298 
PROFESSOR W. J. MACQTTORN RANKINE ON THE 
and if we make and f 2 = f, this becomes Mr. Scott Russell’s rule for the least 
lengths of fore and after body suited to enable a ship to be driven economically at a 
given speed. 
It is well known that in water that is shallow, compared with the length of a wave, 
waves of a given period are retarded according to a certain law (see Airy on Tides and 
W aves). Hence the fact, which has often been observed, that a length which is sufficient 
for a given speed in deep water, becomes insufficient in shallow water — the waves of the 
first class becoming divergent, and the swell under the after body lagging behind, so as 
to make the stern of the vessel “ squat,” as it is called. 
For waves of the second class, the value of W is given by equation (72), putting for 
Z the value given by equation (68) — that is, the mean depth of immersion 13 -j-S. Hence 
we have 
W ! =v A5; (81) 
and this is probably unaltered in shallow water. The period of these waves is the same 
with that of the dipping, or vertical oscillation of the ship, whose value in deep water 
is 
T 2 = 
(82) 
Waves of the third class are observed to have, as theory indicates, a great angle of 
obliquity at and near the bow of the vessel, gradually diminishing as they travel to more 
distant masses of water where the virtual depth is greater. Beyond this general agree- 
ment, their precise laws are not yet known, for want of a sufficient number of precise 
observations. 
The general nature of the phenomena of wave-resistance, as indicated both by theory 
and by observation, are as follows. When either the speed of the vessel is so small, or 
w 
her dimensions so great, as to make the ratio - , of the speed of each set of waves to 
that of the vessel greater than or equal to unity, in other words, to make the ratio 
(l— ^ 2 ^ of the breadth of new wave raised per second to the speed of the ship 
nothing or imaginary, there is no wave-resistance, and the only resistances to be over- 
come in driving the ship at a uniform speed are that due to stiffness or viscosity, and 
that due to friction or “ skin-resistance.” The first of these increases simply as the 
speed ; and at the velocities usual in navigation, it becomes almost inappreciable when 
compared with the resistance due to friction. At very low speeds it is the principal 
resistance. Its laws have been fully investigated by Mr. Stokes. 
The resistance due to friction increases sensibly as the square of the speed. Some 
remarks on this kind of resistance will be added in the next section. 
W 
T 
So soon as the ratio 21 becomes less than unity for any set of waves, wave-resistance 
