Page:On the influence of uneven temperature distribution on the propagation of sound.pdf/3

small oscillation times, I decide in this article to present a dreive differential equations for very small oscillations in an unvenely heated air.

§ 1. Assumptions about the properties and the equation of state of the gas

Let us imagine a mass of air that is at a certain moment $t_{0}$ at rest and in equilibrium and assume that at this moment, the temperature distribution, pressure and density of the gas are known. Determining the position of each spatial points with rectangular coordinates $x,y,z$ and denoting for the moment $t_{0}$ under consideration through $p_{0}$, $\rho _{0}$ and $T$ the pressure, density and absolute temperature of the gas in any element of it at a known point $A(x,y,z)$, we will consider $p_{0}$, $\rho _{0}$ and $T$ as given functional coordinates. If at the same time $X,Y,Z$ are the projection along the axes of the external forces, then from the known equations of hydrostatics, we have the equations

(1)${\frac {1}{\rho _{0}}}{\frac {\partial p_{0}}{\partial x}}=X,\,\,{\frac {1}{\rho _{0}}}{\frac {\partial p_{0}}{\partial y}}=Y,\,\,{\frac {1}{\rho _{0}}}{\frac {\partial p_{0}}{\partial z}}=Z.$

We will assume that none of the quantities

${\frac {\partial \rho _{0}}{\partial x}},\,\,{\frac {\partial \rho _{0}}{\partial y}},\,\,{\frac {\partial \rho _{0}}{\partial z}},\,\,\,\,{\frac {\partial p_{0}}{\partial x}},\,\,{\frac {\partial p_{0}}{\partial y}},\,\,{\frac {\partial p_{0}}{\partial z}},$

${\frac {1}{\rho _{0}}}{\frac {\partial \rho _{0}}{\partial x}},\,\,{\frac {1}{\rho _{0}}}{\frac {\partial \rho _{0}}{\partial y}},\,\,{\frac {1}{\rho _{0}}}{\frac {\partial \rho _{0}}{\partial z}},\,\,\,\,{\frac {1}{p_{0}}}{\frac {\partial p_{0}}{\partial x}},\,\,{\frac {1}{p_{0}}}{\frac {\partial p_{0}}{\partial y}},\,\,{\frac {1}{p_{0}}}{\frac {\partial p_{0}}{\partial z}},$

has an infinitely greater value at any point.

Let us now imagine that, as a result of some shaking, the mass of gas began to move, and that its particles make very small fluctuations around its previous equilibrium positions. Let us denote by $u,v,w$,