*2.2. Frequency-Pressure Dependence*

It is stated frequently in the early experimental papers [51–53] that the behavior of a gas depends on the ratio of frequency and pressure alone. From the point of view of kinetic theory, this scaling property is natural and follows from the Boltmann equation.

However, it is not that straightforward from a continuum point of view. Although this scaling is correct based on the experimental data [50], it requires constant transport coefficients, 1/*ρ* dependence in the new parameters (relaxation times and coupling coefficients) and ideal gas state equation. These assumptions work for the corresponding evaluations; it could not be valid for an extensive pressure (or density, accordingly) domain, e.g., from 10 Pa to 108 Pa as the apparent (It is worth to note again that the continuum models consist apparent (or measurable) coefficients in the constitutive equations.) viscosities and the thermal conductivity do depend on the pressure. Moreover, it is not clear how that scaling would appear using a more general equation of state.

Furthermore, it would worth investigating the scaling of relaxation times as the particle-wall interaction starts to dominate the process instead of the particle-particle collisions. In this kind of process, the characteristics of relaxation times are changed and thus changing the scaling properties of the equations. This could be the validity limit of the pure 1/*ρ* dependence. For example, in the experiments of Meyer and Sessler the lower pressure limit is around 0.2 Pa in which such particle-wall constribution can be especially important to consider.

This scaling property can be easily demonstrated for the classical Navier-Stokes-Fourier model by calculating the dispersion relation using the previous assumptions. Then, there no one will find terms containing the frequency *ω* and the pressure *p* separately, as follows.

Assuming the common *ei*(*ωt*−*kx*) plane wave solution of the system (7) with the usual wave number *k* and frequency *ω*,

$$\begin{aligned} \partial\_t \rho + \rho\_0 \partial\_x v &= 0, \\ \rho\_0 \partial\_t v + \partial\_x \Pi\_d + \partial\_x \Pi\_s + RT\_0 \partial\_x \rho + R\rho\_0 \partial\_x T &= 0, \\ \rho\_0 c\_V \partial\_t T + \partial\_x q + R\rho\_0 T\_0 \partial\_x v &= 0, \\ q + \lambda \partial\_x T &= 0, \\ \Pi\_d + \nu \partial\_x v &= 0, \\ \Pi\_s + \eta \partial\_x v &= 0, \end{aligned} \tag{7}$$

and omitting the detailed derivation, one obtains the following expression for phase velocity *vph* = *<sup>ω</sup> k* :

$$\begin{split} \upsilon\_{ph}^{2} &= \frac{cRT\rho^{2} + R^{2}T\rho^{2} + ic\eta\rho\omega + i\lambda\rho\omega + ic\upsilon\rho\omega}{2c\rho^{2}} + \\ &+ i \frac{\sqrt{\rho^{2}\left(-4c\lambda\omega(-iRT\rho + (\eta + \upsilon)\omega) + (iR^{2}T\rho - \lambda\omega + c(iRT\rho - (\eta + \upsilon)\omega))\right)^{2}}}{2c\rho^{2}}.\end{split} \tag{8}$$

Expanding all the terms within Equation (8), the *vph* = *vph*(*ω*/*p*) dependence becomes visible and all the experimental data can be evaluated without calculating the pressure (or the mass density) independently from the frequency. In the case of the generalized NSF model, the situation is the same, and the previous assumptions ensure such scaling. Here, the final remark is made from an experimental point of view: the frequency and the pressure are separately controlled and should be documented in this way. Then, in a continuum model, the pressure dependence in any parameter could be implemented without any problem, and the model would be free from assumptions that may be made unconsciously. Moreover, it could extend the validity region of this modeling approach.
