angle is 2t - 9 , and the water region is defined by 9 > 9 > and 

 > z > -h . 



5. The velocity field for the wave reflection and diffraction in an 

 ideal fluid can be represented by the velocity potential function $(r,9,z,t) 

 which must satisfy the Laplace equation, where t is the temporal coordinate. 

 We assume that the waves are sinusoidal in time with radian frequency u . 

 Water depth is constant, and the bottom is rigid and impermeable. Therefore, 

 the vertical and temporal components of the velocity potential function, which 

 follow from separation of variables, can be factored out and the velocity po- 

 tential written as 



*/ q ^\ a cosh k(z + h) ,, „v itot ... 



* (r ' 9 ' Z ' t) " A o cosh kh * (r ' 9)e (1) 



where 



A = -iga /oo 

 o o 



i = vCI 



g = gravitational acceleration 



a = incident wave amplitude 

 o 



k = wave number 



<(> = horizontal component of the velocity potential function 



6. Substituting Equation 1 into the Laplace equation and using both 

 the kinematic and dynamic boundary conditions at the free surface, the Laplace 

 equation is then reduced to the Helmholtz equation which is written in polar 

 coordinates as follows: 



2 9 2 <t> 9<f> 9 2 <l> 2 2, 



r — 4+r| i + — 4 + k r ♦ " (2) 



a r 2 3r 89 2 



where k must be a real number and satisfy the dispersion relationship 



2 

 a) = gk tanh kh (3) 



7. The free surface displacement ti from the mean water level z = 

 can be obtained from linear wave theory and is represented as 



n(r,0,t) = - -r- = a <Kr,0)e (4) 



g 9t o 



