249-250] EFFECT OF A PRESSURE-POINT. 455 



250. The investigation by Lord Rayleigh*, from which the 

 foregoing differs principally in the manner of treating the definite 

 integrals, was undertaken with a view to explaining more fully 

 some phenomena described by Scott Russell f and Lord Kelvin J. 



"When a small obstacle, such as a fishing line, is moved 

 forward slowly through still water, or (which of course comes to 

 the same thing) is held stationary in moving water, the surface is 

 covered with a beautiful wave-pattern, fixed relatively to the 

 obstacle. On the up-stream side the wave-length is short, and, as 

 Thomson has shewn, the force governing the vibrations is prin- 

 cipally cohesion. On the down-stream side the waves are longer, 

 and are governed principally by gravity. Both sets of waves move 

 with the same velocity relatively to the water; namely, that 

 required in order that they may maintain a fixed position relatively 

 to the obstacle. The same condition governs the velocity, and 

 therefore the wave-length, of those parts of the pattern where the 

 fronts are oblique to the direction of motion. If the angle between 

 this direction and the normal to the wave-front be called 0, the 

 velocity of propagation of the waves must be equal to v cos 6, 

 where v represents the velocity of the water relatively to the fixed 

 obstacle. 



" Thomson has shewn that, whatever the wave-length may be, 

 the velocity of propagation of waves on the surface of water cannot 

 be less than about 23 centimetres per second. The water must 

 run somewhat faster than this in order that the wave-pattern may 

 be formed. Even then the angle 6 is subject to a limit defined by 

 v cos 6 = 23, and the curved wave-front has a corresponding 

 asymptote. 



" The immersed portion of the obstacle disturbs the flow of the 

 liquid independently of the deformation of the surface, and renders 

 the problem in its original form one of great difficulty. We may 

 however, without altering the essence of the matter, suppose that 

 the disturbance is produced by the application to one point of the 

 surface of a slightly abnormal pressure, such as might be produced 

 by electrical attraction, or by the impact of a small jet of air. 



* I. c. ante p. 393. f " On Waves," Brit. Ass. Rep., 1844. 



J 1. c. ante p. 446. 



