227-228] FORM OF THE WAVE-PROFILE. 401 



The figure on p. 400 shews, with a somewhat extravagant vertical scale, 

 the case where the band (AB) has a breadth K~\ or -159 of the length of a 

 standing wave. 



The circumstances in any such case might be realized approximately by 

 dipping the edge of a slightly inclined board into the surface of a stream, 

 except that the pressure on the wetted area of the board would not be uniform, 

 but would diminish from the central parts towards the edges. To secure 

 a uniform pressure, the board would have to be curved towards the edges, to 

 the shape of the portion of the wave-profile included between the points 

 A, B in the figure. 



If we impress on everything a velocity c parallel to x, we 

 get the case of a pressure-disturbance advancing with constant 

 velocity c over the surface of otherwise still water. In this form 

 of the problem it is not difficult to understand, in a general way, 

 the origin of the train of waves following the disturbance. It is 

 easily seen from the theory of forced oscillations referred to in Art. 

 165 that the only motion which can be maintained against small 

 dissipative forces will consist of a train of waves of the velocity 

 c, equal to that of the disturbance, and therefore of the wave- 

 length 27rc 2 /<7- And the theory of ' group- velocity ' explained in 

 Art. 221 shews that a train of waves cannot be maintained ahead 

 of the disturbance, since the supply of energy is insufficient. 



228. The main result of the preceding investigation is that a 

 line of pressure athwart a stream flowing with velocity c produces 

 a disturbance consisting of a train of waves, of length 27rc 2 /#, 

 lying on the down-stream side. To find the effect of a line of 

 pressure oblique to the stream, making (say) an angle JTT 6 with 

 its direction, we have only to replace the velocity of the stream 

 by its two components, c cos and c sin 0, perpendicular and 

 parallel to the line. If the former component existed alone, we 

 should have a train of waves of length 2?rc 2 /^ . cos 2 6, and the 

 superposition of the latter component does not affect the con- 

 figuration. Hence the waves are now shorter, in the ratio cos 2 6 : 1. 

 It appears also from Art. 227 (21) that, for the same integral 

 pressure, the amplitude is greater, varying as sec 2 6, but against 

 this must be set the increased dissipation to which the shorter 

 waves are subject*. 



To infer the effect of a pressure localized about a point of the 



* On the special hypothesis made above this is indicated by the factor e~ Ml<r in 

 Art. 227 (19), where ^=^c . sec 6. 



L. 26 



