Chapter 1.4, Problem 23P

The pressure $p$, and density, $\rho$, of the atmosphere at a height $y$ above the earth’s surface are related by

$dp=-g\rho dy$.

Assuming that $p$ and $\rho$ satisfy the adiabatic equation of state $p=p_o{\left(\frac{\rho }{{\rho }_o}\right)}^{\gamma }$, where $\gamma \ne 1$ is a constant and $p_o$ and ${\rho }_o$ denote the pressure and density at the earth’s surface, respectively, show that

$p=p_o{\left[\left(1-\frac{\gamma -1}{\gamma }\right)\left(\frac{{\rho }_{o}gy}{p_o}\right)\right]}^{\gamma /\left(\gamma -1\right)}$.