The equation is derived by solving a linearized version of the Navier–Stokes equation for the perturbation velocity field
where  is the unperturbed or basic flow.  The perturbation velocity has the wave-like solution  (real part understood).   Using this knowledge, and the streamfunction representation for the flow, the following dimensional form of the Orr–Sommerfeld equation is obtained:
where  is the dynamic viscosity of the fluid,  is its density, and  is the potential or stream function.  In the case of zero viscosity (), the equation reduces to Rayleigh's equation.  The equation can be written in non-dimensional form by measuring velocities according to a scale set by some characteristic velocity , and by measuring lengths according to channel depth .  Then the equation takes the form
where
is the Reynolds number of the base flow.  The relevant boundary conditions are the no-slip boundary conditions at the channel top and bottom  and ,
: at  and  in the case where  is the potential function.
Or:
: at  and  in the case where  is the stream function.
The eigenvalue parameter of the problem is  and the eigenvector is .  If the imaginary part of the wave speed  is positive, then the base flow is unstable, and the small perturbation introduced to the system is amplified in time.