Bauhaus-Universität Weimar

Titel:
On sound and atmospheric vibrations, with the mathematical elements of music
Person:
Airy, George Biddell
PURL:
https://digitalesammlungen.uni-weimar.de/viewer/image/lit38525/49/
WAVES OF AIR W A TUBE, 
33 
23. General Solution of the Equation. 
The general solution of the equation above (see the 
Author’s Elementary Treatise on Partial Differential 
Equations, Article 35, making a = 0), is 
X= (:n6a . t — x) + yjr {nOa .t + x), 
where the forms of the functions cj) and yjr are abso¬ 
lutely undetermined by the theory of the solution, and are 
to be determined so as to answer to the physical con¬ 
ditions which are to be satisfied. Thus the solution 
admits of infinite variety. If we suppose 
X—mx (nOa .t — x) + m x (nOa .tt + x), 
or 2mnÔa. t, we have simply a uniform current through 
the tube, with equal velocity for all the particles. If 
X = — m x (nOa. t — x) + m x (nda .t + x), 
or 2?nx, so that the original ordinate x is changed into 
x + X or x + 2mx, we have the air in a quiescent state, 
with the original intervals of its particles multiplied by 
1 + 2m, denoting a uniformly increased or diminished 
density throughout the tube, and implying that the 
ends of the tube are stopped. With second or higher 
powers, we should have movements produced by varia¬ 
ble densities. But, for our Theory of Sound, we shall 
most frequently treat each of the functions in a general 
form. 
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