106 Hon. J. W. Strutt on the Theory of Resonance. [Nov. 24, 



II. " On the Theory of Resonance." By the Hon. J. W. Strutt. 

 Communicated by W. Spottiswoode, F.R.S. Received July 2, 

 1870. 



(Abstract.) 



An attempt is here made to establish a general theory of a certain class 

 of resonators, including most of those which occur in practice. When a 

 mass of air or other gas is enclosed in a space bounded nearly all round by 

 rigid walls, but communicating with the external air by one or more pas- 

 sages, there are certain natural periods of vibration or resonant notes 

 whose determination is a matter of interest. If the dimension of the air- 

 space is small compared to the wave-length of the vibration, the dy- 

 namics of the motion is, in its general character, of remarkable simplicity. 

 It is for the most part under this limitation that the problem is considered 

 in the present paper. The formula determining the resonant note is 



a / c 



where n is the number of complete vibrations per second, a the velocity of 

 sound, and S the capacity of the air-space ; c is a quantity proved to be 

 identical with the measure of electric conductivity between the interior of 

 the vessel and the external space, on the supposition that the air is re- 

 placed by a uniform conducting mass of unit specific conducting-power, 

 and the sides of the vessel and passages by insulators. When there is 

 more than one passage the formula is still applicable according to the above 

 definition of c ; and when the passages are sufficiently far apart not to in- 

 terfere with each other, the resultant c is by the electrical law of parallel 

 circuits simply the sum of the separate values for each passage considered 

 by itself. When this condition is not satisfied the value of c, thus found 

 by mere addition, is too great. 



The question thus resolves itself into the determination of the conduc- 

 tivity (or the resistance which is its reciprocal) for different forms of 

 passages or openings. The case of openings, which are mere holes in the 

 sides of the vessel, has been already treated, although in a very different 

 way, by Helmholtz, who, in his celebrated paper on vibrations in open 

 pipes, compared his theory with the observations of Sondhauss and others 

 on the notes produced, when such resonators are made to speak by a 

 stream of air blown across the mouth. Sondhauss has also given an 

 empirical formula applicable when the connecting passages are of the form 

 of long cylindrical necks. These previous results are in agreement, as far 

 as they go, with the formula here investigated, and which is applicable 

 whatever may be the length of the neck. If L be the length and R the 



1 . . L +i R 



radiu« - or the electrical resistance = .. 



3 C 7rll Z 



