227s 
Second order periodic breaker effects 
The Gerstner wave satisfies the nonlinear Lagrangian equations 
exactly. However, a particle never moves far away from its mean position. 
A strong field of vorticity exists. The limiting steepness of a Gerstner 
wave is given by ak =1. For irrotational waves Michell [1893) has shown 
that ee 0.142, and ak = 1 corresponds to <3 = 
much steeper than is possible for irrotational motion. It was mentioned 
0.32 which yields a wave 
above that the Gerstner wave could be made irrotational to second order 
(with third order complications) by adding an appropriate drift current. 
With @ equal to zero, this drift current becomes a wk. 
The parametric representation for the free surface then be - 
comes 
xX = a-a sim(ka - wt) + mae wt 
(62) 
z =a cos(ka - ut) 
for a periodic wave, which can be transformed by solving for a in the 
second equation to become 
li (ha fa 
(63) Cosma - ak sin(cos (z/a)) = kx - wt-a kot 
and therefore the phase speed obtained from the lagrangian equations 
solved for irrotational motion to second order becomes 
(a5, (2 
(64) c=8(ita Ve 
The phase speed obtained by solving the Eulerian equations 
to third order for irrotational motion is given by 
(a 
= & aks 
(65) C = Ihr 5 
The Lagrangian waves to second order are thus not consistent 
with Eulerian waves to third order. The second order periodic irrotational 
Gerstner waves appear to be traveling as if the linear phase speed had 
been simply added to the drift current. Third order periodic irrotational 
Eulerian waves have a drift current that is the same as the Gerstner 
wave, but the phase speed correction is only half of the phase speed cor - 
rection for Gerstner waves. The discrepancy can be explained by noting 
