- 

 tial and the potential resulting from forced sway, heave, and roll mo- 

 tions. 



The incident wave potential is well known and may be expressed 

 directly. Obtaining the diffracted wave and breakwater motion poten- 

 tials requires the solution of boundary value problems. These problems 

 and their solutions are described in Appendix C. Appendix D provides 

 the computer program used to calculate breakwater performance. 



When the velocity potentials have been obtained, the free-surface 

 elevation at any position is found using the linearized free-surface 

 boundary condition: 



n(x,t) = - ^ <^^ (x,0,t). C5) 



Here: 



n(x,t) = free-surface elevation measured from Stillwater 

 level (y = 0), 



g = acceleration of gravity, 



4i (x,0,t) = derivative of the velocity potential with respect to 

 time evaluated at y = 0. 



17 



