Equations (6) are linear. A solution of them in turn determines the 
right hand side of (7). Since the left hand side of (7) is linear, in principle 
at least, a solution can be found. Also in principle (though not in practice), 
it is possible to proceed to as high an order as desired by this procedure. 
The Gerstner wave 
A solution to equations (6), applicable to waves in deep water, when 
added to the zero order solution, yields equation (8). 
ak 
; i Tice 
sq 5S (el, ap Ex a Coane sin(k,a + kB - wt) 
ak 
: ! 7 aie 3 
Vea aiee ys) am ste © sin(k,a + kB wt) 
k6 
(8) z=6+ez,=6tae cos(k,a - k,f - ot) 
P=p +P, =P,- gpd 
aw k4 
=—e 
FY Sis sin(k,a + k,B - wt) 
In order to impose the condition that p(a, B, 6) = P, at © 0); ake 
is required that 
2 
Ww — 
(9) wes 
The continuity equation imposes the condition that 
2 
2 
The first order solution has no vorticity to first order as FY has 
(10) Kk +k 
been found. 
An alternate form for the solution given by (8) is (ll). 
2 2 
x =a-acos@e” 78 sin( S (a cos 6+ 6 sin 6) - at) 
26) we 
(11) y=Bp-a sin Oe” 8 Sin te cos 6 + & sin 6) - at) 
2 2 
z=&+tae’ 5/8 
cos( —— (a cos 8+ B sin®) -at) 
