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At high temperatures, where x = c 2 $_{2}$/c 2 1 ≪ 1 (see figure 1), we have 

<img src='content_image/333313.jpg'>

<img src='content_image/333319.jpg'>

This result was first obtained almost 40 years ago [11, 14, 15, 31] in the context of Brillouin light scattering [13] in superfluid $^{4}$He near the critical region close to T$_{c}$ . For T ≲ 0 . 4 T$_{c}$ , one can see from figure 1 that we can no longer assume x ≪ 1 and hence expand (29). 

As noted above, the result in (30) was obtained in the classic literature on superfluid $^{4}$He. The Landau-Placzek ratio ϵ$_{LP}$ is extremely small (10 − 2 − 10 − $^{3}$) in superfluid $^{4}$He and thus second sound has generally a very small amplitude in the dynamic structure factor. However, under pressure and close to T$_{c}$ , the magnitude of ϵ$_{LP}$ in liquid $^{4}$He can be signif- icantly increased (to about ϵ$_{LP}$ ∼ 0 . 2) such that a second sound doublet in S ( q, ω ) can be measured using Brillouin light scattering techniques [13]. Such studies [16, 17] have verified the correctness of W$_{2}$/W$_{1}$ = ϵ$_{LP}$ in liquid $^{4}$He at a pressure of P = 25 atmospheres in the 

