As a first step, we linearize Eqs. (9)(12), to study smallamplitude harmonic waves of frequency and wavenumber . The linear dispersion relation for electronacoustic waves then reads:
We note the appearance of a normalized dependent screening factor (scaled Debye wavenumber) in the denominator, defined by
From Eq. (14), we see that the frequency , and hence also the phase speed, increases with higher temperature ratio . However, this is usually a small correction to the dominant first term on the righthand side of (15). For large wavelength values (small ), the phase speed is given by
(17) 
However, we should recall from kinetic theory[1,2,3,5] that both for very long wavelengths ( ) and very short wavelengths ( ), the wave is strongly damped, and thus these limits may be of academic interest only. The mode is weakly damped only for intermediate wavelength values, where its acoustic nature is not manifest. [1,2,3,5,29] Here, is the cool electron Debye length.

Restoring dimensions for a moment, the dispersion relation becomes
It appears appropriate to compare the above results with earlier results, in the linear regime. First of all, we note that Ref. [5] has adopted a kinetic description of electronacoustic waves in suprathermal plasmas. For this purpose, Eq. (18) may be cast in the form,
In Figure 1, we depict the dispersion curve of the electronacoustic mode, showing the effect of varying the values of the spectral index and the density ratio . It is confirmed numerically that the phase speed () increases weakly with a reduction in suprathermal particle excess, as the Maxwellian is approached, and that there is a significant reduction in phase speed as the plasma model changes from one in which the cool electrons dominate, to one which is dominated by the hot electron density.
Ashkbiz Danehkar