Page:On the influence of uneven temperature distribution on the propagation of sound.pdf/7

$v=0,\,\,w=0,\,\,X=-g,\,\,Y=0,\,\,Z=0,\,\,f=Const.-(\gamma -1)\mathrm {ln} \,p_{0}.$

Equations (5), (6) and (7) take the following form:

${\frac {\partial q}{\partial t}}-\gamma {\frac {\partial s}{\partial t}}+{\frac {g(\gamma -1)}{kT}}u=0,$

${\frac {\partial u}{\partial t}}=-g(s-q)-kT{\frac {\partial q}{\partial x}},$

${\frac {\partial u}{\partial x}}+{\frac {\partial s}{\partial t}}-{\frac {gu}{kT}}=0.$

Eliminating $s$ and $q$ from here, we find the equation

${\frac {\partial ^{2}u}{\partial t^{2}}}=k\gamma T{\frac {\partial ^{2}u}{\partial x^{2}}}-g\gamma {\frac {\partial u}{\partial x}}.$

This can be satisfied by the integral

(8)$u=Ae^{\varepsilon x}\cos(mx-nt-\delta ),$

in which $A$, $\varepsilon$, $m$, $n$ and $\delta$ are constants; substituting this integral into equation (8) and setting $k\gamma T=a^{2}$, we will find the relationship

$m^{2}={\frac {n^{2}}{a^{2}}}-{\frac {g^{2}\gamma ^{2}}{4a^{4}}},\,\,\varepsilon ={\frac {g\gamma }{2a^{2}}}.$

Expression (8) represents a wave propagating vertically at constant speed

$V={\frac {a}{\sqrt {1-{\frac {g^{2}\gamma ^{2}}{4a^{2}n^{2}}}}}}$

but with varying amplitude. The speed of the sound under these circumstances is increased under the influence of gravity, but it is easy to see that this increase is negligible. So for example, taking the meter as the unit of length,