420 T. H. Havelock 
In general, for any form of the front portion of the model, we have 
R = tgph? | 
b = 
sintady = gph? Bsin® «. (43) 
b 
In (43), « is the angle the tangent to the form makes with Ox, and the bar 
denotes the mean value of sin? « with respect to the beam of the ship. 
Suppose, for instance, that the section of the model is the ellipse 
a2/a2+y?/b? = 1. It is easily shown that in this case 
b2 a2 az— 52 
aot ee wo ae a 
This would be a full form of model. If we take a = 8b, as an average ratio of 
length to beam, we find from (44) that the mean value of sin? is 0-17. 
The mean value is less for models with moderate bow angle; probably an 
average value would be about 0-1, with still smaller values for models with 
fine lines. 
In a recent paper Kreitner (1939) has put forward the proposition that 
the extra resistance to a ship among waves is nothing else than the radiation 
pressure of the ocean waves. The semi-empirical formula given by Kreitner 
for this force upon a ship at rest in a train of waves is 
sinta = 
R = gph? Bsing, (45) 
in the present notation, in which h is the amplitude of the incident waves; 
the last factor is a mean value for the angle of entrance not clearly defined. 
The derivation of this formula is not clear, but it appears to be based upon 
an estimate of the difference of resultant amplitude at bow and stern, and 
upon taking the mean value of the hydrostatic pressure due to the surface 
elevation. This latter assumption is incorrect; and further, we found in (43), 
that the last factor should be the mean value of sin’ «taken across the beam. 
Numerically, for usual ship forms, these differences result in (43) giving 
about one-fifth of the value from (45). 
For a certain model, a ship with full lines, the relevant data are 
B = 69-2 ft., L = 530 ft., h = 24 ft. If we assumed the fore half of the ship 
to be an ellipse and used (44), we should have 0-175 as the mean value of 
sin?a; but this is certainly too large and we take a smaller value, say 0-12. 
With these values, (43) gives a force of 0-6 ton. This is, moreover, an upper 
limit and also assumes the ship to have vertical sides and to be of great 
draft. The recorded extra resistance for this ship is given as about 2-8 tons; 
but this was for a model advancing through the waves. 
The steady pressures we have been considering will certainly be increased 
if the ship is itself in steady motion through the waves, but the problem 
481 
