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A constant-amplitude plane wave has no density derivative and therefore no damping. The complete family iswith arbitrary constant phase $\chi$. The background in the question is the member $A=1$.

Use the exact amplitude–phase equations derived above with ${\mathbf{v}}_n=0$. To first order in the perturbation, $\rho =1+2ϵR$ and ${\mathbf{v}}_s=\mathbf{k}+ϵ\nabla \Theta$, soThese equations retain the quantum potential contribution because no long-wavelength truncation is assumed in this subpart. In particular damping contains $R_t$, not the advective derivative along the moving condensate: its reference is the stationary normal fluid.

For the proposed periodic perturbation, write $q=\mathbf{k}\cdot \mathbf{l}$ and $L=l^2$. The amplitude coefficients satisfyA nonzero perturbation therefore requires the plane-wave dispersion of a material-density-damped condensateor equivalently ${\sigma }^2+(2iq+\zeta l^2)\sigma +l^2+l^4/4-q^2=0$. At zero background flow the roots are $-\zeta l^2/2\pm \sqrt{{\zeta }^2{l}^4/4-l^2-l^4/4}$, whose real parts are negative for $l\ne 0$. As $\zeta \to 0$, $\sigma =-iq\pm i\sqrt{l^2+l^4/4}$ recovers the advected Bogoliubov spectrum with long-wavelength sound speed one in these units. This differs from question 1's time normalization, whose sound speed is $1/\sqrt{2}$.