254 Waves by a Single Impulse in Water of any Depth. 



Example (1). — As a first example take deep-sea waves ; we 

 have 



/H=v/£ ( n >> 



which reduces (4), (3), and (10) to 



Y =i\/i (12) ' 



and *=i\/?' (13) ' 



«= -75 4(cosf + sinf ) = -Lcos(f - J) . (U) ; 



#v^ #A 4,2? Ax/ ^utf \4^ 4/ 



which is Cauchy and Poisson's result for places where x is 

 very great in comparison with the wave-length 2tt//jl ; that is 

 to say, for place and time such that gt 2 /4:X is very large. 

 Example (2). — -Waves in water of depth D, 



m=\/{i)=^} • • • < 15 >- 



Example (3). — Light in a dispersive medium. 

 Example (4). — Capillary gravitational waves, 



f m -\/(i +Tm ) (16) - 



Example (5). — Capillary waves, 



/W = >/(Tm) (17). 



Example (6). — Waves of flexure running along a uniform 

 elastic rod 



f( m )= m \/~ ....... (18), 



where B denotes the flexural rigidity and w the mass per unit 

 of length. 



These last three examples have been taken by Lord Rayleigh 

 as applications of his generalization of the theory of group- 

 velocity ; and he has pointed out, in his " Standing Waves in 

 Running Water'''' (London Mathematical Society, December 13, 

 1883), the important peculiarity of example (4) in respect to 

 the critical wave-length which gives minimum wave-velocity, 

 and therefore group-velocity equal to wave-velocity. The 

 working out of our present problem for this case, or any case 

 in which there are either minimums or maximums, or both 

 maximums and minimums, of wave-velocity, is particularly 

 interesting, but time does not permit its being included in the 

 present communication. 



For examples (5) and (6) the denominator of (10) is ima- 



