Rayleigh's equation (fluid dynamics)

In fluid dynamics, Rayleigh's equation or Rayleigh stability equation is a linear ordinary differential equation to study the hydrodynamic stability of a parallel, incompressible and inviscid shear flow. The equation is:

with ${\displaystyle U(z)}$ the flow velocity of the steady base flow whose stability is to be studied and ${\displaystyle z}$ is the cross-stream direction (i.e. perpendicular to the flow direction). Further ${\displaystyle \varphi (z)}$ is the complex valued amplitude of the infinitesimal streamfunction perturbations applied to the base flow, ${\displaystyle k}$ is the wavenumber of the perturbations and ${\displaystyle c}$ is the phase speed with which the perturbations propagate in the flow direction. The prime denotes differentiation with respect to ${\displaystyle z.}$

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