 Both the observer clock and the particle clock agree on the phase.The frequency and wavenumber of the wave in the observer frame is given by

and

with a phase velocity

The phase of the wave in the observer frame is

The phase of the wave in the observer frame is the same as the phase in the particle frame, as the clock in the particle frame, and the clock in the observer frame.A wave-like picture of particles is thus consistent with special relativity.