STRINGS 73 



If the string be pulled aside through a small space , at 

 a distance a from the end a? = 0, and then be released, the 

 values of the coefficients in 25 (7) are found to be 



/ ,~ . srra . STHZ? STTC 



whence y , y, - x 2 - sin =- sin = cos -- *- . . . .(2) 

 7r 2 a(/ a) ,s* I I I 



The mode of calculation will be explained in the next chapter 

 (see 36). We notice that the harmonic of order 8 will be 

 altogether absent if sin (sirajl) 0, i.e. if the point of plucking 

 be at one of its nodes; this was remarked by Young (1841). 

 Thus if the string be plucked at the centre, all the harmonics 

 of even order will be absent. The formula (1) combined with 

 25 (12) shews that, apart from a trigonometrical factor which 

 lies between and 1, the intensities of the successive harmonics 

 will vary as 1/s 2 . The higher harmonics are therefore relatively 

 feebly represented in the actual vibration of the string. 



The effect of the impact of a hammer depends on the 

 manner and duration of the contact, and is more difficult to 

 estimate. The question is indeed, strictly, one of forced 

 vibrations ( 28); but in the somewhat fictitious case where 

 the duration is so small that the impact has ceased before 

 the disturbance (travelling with the velocity c) has had time 

 to spread over any appreciable fraction of the length, we 

 may treat the problem as one of free motion with given initial 

 velocity concentrated on a short length. The result is 



where a is the distance from the origin to the point struck, 

 and //. represents the total momentum communicated by the 

 impact. Hence 



2u _., 1 . Sira . STTX . Sirct 

 y = -- 2 - sin y- sin = sin = .......... (4) 



TTpC Sill 



As in the previous problem, the sih mode is absent if the 

 origin be at one of its nodes. Apart from the trigonometrical 

 factor on which this circumstance depends, the intensities of 



