GLYPH<s48>GLYPH<s46> GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s50> GLYPH<s48>GLYPH<s46> GLYPH<s52> GLYPH<s48>GLYPH<s46> GLYPH<s54> GLYPH<s48>GLYPH<s46> GLYPH<s56> GLYPH<s49>GLYPH<s46> GLYPH<s48> GLYPH<s45> GLYPH<s50> GLYPH<s45> GLYPH<s49> GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s84> GLYPH<s47> GLYPH<s84> GLYPH<s67> GLYPH<s118> GLYPH<s115> GLYPH<s47> GLYPH<s118> GLYPH<s110> GLYPH<s102>GLYPH<s105> GLYPH<s114> GLYPH<s115>GLYPH<s116> GLYPH<s32> GLYPH<s115>GLYPH<s111> GLYPH<s117>GLYPH<s110>GLYPH<s100> GLYPH<s115>GLYPH<s101> GLYPH<s99>GLYPH<s111> GLYPH<s110>GLYPH<s100>GLYPH<s32> GLYPH<s115>GLYPH<s111> GLYPH<s117>GLYPH<s110>GLYPH<s100> FIG. 10: Superfluid and normal fluid velocity fields associated with first and second sound in a unitary Fermi superfluid. The solid (black) line shows the ratio v (1) s /v (2) n of the superfluid and normal fluid velocity fields involved with first sound, (B8). The dashed (red) line shows the ratio ρ$_{s}$$_{0}$v (2) s /ρ$_{n}$$_{0}$v (2) n of the superfluid and normal fluid currents associated with second sound, (B9). If we set ϵ$_{LP}$ = 0, these ratios would be 1 and − 1, respectively, describing the velocity fields for pure density and temperature oscillations. 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}$$_{0}$ = 0). As we have shown in [8], such in-phase ( v$_{s}$ = v$_{n}$ ) and out-of-phase zero current ( j = 0) solutions correspond to first and second sound velocities given by As we show in figure 2 of [8], these sound velocities are in good agreement with a full calculation including the coupling associated with ϵ$_{LP}$ . In this Appendix and section IV, we have carried out a careful analysis based on expanding to first order in the parameter ϵ$_{LP}$x , which is small ( ≪ 1) at all temperatures [14, 15]. The results in figure 10 show that the u 2 1 = v 2 ¯ $_{s}$, u 2 = T ¯ s 2 0 ¯ c$_{v}$ ρ$_{s}$$_{0}$ ρ$_{n}$$_{0}$ ≡ v $^{2}$. (B12) 29