50 Messrs. C. V. Raman and S. Appaswamaiyar on 



The two series into which the expansion has thus been split 

 up may be summed independently and then added together. 



It will suffice to trace the value of -r over times ranofna 



dt & ■ " 



from t = up to t = T, as the values subsequently repeat 



themselves. At any given point x on the string, the 



dv 

 velocity ~ remains at its initial value eo.v n from £ = up to 

 at rp ^_ x ^ u r 



the time t= — — j 110 ' It then suddenly changes to the 



value — &>(/ — ,r ), at which it remains up to the time 



t= -rr — -^. The velocity then suddenly regains its original 



value g># , which it retains up to the end o£ the complete 

 period T. From these values the configuration of the string 

 may be constructed, and is seen to consist of two straight 

 lines meeting sharply at a point which travels with uniform 



21 

 velocity -^ along two parabolic arcs situated one on either 



side of the string. As the edge or angle in the configuration 

 of the string passes over any point on it, the velocity at that 

 point suddenly alters by the quantity Ico. This sudden 

 alteration of velocity is evidently due to the resultant of the 

 forces acting on the element of the string over which the 

 edge passes being infinite in proportion to its mass, and in 

 this we may trace the propagation of the impulse originally 

 sent out by the impact on the bridge. 



Secondly, by the geometrical method : Since the impulse 

 set up by the sudden stoppage of the motion at the bridge 

 should evidently travel with the ordinary velocity of wave- 

 propagation, the character of the motion can be found from 

 purely geometrical considerations without the aid of the 

 Fourier Analysis, and this is really the more instructive 

 method of considering the problem. The solution of the 

 equation of wave-propagation on an infinite string not subject 

 to damping is 



y=f(x — at) + F(x + at). 

 Differentiating with respect to time, we have 



-lL = -af'(x — at) + aF'(x + at). 



If the two terms on the right-hand side of this equation are 

 periodic functions with wave-length equal to 2/, and are so 



