# Bauhaus-Universität Weimar

### Volltext On sound and atmospheric vibrations, with the mathematical elements of music

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.
3
```

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