  Radio relics are diffuse extended synchrotron sources that originate from
shock fronts induced by galaxy cluster mergers. The actual particle
acceleration mechanism at the shock fronts is still under debate. The galaxy
cluster 1RXS J0603.3+4214 hosts one of the most intriguing examples of radio
relics, known as the Toothbrush. We present new wideband radio continuum
observations made with uGMRT and VLA. Our new observations, in combination with
previously published data, allowed us to carry out a detailed high spatial
resolution spectral and curvature analysis of the known diffuse radio emission
sources, over a broad range of frequencies. The integrated spectrum of the
Toothbrush follows closely a power law over close to 2 decades in frequency,
with a spectral index of $-1.16\pm0.02$. We do not find any evidence of
spectral steepening below 8 GHz. The subregions of the Toothbrush show an
identical spectral slopes, suggesting that observed spectral index is rather
set by the distribution of Mach numbers which may have a similar shape at
different parts of the shock front. Indeed, numerical simulations show an
intriguing similar spectral index, indicating that the radio spectrum is
dominated by the average over the inhomogeneities within the shock, with most
of the emission coming from the tail of the Mach number distribution. In
contrast to the Toothbrush, the spectrum of the fainter relics show a high
frequency steepening. The integrated spectrum of the halo follows a power law
from 150 MHz to 3 GHz with a spectral index of $-1.16\pm0.04$. We do not find
any evidence for spectral curvature, not even in subareas of the halo. This
suggest a homogeneous acceleration throughout the cluster volume. Between the
brush region of the Toothbrush and the halo, the color-color analysis revealed
emission that was consistent with an overlap between the two different spectral
regions.
