Is phase velocity a physical observable ?

There are two types of waves in physics – 

1. Physical waves travelling through a material medium (e.g. sound waves or waves on the surface of water) 

2. Probability waves which are just mathematical functions whose mode square gives the probability of finding a particle in a given region of space. (e.g. EM waves can be thought of as probability waves associated with photons).

From the very definition of probability waves, their phase is not a physically observable quantity as the only physical information is contained in their mode square. Also, even though the phase velocity of probability waves can have observable consequences (for example the bending of light as it goes from one medium to another) it, in general, can’t be associated with the velocity at which the physical information is flowing. 

On the other hand, for physical waves the phase velocity (or group velocity if the wave is a linear combination of several simple waves) is not only a physically observable quantity but (from its very definition) it is also the velocity of the physical information (or disturbance) travelling through the medium.

Also see : https://en.wikipedia.org/wiki/Group_velocity