Page:On the influence of uneven temperature distribution on the propagation of sound.pdf/9

$f=Const.-\gamma \mathrm {ln} \,p_{0}=Const.+\gamma \mathrm {ln} \,T.$

As a result of this, equation (5) takes the following form for the case under consideration:

(10)${\frac {\partial q}{\partial t}}-\gamma {\frac {\partial s}{\partial t}}+{\frac {\gamma }{T}}\left(u{\frac {\partial T}{\partial x}}+v{\frac {\partial T}{\partial y}}+w{\frac {\partial T}{\partial z}}\right)=0.$

Equation (7) in this case will be

${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}+{\frac {\partial s}{\partial t}}-{\frac {1}{T}}\left(u{\frac {\partial T}{\partial x}}+v{\frac {\partial T}{\partial y}}+w{\frac {\partial T}{\partial z}}\right)=0$

or othrewise using equation (10),

(10)${\frac {\partial q}{\partial t}}+\gamma \left({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}\right)=0.$

The system of equations (9), (10) and (11) should serve to resolve small fluctuations in the air mass in the absence external forces. It is remarkable, however, that for any temperature distribution the functions $u,v,w,s$ can be excluded from equations (9), (10) and (11). In fact, by differentiating the first of the equations (9) by $x$, the second by $y$, the third by $z$, adding and taking into account equation (11), we find

(12)${\frac {\partial q^{2}}{\partial t^{2}}}=k\gamma \left[{\frac {\partial }{\partial x}}\left(T{\frac {\partial q}{\partial x}}\right)+{\frac {\partial }{\partial y}}\left(T{\frac {\partial q}{\partial y}}\right)+{\frac {\partial }{\partial z}}\left(T{\frac {\partial q}{\partial z}}\right)\right].$

Once the function $q$ is found as the integral of this equation, the functions $u,v,w$ can easily be determined from equation (9).

In the future, we will consider only those movements that consist of simple oscillations and have them the same period in all places of space occupied by the gas. For such movements, denoting by $n$ some constant, and by $q_{0},u_{0},v_{0},w_{0},\delta ,\delta ',\delta '',\delta '''$ some functions of coordinates, we will have