Serway & Jewett — Fluid Mechanics: Problem 49

Show that the variation of atmospheric pressure with altitude is given by \( P = P_0 e^{-\alpha y} \), where \( \alpha = \rho_0 g / P_0 \), \( P_0 \) is atmospheric pressure at some reference level \( y = 0 \), and \( \rho_0 \) is the atmospheric density at this level. Assume the decrease in atmospheric pressure over an infinitesimal change in altitude (so that the density is approximately uniform over the infinitesimal change) can be expressed from Equation 14.4 as \( dP = -\rho g \, dy \). Also assume the density of air is proportional to the pressure, which, as we will see in Chapter 18, is equivalent to assuming the temperature of the air is the same at all altitudes.