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DYNAMICAL THEOEY OF SOUND 



to a considerable amount of damping. If it were not so, 

 each resonator would go on vibrating, and the corresponding 

 sensation would persist, for an appreciable time after the 

 exciting cause had ceased. A similar interval of time would 

 elapse before the sensation reached its full intensity when the 

 cause first sets in. The effect would be that the sensations 

 due to a sufficiently rapid succession of distinct notes would 

 not be altogether detached from one another in point of time. 

 From considerations of this kind Helmholtz estimated that 

 the degree of damping must be such that the intensity (as 

 measured by the energy) of a free vibration would sink to 

 one-tenth of its initial value in about ten complete vibrations. 

 It follows, as explained in 13, that each resonator will 

 respond to a certain range of frequencies on each side of 

 the one which has maximum effect. It is assumed, further, 

 that the difference of pitch of adjacent resonators is so small 

 that the same simple-harmonic vibration will excite a whole 

 group, the intensity falling off from the centre on either side. 



This is illustrated by the annexed figure, repeated from 13, 

 which may now serve to exhibit the distribution of intensity 

 over a continuous series of resonators under the influence of 

 a given simple-harmonic vibration. The abscissa is p/n 1, 

 where p is now taken to represent the natural frequency of a 

 resonator, and n that of the imposed vibration. The horizontal 

 scale depends on the value of fi, or 1/rw, where r is the 



