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<s53>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s83> GLYPH<s40> GLYPH<s113> GLYPH<s44> GLYPH<s41> GLYPH<s32>GLYPH<s91> GLYPH<s97> GLYPH<s114> GLYPH<s98> GLYPH<s46>GLYPH<s32>GLYPH<s117> GLYPH<s110> GLYPH<s105> GLYPH<s116> GLYPH<s115> GLYPH<s93> GLYPH<s47>GLYPH<s40> GLYPH<s113> GLYPH<s118> GLYPH<s70> GLYPH<s41> FIG. 4: The two-fluid dynamic structure factor given by (22) at unitarity. The plot shows the first and second sound resonances as a function of the frequency ω/qv$_{F}$ (where we take q = 0 . 1 k$_{F}$ and v$_{F}$ is the Fermi velocity) for a series of temperatures from T = 0 to T = T$_{c}$ , in steps of 0 . 05 T$_{c}$ (each temperature is offset). For clarity, the delta functions in (22) are plotted as Lorentzians with a width Δ = 0 . 01 qv$_{F}$ . first and second sound in S ( q, ω ) can be comparable in the temperature region T ≳ 0 . 8 T$_{c}$ . The relative weight of second sound is instead much smaller in Im χ$_{nn}$ (see figure 5) for reasons we have discussed. Figure 3 also shows that W$_{2}$/W$_{1}$ is quite well approximated by ϵ$_{LP}$ . It is useful to derive this result more analytically. Using (14) in (27), we obtain after some algebra This expression was first given by Vinen [30]. Using the expansion for u 2 1 in (24), one can reduce (28) to W$_{2}$ W$_{1}$ = 1 ϵ$_{LP}$ ( u 2 1 v 2 T − 1 ) 2 . (28) W$_{2}$ W$_{1}$ = ϵ$_{LP}$ (1 + γx + · · · ) $^{2}$. (29) 15