The equation is named after Lord Rayleigh, who introduced it in 1880. The Orr–Sommerfeld equation – introduced later, for the study of stability of parallel viscous flow – reduces to Rayleigh's equation when the viscosity is zero.
Rayleighs equation, together with appropriate boundary conditions, most often poses an eigenvalue problem. For given (real-valued) wavenumber  and mean flow velocity  the eigenvalues are the phase speeds  and the eigenfunctions are the associated streamfunction amplitudes  In general, the eigenvalues form a continuous spectrum. In certain cases there may further be a discrete spectrum of complex conjugate pairs of  Since the wavenumber  occurs only as a square  in Rayleighs equation, a solution (i.e.  and ) for wavenumber  is also a solution for the wavenumber 
Rayleighs equation only concerns two-dimensional perturbations to the flow. From Squires theorem it follows that the two-dimensional perturbations are less stable than three-dimensional perturbations.
If a real-valued phase speed  is in between the minimum and maximum of  the problem has so-called critical layers near  where  At the critical layers Rayleighs equation becomes singular. These were first being studied by Lord Kelvin, also in 1880. His solution gives rise to a so-called cats eye pattern of streamlines near the critical layer, when observed in a frame of reference moving with the phase speed