produced by Vibrating Bodies. 35 



In order to demonstrate the effects of the combination of 

 two vibratory movements on a body, we have only to produce 

 movements sufficiently slow to enable us to follow readily their 

 different phases. We shall easily obtain this result by the aid 

 of a lonnj vibrating rod, by striking this rod in a suitable man- 

 ner during its vibrations of totality, so as to produce a double 

 vibratory movement. This experiment is not without in- 

 terest. 



I shall not pursue further the analysis of these different 

 phaenomena, nor shall I speak of the curious appearances 

 presented by a vibrating string when left to itself. What I have 

 developed suffices to render perceptible to the eyes the expla- 

 nation of simultaneous harmonic sounds, as deduced from the 

 fundamental experiments of Pigot, Noble and Sauveur. 



Let us now examine the explanation of multiple sounds pro- 

 posed by M. Duhamel. In this examination, we shall con- 

 sider more specially vibrating strings, as offering the most 

 elegant and interesting example of multiple sounds. To 

 abridge and profit by the details on which we have already 

 entered, we will suppose that the string gives simultaneously 

 the fundamental tone and its octave. 



In the first part of his memoir, M. Duhamel establishes 

 this proposition : " When a body is made to vibrate by seve- 

 ral causes which separately would produce the simple sounds 

 which it can give, its surface generally divides itself into a cer- 

 tain finite number of parts, in each of which the vibrations 

 have unequal durations. These different durations have re- 

 lation to sounds corresponding to the different causes, and we 

 are in the same position as if we had several separate surfaces, 

 each having a particular movement of vibration." 



In the vibrating string which gives the fundamental tone and 

 its octave, the middle is the only point the vibrations of which 

 differ in duration from those of the other points ; I do not 

 find here two finite portions of string in each of which the vi- 

 brations have unequal durations. Perhaps it will be said that 

 one of the two finite portions is reduced to a point or to the 

 contour of the middle of the string, but then the proposition 

 would not be new with regard to strings. Still, however, this 

 interpretation does not appear to me admissible, for, accord- 

 ing to the proposition, we must be in the same case as if we 

 had several separate surfaces, each having a particular move- 

 ment of vibration ; thence the fundamental tone would be solely 

 attributable to the vibrations of the middle of the string, that 

 is to say, of a single point ; they would therefore be insensible 

 with relation to the vibrations which give the octave. It is 

 known that the fundamental tone may be very powerful, and 



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