Howard's semicircle theorem

For an unstable Rayleigh equation for inviscid shear flow mode, Howard's semicircle theorem bounds its complex phase speed by the range of the real base velocity $U(z)$. Assume a smooth parallel flow, real horizontal wavenumber $\alpha >0$, and impermeable channel walls or sufficient decay at infinity to remove boundary terms. Write the mode as $\varphi (z)e^{i\alpha (x-ct)}$, with $c=c_r+i{c}_i$ and $c_i>0$. If $U_{-}\le U\le U_{+}$, thenTo prove this, set $F=\varphi /(U-c)$, which is regular because $c_i\ne 0$. The Rayleigh equation for inviscid shear flow becomes $[(U-c{)}^2{F}^{\prime }{]}^{\prime }-{\alpha }^2(U-c{)}^{2}F=0$. Multiplying by $\overline{F}$ and integrating by parts givesNormalize $W$ to a probability weight. The imaginary and real parts give $⟨U⟩=c_r$ and $⟨U^2⟩=c_r^2+c_i^2$. Since $(U-U_{-})(U_{+}-U)\ge 0$, its average gives $⟨U^2⟩\le (U_{+}+U_{-})c_r-U_{+}{U}_{-}$. Substitution and completing the square prove the bound. It limits where growing modes can occur; being inside the semicircle does not establish an instability.

For a nonreal regular Rayleigh equation for inviscid shear flow mode vanishing at the walls, put $F=\hat{w}/(U-c)$. The equation becomes $[(U-c{)}^2{F}^{\prime }{]}^{\prime }-k^2(U-c{)}^{2}F=0$. Multiplication by $\overline{F}$ and integration by parts gives the displayed identity. With $Q=|F^{\prime }{|}^2+k^2|F{|}^2$, its imaginary and real parts give $\int UQ=c_r\int Q$ and $\int U^{2}Q=|c{|}^2\int Q$. Since $(U-U_{\min })(U_{\max }-U)\ge 0$, these imply Howard's semicircle theorem.