Form of the Irrotational Linear Equations when the Require - 
ment that the Solution be Periodic in Time ts Imposed. 
Definition of periodicity  (x,y,z,t)= O(x,y,z,tt+T) (2.11) 
then @Q(x,y,z,t) = Aatihtanehoe a= (2.12) 
N(x,y,f) = Ren'(x,y) ei (2.13) 
and, potential equation Qxx+ yy + zz = 0 (2.14) 
boundary conditions: 
at Z=-h(x,y) dn=O (2.15) 
at Z=0 ni: - tt ad = (2.16) 
a | 2S) ere (2.17) 
Known solutions if =-h (h constant) 
Straight wave crests 
0 eu ° 
g . Ati eile cos® ty sin®)+ 3] ogn 2 (Zz +h) 
, cosh 22h (ete) 
L 
N= Acos [42 (x cos@ + ysinO@) + § - at] (2.19) 
2 L 
2245-2 tanh 2h. (2.20) 
Circular wave crests 
= const Ho(4=") cosh 22 (z +h) | (2.21) 
joey Ga PE eg alle 
b= © conatwelaee Neko) Sy 5 Ye) @ cosh 2 (Z +h) (2.22) 
V (x- xe)? +(y-y,)2 | 
ne const eos (an ¥(X-X0)* +(y- yo)? -5-271) (2.23) 
V(x-X0)2 + (y= yo)? 
PEATE “TE 
=/2= 
