To find the solution to the linearized equation, a stream function was introduced by Lord Rayleigh for the perturbations of the flow velocity:
These new definitions of the stream function are used to rewrite the linearized barotropic vorticity equation.
Here,  is the second derivative of  with respect to  . To solve this equation for the stream function, a wave-like solution was proposed by Rayleigh which reads . The amplitude  may be  complex number,  is the wave number which is a real number and  is the phase velocity which may be complex as well. Inserting this proposed solution leads us to the equation which is known as Rayleigh's equation.
To get to this equation, in the last step it was used that   cant be zero and neither can the exponential. This means that the terms in the square brackets needs to be zero. The symbol  denotes the second derivative of the amplitude of the stream function,  with respect to  . This last equation that was derived, is known as Rayleigh's equation which is a linear ordinary differential equation. It is very difficult to explicitly solve this equation. It is therefore that Hsiao-Lan Kuo came up with a stability criterion for this problem without actually solving it.