228] 



SHIP-WAVES. 



403 



The values of x and y are both stationary when sin 2 6 = ; this 

 gives a series of cusps lying on the straight lines 



y\x = 2- = tan- 1 19 28' *. 



We have here an explanation, at all events as to the main 

 features, of the peculiar system of waves which accompanies a ship 

 in sufficiently rapid motion through the water. To an observer on 

 board the problem is of course one of steady motion ; and although 

 the mode of disturbance is somewhat different, the action of the 

 bows of the ship may be roughly compared to that of a pressure- 

 point of the kind we have been considering. The preceding 

 figure accounts clearly for the two systems of transverse and 

 diverging waves which are in fact observed, and for the specially 

 conspicuous 'echelon' waves at the cusps, where these two systems 

 coalesce. These are well shewn in the annexed drawing f by 

 Mr. R. E. Froude of the waves produced by a model. 





A similar system of waves is generated at the stern of the 

 ship, which may roughly be regarded as a negative pressure-point. 



* Of. Sir W. Thomson, " On Ship Waves," Proc. Inst. Mech. Eng., Aug. 3, 1887, 

 Popular Lectures, t. iii., p. 482, where a similar drawing is given. The investigation 

 there referred to, based apparently on the theory of ' group -velocity,' has unfortu- 

 nately not been published. See also E. E. Froude, "On Ship Kesistance," Papers 

 of the Greenock Phil. Soc., Jan. 19, 1894. 



t Copied, by the kind permission of Mr Froude and the Council of the Institute 

 of Naval Architects, from a paper by the late W. Froude, " On the Effect on the 

 Wave-Making Kesistance of Ships of Length of Parallel Middle Body," Trans. Inst. 

 Nav. Arch,, t. xvii. (1877). 



262 



