WAVE PATTERNS AND WAVE RESISTANCE. 7 
and diverging waves for different depths or for different speeds. We see that the effect of 
increasing depth, at the same speed, is to diminish relatively the diverging waves. 
But these are perhaps details of purely theoretical interest, and we turn now to some 
cases of ship models. 
MopeELs or Great DRAUGHT. 
7. We consider first a model of great draught, of uniform horizontal cross-section 
throughout and with parabolic lines; this is a model which has been investigated by 
Mr. W. C. S. Wigley, working at the William Froude Laboratory. Fig. 5 shows the hori- 
zontal section. - 
Taking the origin O at the mid-point, the equation of the curve ACB is y = b(1 — 2°//’), 
+ 
MG, 5), 
the beam being 26 and the length 2/. It can be shown that, on the usual mathematical 
theory, the waves in the rear of the model approximate to 
86 IF 
t= | 
This might be regarded as a sine pattern with a somewhat complicated amplitude 
factor; but fortunately we can dissect it into simpler components, for it is identically 
equal to 
9) 4 
«1 cos (x 1 sec 0) — cos @'sin (x l sec @)} sin (w, y)d@ . (12) 
At 
poy 
2 
46 esr eo. 
— sin (« — 1, y) 4 ; sin (x + I, y) 
4 4 
cos 6 cos (x — 1, y) — cos 8 cos (x +1, yao (13) 
ai g [2 g [2 
Here the pattern is seen to be the combined result of superposing four simple patterns, 
two focussed at the bow and two at the stern. The first two are simple sine patterns, 
with constant amplitude factors at a given speed; they may, in fact, be attributed directly 
to the finite angle of the model at the bow A and at the stern B respectively. The other 
two terms in (13) are cosine patterns, with an amplitude factor cos @ in each case; although 
one is focussed at the bow and the other at the stern, it is more appropriate to regard 
these two terms together as representing the resultant effect of the curved sides ACB and 
ADB of the model. 
A matter of great interest is the mutual interference of these four patterns according to 
the speed, the extent to which it is possible to make the crests of one pattern coincide 
with the troughs of another and the speeds at which maximum effects of this kind occur; 
however, these points are better considered in connection with the corresponding wave 
resistance. 
383 
