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S.B. Karavashkin and O.N. Karavashkina

Finally, ''Fisher has rigorously solved the sound propagation problem in an 1D chain of interacting particles, considering a disordered chain of particles located along the axis x '' [16, p. 90]. Fisher has found his model on the state equation, basing it on the potential of forces acting between a pair of particles

where l is the average distance between a pair of nearest particles and T is the absolute temperature, and on the enthalpy equation

Here k is Boltzman constant, and

With it he has yielded an expression for the sound velocity squared as

“In the most general sense the expressions (46) and (47) solve with absolute accuracy the problem of the sound velocity calculation in an 1D liquid as a function of temperature and pressure at the given potential (x). As we see, already in 1D case the expression for sound velocity is very complicated” [16, p. 91]. So it can be rigorously solved only in two extreme cases: T 0 and T . In the first case we yield the results comparable with the solutions by Altenburg and Kudryavtcev.

At the same time, no one of mentioned theories takes into account the following factors studied in our work: the nonlinear affection of resistance on the parameters of vibration process and the features of vibration processes at the overcritical band. But if we compare the experimental data in Fig. 7 with the results presented in Fig. 3, we can easily see that the characteristics are comparable. And some below typical features enable this comparison.

First of all, we see in Fig. 7 that in the low-pressure domain the anomalous dispersion of velocity takes place, increasing as approaching to the critical regime (the point of minimal value of the velocity), the same as in Fig. 3. The curves inclination after the extreme point also is approximately equal.

Second, taking as the basic model a chain of the elastically connected molecules, we may suppose that the increasing constraint stiffness is related to the growing pressure at constant temperature. With it, one and the same external excitation frequency corresponds to a more high-frequency band in Fig. 3, due to the diminishing boundary frequency  ₀. Thus, we can present the plot of c(p) in Fig. 7 as a plot of c().

Third, it is known that the medium temperature increasing at constant pressure can be achieved only if diminishing the medium density. This will cause the increase of intermolecular distance a and fall of the constraint stiffness s. In its turn, according to (15), it causes the growth of boundary frequency and phase velocity in accordance with (27). Thus, the temperature growth will shift the boundary frequency ₀  to the high-frequency band.

And the last, with growing temperature, the elastic system resistance will naturally increase.

Summing up the above key points, we can explain qualitatively the curves behaviour in Fig. 7, using the regularities of the plot in Fig. 3.

Each of dependencies c(p) in Fig. 7 has its distinctive minimum at the boundary frequency ₀. When the temperature grows, this minimum shifts towards the pressure increase (higher frequency), and at the same time the sharpness of maximum is smoothed, because of increasing resistance of a line. With it the wave propagation velocity increases, because of increasing intermolecular distance a. The value of this distance has not an effect on the solution (15), but it effects directly on the phase velocity. And the constraint stiffness effects both on (15) and on the phase velocity through _r. Both these factors effect oppositely on the velocity. This is why we can expect the velocity to be negligibly growing with temperature. It will strongly depend both on the elastic system parameters and on the relationship of the influences in growing intermolecular distance and fall of constraints stiffness.

From the presented brief qualitative analysis we can see that if taking into consideration the effect of resistance on the line vibration processes, we can essentially improve the conventional models and promote them to be better corresponding to the experimental data.