  We propose a polymer quantization scheme to derive the effective propagation
of gravitational waves on a classical Friedmann-Lemaitre-Robertson-Walker
(FLRW) spacetime. These waves, which may originate from a high energy source,
are a consequence of the dynamics of the gravitational field in a linearized
low-energy regime. A novel method of deriving the effective Hamiltonian of the
system is applied to overcome the challenge of polymer quantizing a
time-dependent Hamiltonian. Using such a Hamiltonian, we derive the effective
equations of motion and show that (i) the form of the waves is modified, (ii)
the speed of the waves depends on their frequencies, and (iii) quantum effects
become more apparent as waves traverse longer distances.
