  Wolf-Rayet stars are known to eject winds. Thus, when a Wolf-Rayet star
explodes as a supernova, a fast, $>30,000$ km/s, shock is expected to be driven
through a wind. We study the signal expected from a fast supernova shock
propagating through an optically thick wind, and find that the electrons behind
the shock driven into the wind are cooled efficiently, by inverse Compton over
soft photons that were deposited by the radiation mediated shock that crossed
the star. Therefore, the bolometric luminosity is comparable to the kinetic
energy flux through the shock, and the spectrum is found to be a power-law,
which slope and frequency range depend on the number flux of soft photons
available for cooling. Wolf-Rayet supernovae that explode through a thick wind
have a high flux of soft photons, producing a flat spectrum, $\nu
F_{\nu}=Const$, in the X-ray range $0.1\lesssim T\lesssim 50$ keV. As the shock
expands into an optically thin wind, the soft photons are no longer able to
cool the shock that plows through the wind, and the bulk of the emission takes
the form of a standard core-collapse supernova (without a wind). However, a
small fraction of the soft photons is upscattered by the shocked wind and
produces a transient unique X-ray signature.
