  Phonons are responsible for limiting both the electron mobility and the spin
relaxation time in solids and provide a mechanism for thermal transport. In
view of a possible transistor function as well as spintronics applications in
graphene nanoribbons, we present a theoretical study of acoustic phonons in
these nanostructures. Using a two-dimensional continuum model which takes into
account the monatomic thickness of graphene, we derive Hermitian wave equations
and infer phonon creation and annihilation operators. We elaborate on two types
of boundary configuration, which we believe can be realized in experiment: (i)
fixed and (ii) free boundaries. The former leads to a gapped phonon dispersion
relation, which is beneficial for high electron mobilites and long spin
lifetimes. The latter exhibits an ungapped dispersion and a finite sound
velocity of out-of-plane modes at the center of the Brillouin zone. In the
limit of negligible boundary effects, bulk-like behavior is restored. We also
discuss the deformation potential, which in some cases gives the dominant
contribution to the spin relaxation rate T_1^{-1}.
