Page:On the influence of uneven temperature distribution on the propagation of sound.pdf/10

$u=u_{0}\cos(nt+\delta '),\,\,v=v_{0}\cos(nt+\delta ''),\,\,w=w_{0}\cos(nt+\delta '''),\,\,q=q_{0}\sin(nt+\delta ).$

In this case, equations (12) and (9) will be replaced by the following:

(13)${\frac {\partial }{\partial x}}\left(T{\frac {\partial q}{\partial x}}\right)+{\frac {\partial }{\partial y}}\left(T{\frac {\partial q}{\partial y}}\right)+{\frac {\partial }{\partial z}}\left(T{\frac {\partial q}{\partial z}}\right)+{\frac {n^{2}q}{k\gamma }}=0,$

(14)$u={\frac {kT}{n^{2}}}{\frac {\partial ^{2}q}{\partial x\partial t}},\,\,v={\frac {kT}{n^{2}}}{\frac {\partial ^{2}q}{\partial y\partial t}},\,\,w={\frac {kT}{n^{2}}}{\frac {\partial ^{2}q}{\partial z\partial t}}.$

§ 5. Study of some particular cases of sound propagation in an unevenly heated medium.

1. Let us first assume that the temperature of the medium decreases uniformly along a certain straight line, and that fluctuations in the air mass occur in the same direction. Let us consider, under these assumptions, the propagation of a plane wave corresponding to the sound of a certain tone.

Since in a quiescent atmosphere the temperature usually decreases with increasing elevation above the earth's surface, and since for heights that are not too significant this decrease can be taken to be proportional to the heights,obviously the particular case under consideration is of particular interest, allowing us to indicate circumstances, which can be accompanied in nature by vertical propagation of sound waves.

Taking the direction of the mentioned straight line as the $x$-axis and denoting by $\varepsilon$ a known constant, and by $T_{0}$ the temperature of the medium in the plane passing through origin, we will have

$T=T_{0}-\varepsilon x.$

Equation (13) is accepted for the case under consideration, which once you introduce a new variable $T$, will have the form