text: Consider a parallel shear flow  in the  direction, which varies only in the cross-flow direction  The stability of the flow is studied by adding small perturbations to the flow velocity  and  in the  and  directions, respectively. The flow is described using the incompressible Euler equations, which become after linearization – using velocity components  and 
  with  the partial derivative operator with respect to time, and similarly  and  with respect to  and  The pressure fluctuations  ensure that the continuity equation  is fulfilled. The fluid density is denoted as  and is a constant in the present analysis. The prime  denotes differentiation of  with respect to its argument 
  The flow oscillations  and  are described using a streamfunction  ensuring that the continuity equation is satisfied:
  Taking the - and -derivatives of the - and -momentum equation, and thereafter subtracting the two equations, the pressure  can be eliminated:
  which is essentially the vorticity transport equation,  being (minus) the vorticity.
  Next, sinusoidal fluctuations are considered:
  with  the complex-valued amplitude of the streamfunction oscillations, while  is the imaginary unit () and  denotes the real part of the expression between the brackets. Using this in the vorticity transport equation, Rayleigh's equation is obtained.
  The boundary conditions for flat impermeable walls follow from the fact that the streamfunction is a constant at them. So at impermeable walls the streamfunction oscillations are zero, i.e.  For unbounded flows the common boundary conditions are that