510 O. Grim 
member represents, therefore, a deformed wave which, in the region of the ship body, is in- 
dependent of the coordinate y. The second member in (46), however, mainly represents a 
deformation of the wave which depends on the coordinate y. It may be convenient, although 
not quite exact, to denote G(x) an effective wave amplitude. 
Free Moving Ship Body in Longitudinal Waves 
All hydrodynamic problems relative to the free moving ship body are already solved in 
the foregoing. considerations since a linearized treatment is assumed. 
Some hints will now be given how the motions are determined. All forces or moments, 
respectively, may be found by an integration of the forces per unit length over the length of 
the ship body. The exciting forces are found by an integration of Eq. (47), the restoring 
forces by an integration of Eq. (38). The results for the nondimensional heaving and pitch- 
ing motions (made nondimensional by the wave amplitude or by the amplitude of the wave 
angle, respectively) are as follows: 
A, || BCR + UE) dx + | xB(xR + ULE.) dx = | BG eee dx (49) 
E L L 
and 
z, | xB(R + U,E,) dx + vf xB(xR + UE.) dx =| x BG e*”* E dx. (59) 
L L L ’ 
where E, G, R, and U are complex functions of x. The values of U are nondimensional 
values of the function U(x) which follow from Eq. (34) for the case U,,, = 1 or U,p = *, re- 
spectively. The indices H and P indicate that the equation for U is satisfied either for 
heaving or for pitching. 
Using these values for the amplitudes of the motions two interesting functions* can be 
determined, viz., the resulting wave deformed in a longitudinal direction: 
Ge*”* - z.U,- WU, (51) 
referred to a space fixed zero-line or 
Ge 7s iz .U = ie iz ex (52) 
referred to a zero-line moving with the ship. 
The function (51) appears for the case of the free moving ship body instead of the func- 
tion Ge'¥* for the case of the restrained body. Taking the amplitude of the function (52) 
this function may be conceived as envelope of the wetted ship surface. 
*Of physical significance. 
