64 DYNAMICAL THEOKY OF SOUND 



correction has no importance, the direct influence of the air 

 being quite insignificant; but the solution of (8) when k is 

 small is of some interest from the standpoint of wave-theory, 

 and may therefore find a place here. If the square of k be 

 neglected, the equation may be written 



This is of the same form as 22 (2), and therefore 



y = e~* kt f(ct-x) + e-* kt F(ct+a;) ....... (10) 



This represents two wave-systems travelling in opposite 

 directions with velocity c; but there is now a gradual diminu- 

 tion of amplitude in each case as time goes on, as is indicated 

 by the exponential factor. Again, since the functions are 

 arbitrary, we may replace f(ct x) and F(ct + x) by 



e W-*to f(ct- x ) and e* k(t+xlc) F(ct+x), 

 respectively, so that the solution may also be written 



y = e - lkxlc f(ct-x) + e* lixlc F(ct + x) ....... (11) 



This form is appropriate when a prescribed motion is maintained 

 at a given point of the string. Thus if the imposed condition 

 be that y = (f)(i) for x = 0, the waves propagated to the right 

 of the origin are given by 



(12) 



The exponential shews the decrease of amplitude as the waves 

 reach portions of the string further and further away from the 

 origin. 



24. Reflection. Periodic Motion of a Finite String. 

 If a point of the string, say the origin 0, be fixed, we must 

 have y at this point for all values of t. Hence, in 23 (1), 



f(ct) + F(ct) = 0, or F(z) = -f(z). 

 The solution therefore takes the form 



y=f(ct-x)-f(ct + x) ................ (1) 



As applied for example to the portion of the string which 

 lies to the left of 0, this indicates the superposition of a direct 



