Bauhaus-Universität Weimar

On sound and atmospheric vibrations, with the mathematical elements of music
Airy, George Biddell
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 


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