Page:On the influence of uneven temperature distribution on the propagation of sound.pdf/11

${\frac {\partial ^{2}q}{\partial T^{2}}}+{\frac {1}{T}}{\frac {\partial q}{\partial T}}+{\frac {n^{2}}{\varepsilon ^{2}k\gamma }}{\frac {q}{T}}=0.$

Its integral can be expressed in terms of the Bessel functions.

Denoting by $A,B,A',B'$ the arbitrary constants and introducing the notation

$z={\frac {2n}{\varepsilon }}{\sqrt {\frac {T}{k\gamma }}},$

we have exactly,

$q=[AJ_{0}(z)+BY_{0}(z)]\cos nt+[A'J_{0}(z)+B'Y_{0}(z)]\sin nt.$

Hence, using equation (9), denoting by $C,D,C',D'$ some other constants, and based on the known relation

${\frac {dJ_{0}}{dz}}=-J_{1}(z),$

let us find the expression for the speed

$u=[CJ_{1}(z)+DY_{1}(z)]{\sqrt {T}}\cos nt+[C'J_{1}(z)+D'Y_{1}(z)]{\sqrt {T}}\sin nt.$