sound in S ( q, ω ) as a function of the temperature. We note that ratio W$_{2}$/W$_{1}$ is still given by the expressions in (27) and (28). However, now the parameter ϵ$_{LP}$ is no longer small ( ≪ 1) and thus the expansion based on (24), which leads to (29) and (30), is no longer valid. We also explicitly note that W$_{2}$/W$_{1}$ is not equal to ϵ$_{LP}$ in either the weak or strong coupling limits. 

First and second sound in a dilute, weakly interacting Bose gas involve weakly coupled oscillations of the thermal cloud and condensate, respectively. Both modes contribute to the density response function. Consequently, both first and second sound are sensitive to density probes. This fact has been made use of in a recent experiment involving trapped atomic Bose gases by Meppelink et al . [4]. In this study, a density pulse was generated (as described in section II) and the velocity of the second sound pulse was measured above the hybridization point by tracking the propagation of the density pulse. The inset in figure 6 shows that the amplitude of second sound in S ( q, ω ) and in density pulse propagation experiments is twenty times as strong as first sound for all temperatures above the crossing point at T ∼ 0 . 1 T$_{c}$ . This result may explain the absence of a first sound pulse in the data of [4], which corresponded to n$_{B}$a 3 B ∼ 5 × 10 − $^{6}$. 

In figure 7, we plot first and second sound for a more strongly interacting Bose gas (i.e., with a larger value of g .) This situation is close to the unitarity Fermi gas results shown in figure 1 [36]. In particular, we see that the velocities of first and second sound never become degenerate (cross) and thus the hybridization shown in figure 6 does not occur. In this strongly interacting Bose superfluid, we also show the approximate velocities of u$_{1}$ and u$_{2}$ given by (25). These approximate values give a reasonable first order approximation to the full solutions of the first and second sound velocities, rather than the expression in (32) that applies to a weakly interacting Bose gas as shown in figure 6. The inset shows the ratio of W$_{2}$/W$_{1}$ vs. T , as computed from (27). In this case, the amplitude of second sound in S ( q, ω ) is only 2-3 times larger than first sound for T ≳ 0 . 5 T$_{c}$ . 

In figure 8, we compare the values of W$_{1}$/ ( W$_{1}$ + W$_{2}$ ) and W$_{2}$/ ( W$_{1}$ + W$_{2}$ ) for a weakly interacting Bose gas ( n$_{B}$a 3 B = 10 − $^{4}$) and a strongly interacting Bose gas ( n$_{B}$a 3 B = 0 . 1). One sees that the latter case is qualitatively similar to the result in a Fermi superfluid gas at unitarity given in figure 2. In contrast, in a weakly interacting Bose gas, for the temperature range above the hybridization point, the amplitude of second sound is much larger than first sound in both S ( q, ω ) and density pulse propagation experiments. 

