  Collisions of galaxy clusters generate shocks and turbulence in the
intra-cluster medium (ICM). The presence of relativistic particles and magnetic
fields is inferred through the detection of extended synchrotron radio sources
such as haloes and relics and implies that merger shocks and turbulence are
capable of (re-)accelerating particles to relativistic energies. However, the
precise relationship between merger shocks, turbulence, and extended radio
emission is still unclear. Studies of the most simple binary cluster mergers
are important to help understand the particle acceleration in the ICM. Our main
aim is to study the properties of the extended radio emission and particle
acceleration mechanism(s) associated with the generation of relativistic
particles in the ICM. We measure the low-frequency radio emission from the
merging galaxy cluster Abell 2146 with LOFAR at 144 MHz. We characterize the
spectral properties of the radio emission by combining these data with data
from archival GMRT at 238 MHz and 612 MHz and VLA at 1.5 GHz. We observe
extended radio emission at 144 MHz behind the NW and SE shocks. Across the NW
extended source, the spectral index steepens from $-1.06\pm0.06$ to
$-1.29\pm0.09$ in the direction of the cluster centre. This spectral behaviour
suggests that a relic is associated with the NW upstream shock. The precise
nature of the SE extended emission is unclear. It may be a radio halo bounded
by a shock or a superposition of a relic and halo. At 144 MHz, we detect a
faint emission that was not seen with high-frequency observations, implying a
steep ($\alpha<-1.3$) spectrum nature of the bridge emission. Our results imply
that the extended radio emission in Abell 2146 is probably associated with
shocks and turbulence during cluster merger. The relativistic electrons in the
NW and SE may originate from fossil plasma and thermal electrons, respectively.
