167-168] WAVE- VELOCITY. 275 



168. To trace the effect of an arbitrary initial disturbance, 

 let us suppose that when = we have 



l/c = *(), i/A = *() .................. (11). 



The functions F',f which occur in (10) are then given by 



*"(*) = -If* (*) + *<* 



Hence if we draw the curves y = iji, y = i/s, where 



the form of the wave-profile at any subsequent instant t is found 

 by displacing these curves parallel to x, through spaces + ct, 

 respectively, and adding (algebraically) the ordinates. If, for 

 example, the original disturbance be confined to a length I of the 

 axis of x, then after a time l/2c it will have broken up into two 

 progressive waves of length I, travelling in opposite directions. 



In the particular case where in the initial state f = 0, and 

 therefore < (as) = 0, we have ^ = 773 ; the elevation in each of the 

 derived waves is then exactly half what it was, at corresponding 

 points, in the original disturbance. 



It appears from (11) and (12) that if the initial disturbance be 

 such that f = + ri/h . c, the motion will consist of a wave system 

 travelling in one direction only, since one or other of the functions 

 F' and /' is then zero. It is easy to trace the motion of a surface- 

 particle as a progressive wave of either kind passes it. Suppose, 

 for example, that 



t = F(x-cl) ........................ (14), 



and therefore ^ = crj/h .............................. (15). 



The particle is at rest until it is reached by the wave; it 

 then moves forward with a velocity proportional at each instant 

 to the elevation above the mean level, the velocity being in fact 

 less than the wave-velocity c, in the ratio of the surface-elevation 

 to the depth of the water. The total displacement at any time 

 is given by 



f = 



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