Water- Waves as Asymmetric Oscillations. 775 



and the equation of motion for any additional disturbance, f, 

 is therefore from (1) 



( % + c ll + 4afo?^ cos 2 jbi^ + 4«^ sin 2fo* .~-f- 



- SaPe 2 ^ cos 2fop ^ + SaP sin 2fo* ^ = 0. . . (2) 

 c/j/ ax 



Putting 



f = « cos fee 4- /5 sin fop, 



where a and /? are functions of 3/, we have from (2) 

 Now the pressure at the surface is proportional to 



~ 9 £ + ^-6ak cos 2fop^ +2a£ sin 2kx .^, 

 c «y ay ax 



which is — p (ale — <r) cos kx + <? (a& + <r) sin for, 



where c 2 =f (l + «r). 



This pressure, being nearly of free period, tends to damp the 

 disturbance if its phase is —1/4: that is, if 



p 2 [ak — 0-) = </ 2 (a& + o-) . 



I o- I must therefore be less than ak. Thus there is instability 

 for any value of a numerically less than ak, the standing 

 disturbance of wave-length 27r/{kQ +°")} being magnified 



in one phase — tan"" 1 * / -7—- ■> and diminished in the nume- 

 rically equal phase of opposite sign. 



Interpreting the result in terms of progressive waves in 

 still water, it is evident that if a periodic pressure variation 

 moves uniformly over the surface, the forced train of equal 

 wave-length constitutes an unstable state of motion if the 

 ratio of the wave-length to that of the free wave of equal 

 speed lies within a range 2ak about the value 1/2: for a small 

 disturbance will result in a series of waves of double wave- 

 length, which is continually magnified through the periodic 

 pressure until the amplitude is large compared with the 

 original motion. The process consists essentially in the 

 continual enlargement of an asymmetric oscillation of approxi- 

 mately free type by a direct force of frequency lying within 

 a range about the double frequency of free oscillation. 



For the purposes of experimental illustration it would be 

 simplest to take the case of a stream flowing over a corrugated 

 bed. 



