stylistic:style_transfer
Equations

Consider the sound wave propagating at speed v {\displaystyle v} through a pipe aligned with the x {\displaystyle x} axis and with a cross-sectional area of A {\displaystyle A}. In time interval d t {\displaystyle dt} it moves length d x = v d t {\displaystyle dx=v\,dt}. In steady state, the mass flow rate m ˙ = ρ v A {\displaystyle {\dot {m}}=\rho vA} must be the same at the two ends of the tube, therefore the mass flux j = ρ v {\displaystyle j=\rho v} is constant and v d ρ = − ρ d v {\displaystyle v\,d\rho =-\rho \,dv}. Per Newton's second law, the pressure-gradient force provides the acceleration: d v d t = − 1 ρ d P d x → d P = (− ρ d v) d x d t = (v d ρ) v → v 2 ≡ c 2 = d P d ρ {\displaystyle {\begin{aligned}{\frac {dv}{dt}}&=-{\frac {1}{\rho }}{\frac {dP}{dx}}\\\rightarrow dP&=(-\rho \,dv){\frac {dx}{dt}}=(v\,d\rho)v\\\rightarrow v^{2}&\equiv c^{2}={\frac {dP}{d\rho }}\end{aligned}}}