プラトー・レイリー不安定性
∂ u ~ r ∂ t = − 1 ρ ∂ p ~ ∂ r ∂ u ~ z ∂ t = − 1 ρ ∂ p ~ ∂ z {\displaystyle {\begin{aligned}{\frac {\partial {\tilde {u}}_{r}}{\partial t}}&=-{\frac {1}{\rho }}{\frac {\partial {\tilde {p}}}{\partial r}}\\{\frac {\partial {\tilde {u}}_{z}}{\partial t}}&=-{\frac {1}{\rho }}{\frac {\partial {\tilde {p}}}{\partial z}}\end{aligned}}}
となる。また、線形化された連続の方程式は
∂ u ~ r ∂ t + u ~ r r + ∂ u ~ z ∂ z = 0 {\displaystyle {\frac {\partial {\tilde {u}}_{r}}{\partial t}}+{\frac {{\tilde {u}}_{r}}{r}}+{\frac {\partial {\tilde {u}}_{z}}{\partial z}}=0}
となる。ここで、速度や圧力の攪乱は表面攪乱の式と同じ形をとるとすると、速度と圧力の攪乱は
u ~ r = R ( r ) e ω t + i k z , u ~ z = Z ( r ) e ω t + i k z , p ~ = P ( r ) e ω t + i k z {\displaystyle {\tilde {u}}_{r}=R(r)e^{\omega t+ikz},{\tilde {u}}_{z}=Z(r)e^{\omega t+ikz},{\tilde {p}}=P(r)e^{\omega t+ikz}}