Page:On the influence of uneven temperature distribution on the propagation of sound.pdf/6

${\frac {\partial u}{\partial t}}=X-{\frac {1}{\rho _{0}}}{\frac {\partial p_{0}}{\partial x}}+X(s-q)-kT{\frac {\partial q}{\partial x}},$

${\frac {\partial v}{\partial t}}=Y-{\frac {1}{\rho _{0}}}{\frac {\partial p_{0}}{\partial y}}+Y(s-q)-kT{\frac {\partial q}{\partial y}},$

${\frac {\partial w}{\partial t}}=Z-{\frac {1}{\rho _{0}}}{\frac {\partial p_{0}}{\partial z}}+Z(s-q)-kT{\frac {\partial q}{\partial z}},$

which, following the relations (1) take the form:

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

(6)${\frac {\partial v}{\partial t}}=Y(s-q)-kT{\frac {\partial q}{\partial y}},$

(6)${\frac {\partial w}{\partial t}}=Z(s-q)-kT{\frac {\partial q}{\partial z}}.$

This must be accompanied by the condition of continuity, which, with the accepted degree of approximation, can be represented by the equation

(7)$u{\frac {\partial \mathrm {ln} \,\rho _{0}}{\partial x}}+v{\frac {\partial \mathrm {ln} \,\rho _{0}}{\partial y}}+w{\frac {\partial \mathrm {ln} \,\rho _{0}}{\partial z}}+{\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}+{\frac {\partial s}{\partial t}}=0.$

Equations (5), (6) and (7) serve to determine small oscillation of gas under pressure and any external forces and under the influence of any temperature distribution.

§ 3. On the influence of gravity on vertically propagating plane sound waves.

Let us consider the vertical propagation of a plane sound wave in uniformly heated air, taking in account of the effects of gravity. Directing the $x$ axis vertically up and denoting the gravitational acceleration by $g$, we will have for the case under consideration,