Page:On the influence of uneven temperature distribution on the propagation of sound.pdf/5

for each individual particle. Understanding $h_{0}$ to be value of $h$ for the moment $t_{0}$ and neglecting quantities of second order, we have from equations (2) and (4)

$h=h_{0}(1+q-\gamma s).$

Let us agree to denote by the symbol $D$, the total differentiation with respect to time of any function that characterises the state of each gas particle, i.e., let us define

${\frac {D}{Dt}}={\frac {\partial }{\partial t}}+u{\frac {\partial }{\partial x}}+v{\frac {\partial }{\partial y}}+w{\frac {\partial }{\partial z}}.$

The adiabatic nature of the oscillations can then be expressed by the equation

$h_{0}\left({\frac {Dq}{Dt}}-\gamma {\frac {Ds}{Dt}}\right)+(1+q-\gamma s){\frac {Dh_{0}}{Dt}}=0.$

In here, letting

$\mathrm {ln} h_{0}=f,$

and neglecting infinitesimal terms of higher order, we find

(5)${\frac {\partial q}{\partial t}}-\gamma {\frac {\partial s}{\partial t}}+u{\frac {\partial f}{\partial x}}+v{\frac {\partial f}{\partial y}}+w{\frac {\partial f}{\partial z}}=0.$

§ 2. Differential equations of small oscillations in unevenly heated air mass

Applying the basic formulas of hydrodynamics to the large fluctuations in the air mass and discarding very small terms of the second order, we have the equations

${\frac {\partial u}{\partial t}}=X-{\frac {1}{\rho }}{\frac {\partial p}{\partial x}},\,\,{\frac {\partial v}{\partial t}}=Y-{\frac {1}{\rho }}{\frac {\partial p}{\partial y}},\,\,{\frac {\partial w}{\partial t}}=Z-{\frac {1}{\rho }}{\frac {\partial p}{\partial z}}.$

Substituting here the expressions p and p from (2) and again discarding second-order terms, we have