= Plane-wave dispersion of a material-density-damped condensate
{title2=$(\sigma+i\mathbf k\cdot\mathbf l)^2+\zeta l^2\sigma+l^2+l^4/4=0$}
With the <normal fluid> at rest, linearize about the unit-amplitude <plane wave> $e^{i(\mathbf k\cdot\mathbf x-k^2t/2)}$. For amplitude and phase perturbations proportional to $e^{i\mathbf l\cdot\mathbf x+\sigma t}$, the <continuity equation> gives $(\sigma+i\mathbf k\cdot\mathbf l)R=l^2\Theta/2$, and the phase equation gives $(\sigma+i\mathbf k\cdot\mathbf l)\Theta=-(2+l^2/2+2\zeta\sigma)R$. Their determinant is the displayed dispersion relation. The damping contains the laboratory exponent $\sigma$, not the Doppler-shifted exponent, because the reference flow is the normal component.
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