Plane-wave dispersion of a material-density-damped condensate ($(\sigma +i\mathbf{k}\cdot \mathbf{l}{)}^2+\zeta l^2\sigma +l^2+l^4/4=0$)

With the normal fluid at rest, linearize about the unit-amplitude plane wave $e^{i(\mathbf{k}\cdot \mathbf{x}-k^{2}t/2)}$. 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.