© = f£(0) sin {ko sec? 9(xcos + ysin‘) } dd 
vid 
+ J F(9) cos {ko sec? 0(x cos + ysin®) } dd, 
Tw 
2) 
where c is the velocity of the solid, and ko = g/c?. It was 
shown that the wave resistance is given by 
£(0)? + £(—0)? + F(0)? + F(—9)?} cos? 0.0. 
Es) 
I 
Be 
a 
— 
ic} 
to 
ofSsuls 
ont 
The wave pattern can be considered as made up of elemen- 
tary plane waves travelling in all directions. From our know- 
ledge of the ship wave pattern it appears that the transverse 
waves are made up of plane waves making angles with 0x 
ranging from zero to a certain angle B, while the remaining 
plane waves from B to 90° make up the diverging waves. The 
angle 6 is given by sin?B = 3, and is about 35°16’. With 
this in mind I suggested (I.N.A., 1934) that one might pos- 
sibly divide up the wave resistance integral similarly; thus 
the value of the integral in the range 0 to 6 would represent 
the part due to the transverse waves, and the part from f to 
90° that due to the diverging waves. Of course this, as it 
stands, is no more than a fairly plausible assumption. I have 
been examining the possibility of putting it on a better basis 
by a different analytical approach; however I leave that mean- 
time with the remark that I think it can be justified as a 
fairly reasonable assumption, which can be used to give some 
interesting results. Taking some simple cases, consider a 
sphere with its centre at a depth f; the total resistance is 
1 
= 
R=4nxgo0 ko? a® J sec® } e-2 Ky fsec* 0 do. 
0 
We see by inspection that for low speeds the greater part 
of the integral comes from the range 0 to B, while for high 
speeds the greater part comes from the range B to 90°; a 
direct calculation shows that at c/ Y(gf) = 2, the diverging 
waves account for about 80 per cent. of the total resistance. 
From another point of view, this illustrates the fact that 
diminishing draft increases the relative importance of the 
diverging waves, and vice versa. 
We may illustrate interference effects by taking a system 
of a source and sink each of numerical strength me, at a depth 
f, and at a distance { apart. The total resistance is given by 
R=32 20 m?ko?c? f {1 —cos(2 ko fsec 8) }e? Fn F8€C* sec? Hdd. 
0 
Consider the oscillating part of the integral due to the 
factor cos (2kolsec®) in the two parts of the range of inte- 
gration. Approximately, the last hump on the resistance curve 
in each case will be near a value of ko! given by 2kolsect 
=x. For the range 0 to B, sec 0 is not much different from 
unity; so the last hump on the transverse wave resistance 
curve will be near ko! = a/2, or a Froude number F = 0.56. 
On the other hand, on the range B to 90° we may take sec} 
as about 2 to give the maximum result; so the last hump on 
the diverging wave resistance curve will be near ko! = 1/4, 
or F = 0.78. The interference effects due to the superposition 
of two sets of transverse waves is a familiar idea; it is not 
so well-known that we may have interference of the diverging 
waves of two systems. 
In conclusion I may refer to some calculations which have 
been made for simple ship forms on this assumption for 
separating the contributions of the transverse and diverging 
waves, 
Wigley (I.N.A. 1942) has given numerical results for a 
simple parabolic model with two ratios of length to draft and 
up to a Froude number of 0.6. Inui (Intl. Conf. Ship Hydro 
1954) refers to some similar unpublished calculations by him- 
self, and gives an interesting diagram for water of finite 
depth: in which case there are only diverging waves above a 
critical speed. Finally, I would refer in particular to Lunde 
(S.N.A.M.E., 1951) who gives a diagram of curves of trans- 
verse and diverging wave resistance for a parabolic model. 
These curves are very interesting, bringing out clearly the 
humps and hollows on the two curves; for instance, the last 
hump on the transverse wave curve is at about F = 0.45, while 
that for the diverging waves is at some value greater than 
F = 0.6, outside the range shown on the diagram. It might be 
of interest to have calculations for other models, to show how 
the various elements of form affect the relative importance 
of the transverse and diverging waves. 
616 
