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PEOFESSOE W. J. MACQUOEN EANKINE ON THE 
§ 18. Probable Laws of Wave-resistance. — It has been proved by observation that a 
boating solid, such as a ship, is accompanied by waves, originating in the disturbances 
which it produces in the level of the water. None of those waves (at all events none 
whose energy is appreciable) travel faster than the floating solid. Some travel at the 
same speed, and some slower — each wave having its velocity in a direction normal to its 
crest regulated by its virtual depth, according to equation (72). 
Those waves maybe divided into three classes. The first class, whose properties were 
pointed out by Mr. Scott Russell about twenty-five years ago, have a speed, and there- 
fore a virtual depth, depending on the periodic time which elapses between the raising 
of a swell by the fore body and after body of the vessel respectively. 
In the second class the virtual depth is regulated by the mean virtual depth of the 
whole longitudinal disturbance (68)— that is, by the mean depth of immersion of the 
vessel ; the existence of these waves has been proved by observations of several actual 
vessels, some of which are described in a paper read to the British Association in 1868 
(see the Reports for that year, p. 194 ; also the Transactions of the Institution of Naval 
Architects, 1868, p. 275; and the ‘Engineer’ for the 28th August and 30th October, 
1868). 
The waves which have been found by observation most distinctly to follow this law, 
are a pair of diverging waves which closely follow the stem of the vessel. 
The third class of waves appear to depend on the several virtual depths of disturbance 
at various points in the neighbourhood of the vessel, especially at and near the bow. 
They diverge at various angles ; and travelling into water in which the virtual depth 
increases, they become accelerated, so that their ridges are gradually curved forward. 
The general theory of this class of waves has been stated in the papers already referred 
to in connexion with the second class ; but, so far as I know, they have not yet been 
subjected to exact observation, for which perfectly smooth Avater is necessary. 
When a wave accompanies a disturbing body whose speed is greater than that of the 
wave, the direction of advance of the wave, which is perpendicular to its ridge-line, 
adjusts itself so as to make with the direction of advance of the vessel an angle whose 
cosine is the ratio borne by the speed of the wave to the speed of the ship ; that is to 
say, let W be the speed of the wave, V that of the ship, a the angle of obliquity of the 
advance of the wave, then 
cos« = y (73) 
(see Transactions of the Institution of Naval Architects for 1864, vol. v. p. 321; also 
Watts, Raxkine, Napiee, and Baexes, on ‘ Shipbuilding,’ p. 79). The effect of the 
divergence of a wave is to disperse, to distant parts of the water, a certain quantity of 
energy which is never restored to the vessel, and thus to cause a kind of resistance which 
may be called wave-resistance. It has been suggested, as a probable law of the rate at 
which a diverging wave disperses energy, that this rate is proportional to the breadth of 
new wave raised in a second ; which breadth is equal to the speed of the vessel multiplied 
