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XXV. On the Theory of Sound. By R. Moon, M.A., Honorary 

 Fellow of Queen's College, Cambridge*. 



THE transmission of sound through air confined within a 

 cylindrical tube, when the axis of the tube is the direction 

 of transmission, may be represented by a single partial differen- 

 tial equation, which in its rudest form is of a very simple cha- 

 racter. 



If, when the air is undisturbed, ce denote the distance from 

 the origin of an element of the thickness dx made by planes per- 

 pendicular to the axis, if y be the same distance at the time t 

 after disturbance,^ the pressure at the time / at the point whose 

 ordinate is?/, and D denotes the density of equilibrium, then, the 

 pressure at the time t on the one side of the element being p 



and on the other side being p + -f-dx, the moving force on the 



dp 

 element, estimated without reference to sign, will be -j- dx; and 



dividing this last by Ddx, the mass of the element, we get 



YT * -J- for the accelerating impressed force acting on the element 



at the time t. 



The corresponding effective force on the element will of course 



d 2 y 

 be -7-f ; hence equating these in accordance with D'Alembert's 

 az 



principle, account being now taken of signs, we get for the dif- 

 ferential equation of motion, 



U dt* + D dx [l) 



If we assume that Mariotte's law, which has been proved to 

 hold very approximately in the case of equilibrium, holds also 



when the air is in motion (observing that — = -^, and therefore 



dl~* P 



, where p denotes the density), (1) be- 



comes 



.... (2) 



dt 2 dx 



* dx* 



When the motions are small, (2) induces to 



0-SF--& < 3 > 



from which approximate equation the theoretical value of the 

 * Communicated by the Author. 



