402 Lord Rayleigh on the 



character of the dispersive medium — the relation of velocity 

 to wave-length — without enquiring farther as to its con- 

 stitution. For example, we find the resolving-power of a 

 prism to be given by 



dk d\ •••••• • w> 



in which A, denotes the wave-length in vacuo, T the "thickness" 

 of the prism, /n the refractive index, and d\ the smallest differ- 

 ence of wave-length that can be resolved. A comparison 

 with the corresponding formula for a grating shows that (1) 

 gives the number of waves (X) which travel in the prescribed 

 direction as the result of the action of the prism upon an 

 incident pulse. 



But, although reasoning on the above lines may be quite 

 conclusive, a desire is naturally felt for a better understanding 

 of the genesis of the sequence of waves, which seems often to 

 be regarded as paradoxical. Probably I have been less 

 sensible of this difficulty from my familiarity with the 

 analogous phenomena described by Scott Russel and Kelvin, 

 of which I have given a calculation*. "When a small 

 obstacle, such as 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 shown, the force governing the vibrations 

 is principally 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 wave-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 6, the velocity of pro- 

 pagation of the waves must be equal to v cos 6, where v 

 represents the velocity of the water relatively to the (fixed) 

 obstacle. 1 " In the laboratory the experiment may be made 

 upon water contained in a large sponge-bath and mounted 

 upon a revolving turn-table. The fishing-line is represented 

 by the impact of a small jet of wind. In this phenomenon 

 the action of a prism is somewhat closely imitated. Not only 

 are there sequences of waves, unrepresented (as would appear) 



* " The Form of Standing Waves on the Surface of Running Water," 

 Proc. Lond. Math. Soc. xv. p. 69 (1883) ; Scientific Papers, ii. p. 258. 



