Bauhaus-Universität Weimar

On sound and atmospheric vibrations, with the mathematical elements of music
Airy, George Biddell
on sound: 
21. Algebraical treatment of Waves of Air tra¬ 
velling along a tube; and formation of the Partial 
Differential Equation, neglecting small quantities. 
The algebraical method of treating the problem will 
be as follows. Let x be the distance (measured parallel 
to the axis of the tube) of any particle from the origin 
of measure of x, in the quiescent state, that is, in the 
state in which the pressure of the air in every part 
of the tube is represented by D ; and at any time t 
let the particle be in a place advanced beyond its 
original place by the quantity X, so that its ordinate is 
now x + X; X is different for different particles at the 
same time, and is different for the same particle at 
different times, and therefore is a function of both x 
and t. Now consider the place of a particle whose 
primary distance from origin was x + h ; its present 
distance from origin is 
x+X+h + ~h + 
dx2 * 1.2 
+ &c. ; 
its present distance from the particle before men¬ 
tioned is 
hi 1 + 
dx\ æx ii 
dx2 ’ 1.2 
+ &c. ; 
the mass of air which, with density D, did occupy the 
length A, now with density A occupies the length 
11 + 
, d2X A2 , e 
+ ^.^ + &c.; 
dx) dx*


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