Page:On the influence of uneven temperature distribution on the propagation of sound.pdf/4

the velocity of the particle along the axes of the coordinates passing, at a known time $t$, through the point $A(x,y,z).$ Pressure and density at the same point for the moment $t$ are denoted by $p$ and $\rho$.

We will also assume that the changes in gas densities occurring during oscillations in each individual points of the space are very small in relation to equilibrium density. A similar assumption for pressure is also assumed. As a result, letting

$s={\frac {\rho -\rho _{0}}{\rho _{0}}},\,\,\,q={\frac {p-p_{0}}{p_{0}}},$

or in other words,

(2)$\rho =\rho _{0}(1+s),\,\,\,p=p_{0}(1+q),$

we assume $s$ and $q$ to be small quantities of the first order, on par with the values $u,v,w$ and therefore, we agree to discard in the equations any term containing the product of any pair of the these five quantities.

In addition, we will accept, as is usually done when deriving the basic equations of the theory of sound, that the oscillating gas is subject to the laws of Mariotte and Gay-Lussac, and that each particle of gas retains all its warmth. Mariotte and Gay-Lussac law for the moment $t$ is expressed by the equation

(3)$p_{0}=k\rho _{0}T,$

in which $k$ is the known constant; for the adiabatic process accompanying the vibration of particles, we will use the known equation

(4)${\frac {p}{\rho ^{\gamma }}}=h,$

in which $\gamma$ represents the constant ratio of specific heats and $h$ is a constant quantity