52 THE APPROXIMATE CALCULATION OF WAVE RESISTANCE AT HIGH SPEED 
Calculations have been made for the standard model with length = 16 ft., 
beam, = 1-5 ft., draught = 1 ft. We note that k = g/v? = 1/f*L, where f is 
Froude’s number. The integral was evaluated by direct quadrature, and 
no attempt was made to attain any high degree of accuracy in the numerical 
values as the work is regarded mainly as an exploration of possibilities ; if 
necessary, more systematic methods of computation could be devised, but 
meantime it is hoped there are no errors serious enough to invalidate the general 
deductions. ' 
For a given value of f, the sines in (14) were calculated for values of wu in- 
creasing by 0-2, or in some cases by 0-1 ; it was not found necessary to go beyond 
u = 4, because of the decrease in the exponential factor. The integrand in (13) 
was then calculated for these values and graphed as a function of u, so that 
additional values could be inserted where needed ; finally the value of the 
integral was obtained by the usual rules for the area under the graph. In 
Table 1, the wave resistance in lb. calculated in this way from (13) and (14) is 
denoted by R,; the corresponding values R- have been obtained from the 
tables given by Wigley, using the complete theoretical integrals and omitting 
any correction for viscosity. 
TABLE 1 
i 265 -303 341 404 522 607 884 
Ra 1:93 3-48 2:44 8:0 20:7 23-7 27-1 
Re 1-08 3:25 2:43 Ve 20-4 23:6 26:1 
The agreement in the range -341 to -607 is surprisingly good ; the differences, 
it should be stated, are well within the limits of possible error in the present 
numerical computations. At lower speeds it was expected that the subdivision 
would be too coarse-grained, and the approximation gives unreliable results 
due to accidental coincidences between the various sine terms. One way of 
expressing it is that replacing the model by a small number of finite sources 
introduces interference effects between these sources taken in pairs and these 
become important at the lower speeds ; whereas in the actual model with 
its continuous lines these are smoothed out. To obtain the same result by 
calculation we should have to increase the subdivision in length. Suppose 
we take, in addition, a horizontal section at half-draught, then, considering 
any transverse section, 11/16 of the area is above and 5/6 is below this level ; 
further the centroid of the upper portion is at a depth 21d/88, and that of the 
lower portion is at a depth 27d/40 below the water plane. Hence all that is 
necessary, for this model, is to replace the exponential factor in (14) by 
—: kd cosh?u — 4ikd cosh*u 
1be a8 a re BU 
Various calculations have been made in this way, and also with different trans- 
verse sections. In general it may be said that numerical values are increased 
by greater subdivision in depth and diminished by additional transverse sections ; 
increasing both enables one to increase the range of speed for which effective 
agreement can be obtained. 
9. We have now to examine the extent to which the approximation reflects 
changes in form and whether it is sufficiently sensitive in that respect. In 
the paper already quoted, Wigley compares calculated and experimental values 
for a set of models defined by two parameters, the general equation of the 
forms being 
y= 6b (1 — 23/d?) 1 — x2/l?) + apx®/l2? + ayxt/I) . . . . (15) 
It is a simple matter to obtain general formule for the sectional areas and 
their differences for any scheme of subdivision, and for the positions of the 
respective centroids in accordance with (10) and (11), and they need not be 
reproduced here. We shall take three particular cases, for two of which experi- 
mental results are also given in the paper quoted. 
505 
