to a Travelling Disturbance. 389 



according as x is positive or negative. Hence, for x>0, 



„= ___>) e -pz /&■-«*+ £«■ ri5 N 



and, for x<0, 



4* M Bt i(K-tP)x+lin 



- _______ /?px p 4 



^--vw^mjdkY - e 4 • • • (16) 



The wave-system now extends on both sides the origin. 

 For a moderate range of x we may put /3=0 in the 

 exponentials; thus 



77-4- <H*) /*+;»> ( 17) 



If dTJ/dk is negative we must take the lower sign in (13). 

 The result is, with the simplification referred to, 



' =± 7(=ppj' «..-.. (18) 



Here, as in (17), the upper or lower sign is to be taken 

 according as x^Jd. 



3. The formula (12) may be used to reproduce a number 

 of known results. Thus, in the case of an impulsive pressure 

 cos kx on the surface of deep water, the consequent surface- 

 elevation is 



k 



v= — — sin at cos kx (19) 



pa 



To conform to (1) we must put 



#*).= £* (20) 



and take the real part. Hence, for the train of waves 

 following a unit concentrated pressure, we find 



9 



^=--^.sin^, (21) 



which is the known result. We have here made use of the 

 relations a = /cc, \J=^e. 



Again, in the case of water of finite depth A, the result 

 of a surface-impulse cos kx is easily found to be 



7i= tanh kh sin at cos kx. -. . . (22) 



pa 



We therefore write 



(/)(£) = ^ tanh kh (23) 



