formulated in this report with potential for expansion and generalization to the more 
complicated case. The simpler problem is that of a V-shaped prismatic body with hard chines 
and constant deadrise planing at high speed in regular head waves. 
The mathematical formulation is analogous to low-aspect-ratio wing theory with 
provisions for including hydrodynamic impact loads, essentially a strip theory. Surface wave 
generation and forces associated with unsteady circulatory flow are neglected, and the flow is 
treated as quasi-steady. The mathematical formulation is an empirical synthesis of several 
theoretically derived flows describing the overall craft hydrodynamics. Wave input is restricted 
to monochromatic linear deepwater waves with moderate wavelengths and low wave slopes. 
MATHEMATICAL FORMULATION 
GENERAL 
Consider a fixed coordinate system (x,z) (Figure 1) with x axis in the undisturbed free 
surface, pointing in the direction of craft travel, and the z axis, pointing downward. If the 
motions of the craft are restricted to pitch 0, heave Zcq> and surge Xcq, the equation of 
motions can be written as 
MXcg = T,,-N sin @.—Djcos 0 
MZcc = T,-Ncos#+Dsin@ +W 
10 = Nx, + Dx Tx, (1) 
where M is mass of craft 
I is pitch moment of inertia of craft 
N_ is hydrodynamic normal force 
D_ is friction drag 
W is weight of craft 
is thrust component in x direction 
is thrust component in z direction 
x, is distance from center of gravity (CG) to center of pressure for normal force 
Xq is distance from CG to center of action for friction drag force 
x. is moment arm of thrust about CG. 
Equation (1) is exact; however, defining the hydrodynamic forces and moments in waves 
can be extremely difficult. 
