306 T. H. Havelock 
have assumed the free waves to be small, the corresponding quantities being 
of the same order as those of the forced waves due to the ship; moreover, we 
have neglected any direct disturbance of the free waves by the surface of 
the ship for the same reason as for omitting terms of a like order in obtaining 
expressions for the forced waves. The additional pressures are therefore 
simply those due to the free waves, and the additional resistance is the 
horizontal resultant of these pressures acting upon the ship’s surface. Taking 
as an example the model described in (10), the resistance is given by R, +f’, 
where &, is the resistance in still water, and 
pSV CORDON BNA eG eho Noles 
w= ae alata) eat aa) | 
x {sin «(1+ a)—xl cos(k+a)—sinxa}cosf, (25) 
where Ve= g/k, P=KC+V)t = 27/2. 
In this expression, 7’ is the period of encounter of the ship with the waves. 
In experiments on models in artificially produced waves, a critical con- 
dition occurs when the wave-length is about equal to the length of the model. 
We take therefore as a numerical example A = 27/K = 2(/+a), and 
V2 = gA/2m = g(l+a)/rn. 
Putting in the numerical values for this model, we get from (25) 
R’ = 89-1hcos(2nt/T), (26) 
R’ being in lb. with the amplitude h in ft. 
In experiments a usual assumption is a wave-height of 6 ft. for a wave of 
length 400 ft.; this ratio gives for the wave-length 16 ft. the value h = 0-12 ft. 
In that case we have 
R’ = 10-7 cos(27t/T) lb. (27) 
Values of R, for this model are known. For instance, for model speeds of 
7-08, 9-22, 11-04 ft./sec. we have R, = 4:9, 13, 29 lb. respectively; the total 
resistance, wave-making and frictional, at these speeds was 14-54, 30-11, 
52-15lb. respectively. We see that, for quite a moderate ratio of wave-height 
to wave-length, #’ represents an alternating force of relatively large ampli- 
tude. It should be noted, however, that this is for the particular case when 
the wave-length of the free waves is equal to the length of the model. 
8—It is necessary to emphasize the basis of the present calculations. It is 
assumed that the model is maintained in the same relation to the un- 
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