Context : 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 .

Aims : To understand the mechanism ( s ) that accelerate relativistic particles in the intracluster medium ( ICM ) , we investigate the spectral properties of large scale diffuse extended sources in the merging galaxy cluster 1RXS J0603.3+4214 .

Methods : 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 .

Results : 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 \pm 0.02 .
We do not find any evidence of spectral steepening below 8 GHz .
The subregions of the main Toothbrush also exhibit near-perfect power laws , implying a very regular combination of shock properties across the shock front .
Recent numerical simulations show an intriguing similar spectral index , suggesting 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 .
Moreover , also the integrated spectrum of the halo follows a power law from 150 MHz to 3 GHz with a spectral index of -1.16 \pm 0.04 .
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 .

Conclusions : None of the relic structures , the Toothbrush as a whole or its subregions or the other two fainter relics , show spectral shapes consistent with a single injection of relativistic electrons , such as at a shock , followed by synchrotron aging in a relatively homogeneous environment .
Inhomogeneities in some combination of Mach number , magnetic field strengths and projection effects dominate the observed spectral shapes .