Page:On the influence of uneven temperature distribution on the propagation of sound.pdf/8

second as a unit of time, counting the number of oscillations per second $N=200$, and putting approximately $\gamma =1.4$, $g=9.8$ $a=330$, $\pi =3.14$, we find

$V=a(1+\omega ),$

where

$\omega <{\frac {1}{10^{8}}}.$

Slightly more significant and influential is the $e^{\varepsilon x}$ multiplier, which is with the numerical values that we took, is approximately

$\varepsilon ={\frac {1}{15875}}.$

Along the upward propagation of a sound wave, the sound strength increases slightly. But this conclusion obviously relates to the proposition that all along the way sound waves and gravity do not change, temperature is everywhere is the same, and gas particles, apart from elasticity and gravity, do not experience any other influences.

§ 4. Effect of temperature on sound propagation in the absence of external forces.

Having thus become convinced of the very weak effect gravity on the propagation of sound, let us return to the general equations (6) and, discard terms in them that depend on external forces but retain what was made earlier by assuming that the temperature is a given function of coordinates. Then instead of equation (6) we will have the following

(9)${\frac {\partial u}{\partial t}}=-kT{\frac {\partial q}{\partial x}},\,\,{\frac {\partial v}{\partial t}}=-kT{\frac {\partial q}{\partial y}},\,\,{\frac {\partial w}{\partial t}}=-kT{\frac {\partial q}{\partial z}}.$

As for equation (5), it should be noted that in the absence of external forces, the pressure has the same value in all places of the gas and therefore