= Howard's semicircle theorem
{c}
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 $\phi(z)e^{i\alpha(x-ct)}$, with $c=c_r+ic_i$ and $c_i>0$. If $U_-\leq U\leq U_+$, then
$$
\left(c_r-\frac{U_++U_-}{2}\right)^2+c_i^2
\leq\left(\frac{U_+-U_-}{2}\right)^2.
$$
To prove this, set $F=\phi/(U-c)$, which is regular because $c_i\ne0$. The <Rayleigh equation for inviscid shear flow> becomes $[(U-c)^2F']'-\alpha^2(U-c)^2F=0$. Multiplying by $\overline F$ and integrating by parts gives
$$
\int(U-c)^2W\,dz=0,\qquad W=|F'|^2+\alpha^2|F|^2.
$$
Normalize $W$ to a probability weight. The imaginary and real parts give $\langle U\rangle=c_r$ and $\langle U^2\rangle=c_r^2+c_i^2$. Since $(U-U_-)(U_+-U)\geq0$, its average gives $\langle U^2\rangle\leq(U_++U_-)c_r-U_+U_-$. Substitution and completing the square prove the bound. It limits where growing modes can occur; being inside the semicircle does not establish an instability.
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