Yes, it is possible to determine the phase velocity of electromagnetic (EM) waves traveling through a quarter-wave matching network given the input and load impedances and the frequency. However, it's important to note that the phase velocity of EM waves does not depend on the specific matching network configuration. The phase velocity is a fundamental property of the medium through which the waves are propagating and is independent of any matching network.
The phase velocity of an EM wave in a medium is given by the formula:
v = c / sqrt(εrμr)
where: v is the phase velocity, c is the speed of light in a vacuum (~3 x 10^8 meters per second), εr is the relative permittivity of the medium, and μr is the relative permeability of the medium.
In most practical situations, the relative permittivity and relative permeability of the medium are assumed to be approximately equal to 1, unless you're dealing with specific materials or situations that require consideration of these factors.
Therefore, if you know the frequency of the EM wave and assume the medium has a relative permittivity and permeability of 1, the phase velocity can be approximated as the speed of light in a vacuum (c) for most practical purposes.