= Solution
The fixed observer is the ray $V=0$. The physically selected zero-<group velocity> saddle defines the <absolute wavenumber> $k_0$ and <absolute frequency> $\omega_0$:
$$
\boxed{\omega'(k_0)=0,\qquad D(k_0,\omega_0)=0,\qquad\omega_0=\omega(k_0).}
$$
For a smooth frequency branch with $D_\omega\ne0$, the saddle condition is equivalently $D_k(k_0,\omega_0)=0$. The <absolute growth rate> is
$$
\boxed{\sigma(0)=\operatorname{Im}\omega_0.}
$$
Here $k_0$ and $\omega_0$ can be complex. A physical <saddle point> is selected by the localized impulse response and deformation of its <Fourier transform> contour; an arbitrary algebraic stationary point need not determine <absolute hydrodynamic instability>.
Back to article page
