content_image/351657.jpg content_image/351662.jpg content_image/351663.jpg
<img src='content_image/351657.jpg'>

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 

<img src='content_image/351662.jpg'>

This expression was first given by Vinen [30]. Using the expansion for u 2 1 in (24), one can reduce (28) to 

<img src='content_image/351663.jpg'>

