Wave Forees on a Restratned Shtp tn Head-Sea Waves 
Here w, is the frequency of the wave and U is the for- 
ward speed of the ship. x =- > is the x-coordinate of the forward 
perpendicular. Positive x is in the direction of the after perpendi- 
cular. 
We do not want to go into detail in this chapter about the ma- 
thematical expression for ~,) (or wy’ ). For more details see the 
chapters about the zero-speed and the forward-speed problems. But 
we should note the following : wg is the solution to the zero-speed 
problem. So the solution at a certain forward speed U can simply be 
obtained from the solution of the zero-speed problem by multiplying 
with 
Further wy' will not vary with x if the submerged cross-sectional 
area of the ship is not varying. The second expression of y is then 
telling us that there is a decaying factor (x + L/2) — as the wave 
propagates along the ship. 
The presentation in this paper is divided into the following 
steps. First we set up a general formulation valid for both the zero- 
speed and the forward-speed problem. Then we study the zero-speed 
problem separately. We derive a far-field expansion for a source 
distribution located on the x-axis between -L/2 and L/2. We 
obtain an inner expansion of the far-field source solution and study 
then finally the near-field solution and the matching between the near- 
field and the far-field solution. Then we have a chapter for the 
forward-speed problem, which is presented in a similar way as the 
zero-speed problem. Finally we have a chapter about numerical cal- 
culations. A computer program has been developed for a ship having 
circular cross-sections, and comparisons between experiments and 
calculations have been done. The agreement is shown to be good, 
II. GENERAL FORMULATION 
We assume that the ship is moving with constant speed U in 
the direction of the negative x-axis. The z-axis is upwards, and the 
y-axis extends to starboard. The origin of coordinates is located in 
the undisturbed free surface at midship, so that the forward-speed 
effect appears as an incident, undisturbed flow with velocity U in the 
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