19 



When atmospheric pressure is defined as zero, and dynamic pressure is substituted 

 into Eq. 2.8. the result is: 



dt 

 The Bernoulli Constant 



+ l{u^ + w^) + ^-Br=0 (2.10) 



In a potential flow, the Bernoulli constant is the same throughout space and 

 time. For the special case of wave motion, the value of the Bernoulli constant can 

 be computed if the kinematics are known (Longuet-Higgins 1975). The Bernoulli 

 equation (2.8) and the bottom boundary condition are applied at the bottom: 



| + 5«? + <;^-. + ^-B = o (2.11) 



where uj, pb, and zi, are the velocity, pressure, and elevation at the sea bed. If the 

 flow is periodic, a time average over a period results in: 



^^ + ^Zfc + ^-5 = (2.12) 



2 p 



where the over-bars indicate time averaging. 



In the case of steady periodic wave motion, the total vertical momentum at any 

 horizontal location must be the same at the beginning and end of a period. In order 

 for this to be the case, the vertical momentum averaged over a period must be a 

 constant. In order for the momentum to remain constant, the time averaged net 

 vertical force on the water column at that point must be zero. If the pressure at the 

 water surface (pa) is taken to be zero, then the force of the pressure at the bed must 

 be equal to the gravitational force on the column of water, so that the mean pressure 

 on the bed is hydrostatic, 



Pb = P9{v-Zb) (2.13) 



resulting in a simple and exact expression for the Bernoulli constant. 



B = grj^^^, (2.14) 



