Computation of the Motion of Long Water Waves 



typical normal and tangential fluid velocities at the wall, then in 

 inviscid flow 



^n = (7) 



U = (8) 



Vn = (9) 



Unn=0 (10) 



Accordingly, for Wall 1 shown in Fig, 1 we have for < y < L^, 

 < t: 



T1x(0,y,t) = 0, Vx{0,y,t) = 0, u(0,y,t) = 0, u^x(0»y't) = 



Similar conditions hold on the other walls . 



If we choose to prescribe ti = r]{t) edong Wall 1 where x = 0, 

 then a special set of conditions must be derived (cf, , Madsen and Mei 

 [ 1969a] who point out that this prescription corresponds physically 

 to the situation of having measured the incoming wave height at a 

 particular station). Peregrine [ 1967] gave the irrotationality con- 

 dition 



Uy - V, = i dy[V • (dlT)], - -^ d,[ V . (dir)]y 



-jddyLV-"^] +iddjV."jr]y (11) 



where "u = (u,v) and V = (8/9x, d/dy). If the bottom is flat in the 

 neighborhood of Wall 1, d^ = dy= and Eq. (11) becomes 



Uy = V^ (12) 



But, if ■n(0,y,t) = 'n(t) , then until some reflection from a shoal or 

 beach in the tank returns to x = 0, 



Uy(0,y,t) = (13) 



and, hence, 



vjO,y,t) = (14) 



155 



