Since the definition of sound speed, ($c_{s}$), in each ion s$th$ species ($is$) expressed that:
\begin{align*} c_{s,is} = \sqrt{\frac{k(T_{e}+T_{is})}{m_{is}}}. \end{align*}
How can we determine sound speed for multi-component plasma when the multi-component plasmas containing with $\text{e}^{-},\, \text{p}^{+}$, and $\text{Ar}^{+}$, and entering in the sheath region? Should it be a linear combination of each ion species for ion sound speed?
The formal definition for the speed of sound is given as: $$ C_{s}^{2} = \frac{ \partial P }{ \partial \rho } \tag{0} $$ where $P$ is the total thermal pressure and $\rho$ is the total mass density. Note that velocity moments are additive, so if you have a multi-species plasma, the total pressure and total mass density are given by: $$ \begin{align} P & = \sum_{s} P_{s} \tag{1a} \\ \rho & = \left( \frac{ \sum_{s} m_{s} \ n_{s}^{2} }{ \sum_{s} n_{s} } \right) \tag{1b} \end{align} $$ where $n_{s}$ is the number density of species $s$ and $m_{s}$ is the mass of species $s$. Note that mass density is not a velocity moment, number density is, thus why mass density does not add linearly.
In the case of a sound wave, one assumes that during the longitudinal and compressional oscillation of the particles, the dynamics are said to be adiabatic so the pressure of any given species goes as: $$ P_{s} \propto \rho_{s}^{\gamma_{s}} \tag{2} $$ where $\gamma_{s}$ is the polytropic index of species $s$. In a weakly collisional plasma, each particle species can have a unique $\gamma_{s}$. If the system is collisionally dominated (i.e., collision frequencies are comparable to the cyclotron frequency and/or the plasma frequency), then the plasma will behave more like an ideal gas with a single $\gamma$.
Thus, the speed of sound will go to: $$ C_{s}^{2} = \frac{ \sum_{s} \gamma_{s} \ P_{s} }{ \rho } \tag{3} $$
If we further assume an ideal gas, we can say each pressure is roughly $n_{s} k_{B} T_{s}$, where $n_{s}$ is the number density of species $s$, $k_{B}$ is the Boltzmann constant, and $T_{s}$ is the temperature of species $s$. The the sound speed will take the form of: $$ C_{s}^{2} = \frac{ \sum_{s} \gamma_{s} \ n_{s} \ k_{B} \ T_{s} }{ \rho } \tag{4} $$
In a two population plasma with quasi-neutrality satisfied, note that $n_{e} = n_{i} = n_{o}$ such that the right-hand side of Equation 1b simplifies to $n_{o} \sum_{s} m_{s}$. If there are multiple ion species, then it remains in its less trivial form since $n_{i}$ would be the total ion density given by: $$ n_{i} = \sum_{s} Z_{s} \ n_{s} \tag{5} $$ where $Z_{s}$ is the charge state of species $s$. It is still true that $n_{e} = n_{i}$ but note that $n_{e} \neq n_{i,j}$ where $j$ refers to the ion subpopulation (i.e., protons or argon ions in your example).