I was assigned a problem that stated pressure, p, and density, ρ, are related by the equation dp = -gρ dy. It said it follow the adiabatic equation p=p0(ρ/ρ0)^γ, where γ is not equal to 0, and p0 is pressure at earths surface and ρ0 is density at earths surface. It then wants me to show that
p = p0(1-((γ-1)/γ)*((ρ0gy)/p0))^(γ/(γ-1))
I've tried for an hour or more, and cannot figure out this problem. If anyone has any insight or even knows where to start, it would be greatly appreciated.
Integrating this gives:
The constant of integration can be found by the boundary condition that at y=0, the density [imath]\rho=\rho_0[/imath]. This gives:
The solution is now -after some algebra- the one given in the post as: