MATHEMATICAL THEORY OE STREAM-LINES. 
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begins to be felt, and shows its nature by increasing more rapidly than the square of 
the speed ; and its effects become more and more conspicuous as additional sets of waves 
come successively into operation as means of dispersing energy. 
When the speed of the disturbing body becomes so great that, for all or for most of the 
W 
sets of waves, the ratio y becomes a very small fraction, the whole, or nearly the whole 
of the energy of disturbance is dispersed and wasted, and wave-resistance becomes the 
principal, or it may be the only appreciable resistance. In this extreme case it is 
possible to make a theoretical estimate of the amount of that resistance, as follows. 
The whole energy of disturbance is expressed, in absolute units, by 
V 2 ^E d , 
a function of which values have been given in equations (47 a), (53), (60), &c. 
The total dispersion of that quantity of energy, and its reproduction by the disturbing- 
action of the solid, may be considered as taking place while the midship section M sweeps 
through a space equal to the displacement D of the solid — that is, while the solid 
advances through the distance ^ ; and hence the propelling force required to overcome 
wave-resistance will probably have the following value, in units of weight, 
^ p ,_V 2 ? E d M. 
’ 
and the resistance will again increase as the square of the velocity. 
The only solid of continuous figure on which experiments have been made suitable 
for comparison with this formula is the sphere. For that body, equation (60b) informs 
us, we have = L ; and-Fk =|-; therefore the extreme wave-resistance is 
£.R'=rVM; ( 83 a) 
that is to say, it is equal to the weight of a column of liquid of half the height due to the 
speed, on a base equal to the midship section, — a result which agrees very closely with ex- 
periment. 
Since a propelling instrument which acts by the reaction of the water, as a paddle, 
a screw, an oar, or a jet, drives the particles of water astern, it tends to diminish the 
height of the crest of a wave, and to increase the depth of a trough or hollow ; — in 
the former case diminishing, and in the latter increasing the energy of the wave, which 
partly goes to waste in the case of divergence ; and hence it follows that it is favourable 
to economy of power that such a propelling instrument should act on the crest, rather 
than on the hollow of a wave. This fact is well known in practice. 
The production of diverging waves is not prevented by totally submerging the dis- 
turbing body ; but those waves are of less height at the surface of the water, the more 
deeply the body is covered. The virtual depth, and consequently the speed, of the waves 
of the second and third classes increases, and their angle of divergence diminishes, with 
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