Squire transformation ($k^{\prime }=\sqrt{k^2+l^2},R^{\prime }=kR/k^{\prime },c^{\prime }=c$)

For positive streamwise wavenumber $k$, replace a three-dimensional Orr-Sommerfeld equation by a two-dimensional one with the displayed parameters. Both the transverse operator and the coefficient $1/(ikR)$ remain unchanged, preserving the eigenfunction and phase-speed eigenvalue. This proves the modal comparison in Squire's theorem.