Figure 3 shows how the sound velocities change with temperature at saturated vapor pressure. This graph is of fundamental importance in this lab and it will be attempted to be recreated with the data.
For first sound, it can be seen from figure 3 that there is a slow decrease in \(C_1\) from T=0 onward except at the \(\lambda\) point where there is a cusp due to theromodynamic behavior close to a transition temperature [1]. First sound also experiences attenuation peaks at \(T_\lambda\) and near 1 K.
The second sound mode is special to the superfluid state of helium 4 in that its a propagating temperature and entropy wave, different from normal liquids where waves are diffuse. Here the normal fluid and superfluid move in opposite directions so that \(J=0\) so \(v_s=-(\frac{\rho_n}{\rho_s})v_n\) Second sound can be detected using a transducer with small pores in it where only the superfluid component, having zero viscosity, can move through the pores. Again referencing figure 3, second sound starts with dashed lines that are not observed in practice. Above 1.2 K, \(c_2\) has a max at about 20.4 m/s near 1.65 K and then falls to zero at \(T_{\lambda}\).
Fourth sound is a pressure-density wave that propagates in helium contained in a porous superleak material [1]. The pores of the material in question must be small enough so the normal fluid cant get through and only the superfluid component can be used for wave motion. The velocity of fourth sound for an ideal superleak is \[c_4^2=\frac{\rho_s}{\rho}c_1^2\] neglecting a \(c_2^2\) term. Again looking at figure 3, we see that fouth sound drops to zero at \(T_{\lambda}\). This is what will be looked for in the data when the temperature is increasing.