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 $-1.16pm0.02$. 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 $-1.16pm0.04$. 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.
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