In this hydrodynamic region, the related dynamic structure factor is given by S ( q , ω ) ∝ Im χ$_{nn}$ ( q , ω ) /ω . In studies of superfluid $^{4}$He, S ( q , ω ) is directly measured in Brillouin light scattering experiments [13, 14]. In dilute gases, density pulse propagation experiments ef- fectively measure S ( q , ω ) as noted in [10]. In addition, experiments using two-photon Bragg scattering have a cross-section proportional to Im χ$_{nn}$ ( q , ω ). While all these experiments probe the same density response function, Im χ$_{nn}$ ( q , ω ), the relative weight of the low fre- quency resonances are strongest in S ( q , ω ) because of the extra factor of 1 /ω noted above. This fact has significant implications for measuring second sound since the frequency of second sound becomes much smaller than first sound as T$_{c}$ is approached. We relate our analysis to classic discussions of the dynamic structure factor S ( q , ω ) in the two-fluid region of superfluid $^{4}$He in the critical region [13–17]. However, many new features arise when dealing with quantum gases. New aspects of our work include the use of the compressibility sum rule (sections III and IV) in understanding the relative weights of first and second sound in the density response function. We also discuss the role played by the Landau–Placzek ratio ϵ$_{LP}$ ≡ (¯ c$_{p}$ − ¯ c$_{v}$ ) / ¯ c$_{v}$ in determining these weights, with emphasis on new aspects that arise in superfluid gases. Here, ¯ c$_{p}$ and ¯ c$_{v}$ are the specific heats per unit mass at constant pressure and volume, respectively. We discuss the differences between S ( q , ω ) of a unitary Fermi gas and that of a Bose-condensed gas (sections IV and V), and present results for Bragg scattering with localized beams applied to the centre of a trapped unitary Fermi gas using the local density approximation (section VI). In B, we provide a detailed analysis of the superfluid and normal fluid velocity fields associated with first and second sound. In this section, we review some basic properties of the density response function χ$_{nn}$ ( q , ω ), the related dynamic structure factor S ( q , ω ), and the experiments that measure these quan- tities. The density response function involves the correlation between the density at two different points at different times: II. DENSITY RESPONSE FUNCTION χ$_{nn}$ ( r , t ) ≡ − iθ ( t ) 〈 [ˆ ρ ( r , t ) , ˆ ρ ( 0 , 0)] 〉 . (1) 4