content_image/344154.jpg content_image/344158.jpg
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and second sound are significantly different from those given in (B2). This is a reflection of the fact that first and second sound are not pure uncoupled density and temperature waves, respectively, which would be described by (B2) and correspond to ϵ$_{LP}$ = 0 (and ρ$_{n}$$_{0}$ ̸ = 0). As we have shown in [8], such in-phase ( v$_{s}$ = v$_{n}$ ) and out-of-phase zero current ( j = 0) solutions correspond to first and second sound velocities given by 

<img src='content_image/344158.jpg'>

As we show in figure 2 of [8], these sound velocities are in good agreement with a full calculation including the coupling associated with ϵ$_{LP}$ . In this Appendix and section IV, we have carried out a careful analysis based on expanding to first order in the parameter ϵ$_{LP}$x , which is small ( ≪ 1) at all temperatures [14, 15]. The results in figure 10 show that the 

