198 A. Hadjidakis 
As the pitch increases with increasing wave height, it will be clear that at a certain 
wave height the combination of a high value of negative pitch, combined with the orbital 
effect, makes the bow touch water. This phenomenon introduces a braking effect which 
causes the craft to slow down, so this wave height represents the limit up to which the 
craft can remain foilborne on a following sea. The lift reserve of the bow foils of a certain 
hydrofoil system evidently determines the limiting wave height. 
However, in Fig. 3 the dynamic pitch amplitudes of two similar hydrofoil craft differing 
only in size have been indicated as functions of the relative wavelength. The critical con- 
dition for the craft with the lower Froude number is seen to occur at a greater relative 
wavelength. Consequently, of two similar craft the bigger one (having the lower Froude 
number) will encounter its critical condition on a following sea on relatively higher waves. 
When going along the waves, i.e., when y = 7/2, a hydrofoil craft will attain its 
greatest roll angle 9, - However, experience has shown that this is not a critical condition 
at all; therefore it will not be discussed in more detail. 
When the craft is going against the waves, i.e., y = 7, the pitch amplitude is not criti- 
cal, because the orbital moment will help to surmount the next wave, but the maximum value 
of the excitation frequency is obtained. Therefore this can be called a critical condition as 
far as vertical accelerations are concerned, these accelerations being proportional to the 
square of the excitation frequency. While discussing vertical accelerations, y will hence- 
forth be assumed to be equal to 7z. 
Applying the foregoing theory to an example, a comparison is made between an existing 
small hydrofoil craft (J = 6.73 m, V = 30 knots) and a designed bigger craft (J = 36 m, 
V = 45 knots). Of the existing craft (Fig. 6), the natural frequencies for pitch and heave 
and the damping ratio are known. With the aid of Figs. 3 and 4 a curve could be obtained 
indicating the vertical acceleration at the bow as a function of the relative wavelength. The 
natural frequencies and the damping ratio of the designed bigger craft have been calculated 
according to Eqs. (7), (8), (13), and (14), which made it possible to establish an equivalent 
curve for the bigger craft. Both curves are shown in Fig. 7. 
It will be noted that there are two critical values of the relative wavelength A // where 
the vertical acceleration at the bow Z;/g attains peak values. These critical wavelengths 
correspond with 2/3 and 3/2 times the craft’s length. 
Contrary to what might be expected, Fig. 7 shows that critical values of acceleration 
are not to be expected at a relative wavelength of more than two. This means that vertical 
acceleration decreases with increasing wavelength. In other words, the comfort of the small 
craft will be much better on waves with a length of 20 m and a height of 1 m, than on waves 
with a length of 10 m and a height of 0.5 m. 
Similarly, the big craft will be more comfortable on waves of 100-m length and 5-m 
height, than on waves of 50-m length and a height of 2.5 m, although the comfort in the 
latter case is much better than the best that can be obtained with the small craft. 
In reality the wave pattem is never found to be so regular as the theory supposes it to 
be. On a seaway, wavelengths vary considerably, which tends to level off the extreme 
accelerations. Furthermore small waves are always superimposed on the longer waves, 
which will tend to raise the maximum accelerations to be found for values of \/I higher than 
two. These two effects have been taken into account in the dashed curves shown in Fig. 7. 
