  We investigate collective effects in the strong pinning model of disordered
charge and spin density waves (CDWs and SDWs) in connection with heat
relaxation experiments. We discuss the classical and quantum limits that
contribute to two distinct contribution to the specific heat (a $C_v \sim
T^{-2}$ contribution and a $C_v \sim T^{\alpha}$ contribution respectively),
with two different types of disorder (strong pinning versus substitutional
impurities). From the calculation of the two level system energy splitting
distribution in the classical limit we find no slow relaxation in the
commensurate case and a broad spectrum of relaxation times in the
incommensurate case. In the commensurate case quantum effects restore a non
vanishing energy relaxation, and generate stronger disorder effects in
incommensurate systems. For substitutional disorder we obtain Friedel
oscillations of bound states close to the Fermi energy. With negligible
interchain couplings this explains the power-law specific heat $C_v \sim
T^{\alpha}$ observed in experiments on CDWs and SDWs combined to the power-law
susceptibility $\chi(T)\sim T^{-1+\alpha}$ observed in the CDW o-TaS$_3$.
