Wave Forces on a Restratned Shtp tn Head-Sea Waves 
neglecting surface tension, 
(A) the dynamic free surface condition 
2 
2 2 
Bie eat ar ee ALP ye ee Th SD wetiy 
On. 2, =" &\ sy ve t.) 
(B) the kinematic free surface condition 
+ a + = = 
Ob ta Oe T= 0 on a= Hey t) 
Here g is the acceleration of gravity. 
@ must also satisfy a radiation condition, which will be discussed 
more later. 
We will assume that there are incoming, regular gravity 
waves propagating along the positive x-axis. The wave amplitude is 
assumed to be small so that the classical linear free-surface theory 
is applicable. We will later linearize the problem with respect to 
the wave amplitude. The potential $, of the incoming waves will be 
given by 
$ 4 Re | F< VZ ei (vt ~ hin (5) 
Re means the real part*® h is the wave amplitude of the incomin 
RE Pp Pp g 
waves, , is the wave frequency, vy is the wave number and t 
* As is usual we are going to drop the notation Re. We will write 
the potential in complex form, and it should be understood that we 
should take the real part. 
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