THE APPROXIMATE CALCULATION OF WAVE RESISTANCE AT HIGH SPEED 51 
depth of the centroid of the area between the corresponding traces of the sec- 
tions on the body plan of the ship. The longitudinal location of the centroid 
may be specified by a kind of local prismatic coefficient for the increase in 
volume in relation to the increase in cross-sectional area. It is readily seen, 
from the argument in the previous sections, that if x is the distance of the 
centroid ahead of the station 2, x,. the distance between the stations and V,, 
the volume between them, we have 
B= (he — Seta — Sd) s 92 6 My eos ee Meee CLL) 
The same construction holds if we take porontal sections in addivon and 
subdivide in depth as well as in length. We replace each subdivision so formed 
by a single source at a certain point ; the strength of the source and its location 
are easily derived from the usual data for the ship, for example the curves of 
sectional areas and volumes, the body plan and principal dimensions. We may 
exhibit this information in the form of a diagram representing the longitudinal 
section of the ship divided into compartments ; in each compartment is placed 
a number for the strength of the source at a given point in that compartment. 
The diagram gives quantitative information about the wave-making quality 
of the ship, and may be useful even if we do not carry out the subsequent calcu- 
lation for the wave resistance. It may be noted that we have tacitly assumed 
a normal form of ship, with the sources all positive on the fore part and all 
negative on the after part. For a bulbous bow, for instance, we should have 
a superposed source and sink combination which could be calculated by the 
same procedure. Of course, if we pursue this process far enough to arrive 
at very small subdivisions, we are back at the original problem of approximate 
evaluation of the complete theoretical integrals ; in particular, the precise 
location of the elementary source within its compartment would lose signi- 
ficance. It remains to be seen whether, with the particular method described 
above, a relatively small number of subdivisions will give any accuracy in calcu- 
lation. It is obvious in advance that high speeds will give conditions most 
suitable for comparison ; roughly speaking, the deciding factor is the relation 
between the distance between stations and the predominant wave length, and as 
we come down to lower speeds it will be necessary to increase the number 
of stations. 
Comparison with Experimental Models 
8. Before applying the method to models with non-mathematical lines, we 
test it by comparison with experimental models of simple form. We take 
first the parabolic form, the equation of whose surface has been given in (8). 
Extensive calculations have been made for this form from the usual complete 
integrals and tables of the various integrals have been given by Wigley in a 
recent paper.* We shall take, at first, complete transverse sections at 
x = 0, + 4/, + #/. The sections are all similar and their centroids, and there- 
fore those of their differences, are all at the same depth $d. The sectional area 
is given by S = M(1 — x°//*). Using the formule (10) and (11), we obtain 
sources of strengths, omitting the common factor vM//4z, 
4 it is ‘ 
Eye Sees YL NE a Tg dy eae 1 gciay ited OLD) 
respectively. The model being cymmetrical fre ana aft and alecting vis- 
cosity, there are corresponding negative sources at similar negative values of x. 
urs to (4) and (5), the cosine terms cancel out, and we are left with 
R= (ge/)kM*| F2COSH AIAN: WPS yin <ul WA ee Mae pes lanes CLES) 
T = 2e — 8kd cosh'u! .95 sin (-333 kl cosh u) 
+ +3125 sin (-633 kl cosh u) + -4375 sin (-881 kl cosh uw) } : . (14) 
*“Calculated and Measured Wave Resistance of a Series of Forms defined Algebraically, the Prismatic Coeffici- E 
ent and Angle of Entrance being Varied independently,’’ by W. G. S. Wigley, M.A. J.N.A. Vol. 84, p. 52, 
1942. 
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