and Combining the above results, the Landau–Placzek ratio at low temperatures is given by and hence, recalling that x = v $^{2}$/γv2 ¯ s and γ = v 2 ¯ $_{s}$/v 2 $_{T}$, This result is valid for both the superfluid unitary Fermi gas and superfluid $^{4}$He. Further simplifications are not possible without ab-initio or experimental information about c . At low temperatures, however, where ρ$_{s}$$_{0}$ ≃ ρ$_{0}$ and v$_{¯}$ s ≃ c (see figure 1), (A9) reduces to Equations (A6) and (A7) also give us the result that γ → 1 and x = v $^{2}$/γv2 ¯ s → 1 / 3 as T → 0. Finally, for a Fermi gas at unitarity, ( ρ/c )( ∂c/∂ρ )$_{T}$ = 1 / 3 and (A10) is further reduced to the result given in (B10). In superfluid $^{4}$He, the Gr¨ uneisen constant is almost a factor of 9 larger, ( ρ/c )( ∂c/∂ρ )$_{T}$ ≃ 2 . 84 [42]. As Landau [2] originally noted, when the thermal expansion coefficient can be neglected (i.e., ϵ$_{LP}$ = 0), one can show that second sound is a pure temperature oscillation, with no associated density fluctuations. This last feature follows from the fact that when ϵ$_{LP}$ = 0, second sound involves no mass current ( j ≡ ρ$_{n}$$_{0}$ v$_{n}$ + ρ$_{s}$$_{0}$ v$_{s}$ = 0) and thus the continuity equation requires that δρ = 0. Likewise, first sound is a pure density oscillation, and involves no fluctuations in the entropy per unit mass ¯ s . The non-dissipative two-fluid equations give the following equation for this quantity: This shows that for pure adiabatic motion ( δ ¯ s = δT = 0) such as first sound, the superfluid and normal fluid velocities are identical: v$_{s}$ = v$_{n}$ . Putting these results together, the velocity 26 ∂δ ¯ s ∂t = ¯ s$_{0}$ρ$_{s}$$_{0}$ ρ$_{0}$ ∇ · ( v$_{s}$ − v$_{n}$ ) . (B1) √ ρ$_{s}$$_{0}$ ρ$_{n}$$_{0}$ ϵ$_{LP}$x ≃ 1 3 [ 1 + 3 ρ c ( ∂c ∂ρ ) T ] . (A10) √ ρ$_{s}$$_{0}$ ρ$_{n}$$_{0}$ ϵ$_{LP}$x = 1 3 ρ$_{s}$$_{0}$ ρ$_{0}$ c 2 v 2 ¯ s [ 1 + 3 ρ c ( ∂c ∂ρ ) T ] . (A9) ϵ$_{LP}$ = 2 π $^{2}$T 4 135 ρc $^{3}$v 2 T [ 1 + 3 ρ c ( ∂c ∂ρ ) T ] 2 (A8) v 2 ≡ ¯ s 2 0 T ¯ c$_{v}$ ρ$_{s}$$_{0}$ ρ$_{n}$$_{0}$ = c 2 3 ρ$_{s}$$_{0}$ ρ$_{0}$ . (A7) Appendix B: Superfluid and normal fluid velocity fields in second sound