WAVE RESISTANCE THEORY AND APPLICATION 17 
cal problem which presents interesting difficulties, 
worthy of serious work for anyone who may care 
to undertake it.” 
It may be added that such work would give a 
better idea of what has been neglected in the pres- 
ent approximate theory, and might lead to a fresh 
approach to the problem of the ship with more 
usual values of the ratio of beam to length. 
The approximate solution for a slender ship 
form was given more than fifty years ago by 
Michell in a classical paper, which unfortunately 
was overlooked and forgotten for many years. 
Michell’s approach was different from that out- 
lined in the previous section. He considered a 
semi-infinite uniform stream of water with a free 
upper surface and bounded by a vertical plane 
parallel to the stream; and he solved the prob- 
lem of the motion due to a given distribution of 
normal velocity over this vertical plane. A ship 
of narrow beam placed in the stream was pictured 
as producing a normal velocity outwards on both 
sides of amount given approximately by the prod- 
uct of the stream velocity and the horizontal 
gradient of the level lines of the form; finally, this 
was treated as a given distribution of horizontal 
velocity outwards on the two sides of the longi- 
tudinal vertical section of the ship. Such a dis- 
continuity of normal velocity is equivalent, of 
course, to a corresponding distribution of sources 
and sinks over this vertical plane; and so we ar- 
rive at Michell’s results as a particular case of 
the source distributions we have considered in the 
previous section. In particular, we may quote 
for reference the well-known resistance integrals. 
With one-half of the submerged form given by 
y = f(x, 2) we have 
R = Hee f° (7 + J) sect 08; by = & 
0 
T ge 
with 
T+ iJ = Hui [sf e Koz sec? 9 + ikor sec @ dy dz 
Ox 
taken over the longitudinal vertical section of the 
ship. 
Although, as might be expected, this formula 
does not enable us to predict with certainty the 
resistance of a given model at a given speed, it 
proved to be near enough to the general run of 
the resistance-velocity curve to give much in- 
teresting qualitative information: in particular, 
in the changes produced by small variations in 
the form of the model and the general explanation 
of such changes. 
Fig. 5 shows the resistance curve A for the 
simple parabolic model given by y = 6(1 — x?//?) 
(1 — 2?/d?), for the case with the draft one- 
sixteenth of the length, showing the humps and 
hollows which are so much exaggerated at low 
speeds compared with experimental curves. It 
is interesting to recall that Kelvin ended his lec- 
ture on ‘Ship Waves’ (1887), in which he first 
described the ship-wave pattern, by making 
“with some diffidence’’ a practical suggestion. It 
was to the effect that since wave disturbance is 
so much a surface effect, it might be an advan- 
tage to put as much displacement as possible be- 
low the waterline, assuming no doubt that one 
would not then increase other resistance by a 
greater amount. It is, of course, well known that 
the form of the lower level lines has compara- 
tively little influence on the wave resistance as 
compared with the form of the level lines near 
the LWL; and one can see this confirmed by 
work on pressure distribution round the hull, such 
as that of Eggert. This point may be illustrated 
quite simply in Fig. 5 by inverting the parabolic 
model and putting the keel in the surface; thus 
the equation of the form is now 
y = 0(1 — x2/l?)(22/d — 2?/d?) 
Curve B shows the result. The operative fac- 
tor is the ratio of draft to wave length at each 
speed. As one would expect, the wave resistance 
in the second curve is negligible at low speeds, 
but ultimately would rise to equality with curve 
A; the difference is rather striking even when the 
wave length is several times the draft. 
I wish to refer now to some attempts which 
have been made to improve the theory and to ex- 
tend its range of application. It may be remarked 
that the Michell resistance integrals can be ap- 
plied to a much greater range of forms than was at 
567 
