Homework VI.1 (a)

assume \[T = T_m + bP^\kappa\]



then \[ {{d ln \theta }\over{dz}} = {{1\over{T_m + bP^\kappa}} {{dT}\over{dz}} } - {\kappa{dlnP}\over{dz}} \]

\[ = {{{\kappa b P^{\kappa -1}}\over{T_m+bP^\kappa}} {{dP}\over{dz}}} - {\kappa{dlnP}\over{dz}} \]

\[ = {{{\kappa b P^{\kappa}}\over{T_m + bP^\kappa}} {{dlnP}\over{dz}}} - {\kappa{dlnP}\over{dz}} \]

\[ = {{- T_m\kappa}\over{T_m + bP^\kappa}} {{dlnP}\over{dz}} \]

\[ = {{- T_m\kappa}\over{T}} {{dlnP}\over{dz}} , \]



where \[ {{d ln P }\over{ dz}} = {- g\over{RT}} \]



\[ => N^2 = {g {{dln\theta}\over{dz}}} = {{g^2T_m\kappa}\over{RT^2}} \]


let
\[ H = {{RT}\over g} \]


\[ => N^2 H^2 = RT_m\kappa\]