74 DYNAMICAL THEOBY OF SOUND 



the successive modes are, according to 25 (12), now of the 

 same order of magnitude. The unreal character of the pre- 

 ceding hypothesis betrays itself in this result ; but we may at 

 all events infer that in the case of a very brief impact the 

 higher harmonics are relatively much more in evidence than 

 in the former problem. 



In reality the impact, even in the case of a metallic 

 hammer, is far from instantaneous, the time of contact, though 

 very short as measured by ordinary standards, being at all 

 events comparable with the period of vibration of the string*. 

 The effect of an impulse of finite duration has been calculated 

 by Helmholtz, to whom most of the present theory is due, on 

 the supposition that the pressure begins at the instant t = 0, 

 and lasts for a time T, during which it rises from zero to a 

 maximum and falls to zero again, according to the law sin (TT/T). 

 A somewhat simpler result is obtained if we imagine the law 

 of pressure to be 



where /* represents the time-integral of the force from t = oo 

 to t + oo . This law, whose graphical representation has the 

 form of the curve in Fig. 14, p. 33, has the defect that there is 

 no definite instant of beginning or ending, but as the true law 

 is in any case unknown, it may serve for purposes of illustration. 

 The interval of time during which the force is sensible is 

 comparable with r, and can be made as narrow as we please 

 by diminishing T. The details of the calculation will more 

 conveniently find a place in the next chapter ( 38). The 

 result is 



(6) 



When T is infinitesimal this agrees with (3). In other cases 



the intensities of the higher harmonics vary as e ~ 8irCT ' , if we 

 omit the trigonometrical factor. 



Although the pressure is thus rendered less abrupt as 

 regards its variation with the time, it is still assumed to be 



* Kaufmann, Wied. Ann., vol. LIV. (1896). 



