50 THE APPROXIMATE CALCULATION OF WAVE RESISTANCE AT HIGH SPEED 
This integral can be expressed in terms of Bessel functions, and its value obtained 
readily from tables of these functions. 
It is clear that this extreme simplification can only be an ideal limit for very 
high speeds, and it is no use comparing it with experimental results. It is, 
however, of interest as the limit towards which the usual complete theoretical 
expressions should tend. Consider, for instance, the simplest type of experi- 
mental model with parabolic lines, the surface being specified by 
y=b0 — 2a) (1 — x /P?) . : : ; P i 5 , a(S) 
Calculations, meantime unpublished, from the complete integrals (4) and (6) 
have been made recently for very high speeds by Mr. W. C. S. Wigley, who 
has placed his results at my disposal. Taking the Froude number f = v/1/(gL), 
the highest value for which calculations were made was f= 1-77. With length 
— [ = 2/= 16 ft., beam = 25 = 1-5 ft., draught = d= 1 ft., at this value 
of f the complete formula gives a wave resistance of 31:8 lb. Calculating from 
(7) with V = 103 cub. ft., h = 3 ft., we obtain a resistance of about’40 lb. The 
comparison is not so far out as might have been anticipated, and to that extent 
it may be taken as confirming the argument by which the simple formula was 
obtained. 
6. The next simplest dissection of the ship is to divide it into two by the 
midship section. We consider the fore and aft parts separately, replacing 
each part by a single source at the centroid of the distribution in each case. 
For the positive sources on the fore part of the ship we have seen that if M is 
the area of the midship section the total source strength is Mv/4x. From the 
argument in the previous sections it is readily seen that the moment of the 
distribution with reference to the midship section is V, 0/4x, where V, is the 
volume of the fore part ; hence the centroid is at a distance V,/M, or Pil, ahead 
of the midship section, where /, is the length of the fore part and p; its prismatic 
coefficient. Similarly the controid of the negative sources on the after part 
is at a distance p./, astern of the midship section, where /, is the length of the 
after part and p, its prismatic coefficient. Thus we have a pair of sources, 
positive and negative, each of numerical strength Mv/4z, at the depth A of the 
centroid of the midship section, and at a distance pL apart, where L is the 
length of the ship and p its prismatic coefficient. Applying the formule (4), 
(5) to this combination, we obtain for the wave resistance 
co 2 
R = (4ge/n) ke, e — 2kh-cosh’u sin? (4 kpL cosh w) cosh*u du (9) 
This is an interesting expression from a theoretical point of view, as it brings 
in factors which are adinittedly of the first importance in wave resistance : the 
area of the midship section and the depth of its centroid, or roughly the depth 
of the centre of buoyancy of the ship, the length of the form and its prismatic 
coefficient. But it will clearly exaggerate, in general, the interference between 
bow and stern systems ; and it is too simplified for practical purposes, except 
possibly for special types of model over a limited range of speed. 
General Sub-division of the Ship 
7. The total moment of the ship is distributed in a continuous source 
distribution over the surface of the ship: distributed in length, in depth, and 
in beam. The last of these is neglected in the usual theory and we leave it on 
one side meantime, noting the possibility of including it in further developments. 
Of the other two, the distribution in length is specially important. _ We now 
divide the ship by taking transverse sections at any required number of stations ; 
for simplicity at first we consider complete sections, leaving subdivision in 
depth till later. Let S,, S, be the areas of any two transverse sections, say 
in the fore part of the ship with S, > S,. The total source strength on the 
ship’s surface between these stations is 
(Sea = Si) OB 9 oo . (10) 
The ship being symmetrical with respect to the vertical longitudinal section. 
the centroid of the distribution lies on this median plane. Its depth is the 
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