The propagation of elastic plane waves in a 2D model of a weakly cohesive powder is numerically simulated by the discrete element method (DEM). Isotropic samples of disks interacting by elasticity, friction and cohesion in their contacts are first assembled in equilibrium states under isotropic pressure. Contact laws are linearized in elastic form, with fixed normal (K N) and tangential (K T) stiffness constants, to model the reponse to a sine-shaped impulse (main frequency ω) imparted to the sample boundary. The shape of this disturbance depends on the value of ω, relative to basic frequency ω * ≡ K N /m (m denoting the grain mass). Longitudinal and transverse waves triggered at low ω/ω * propagate with a coherent wavefront followed by an incoherent tail, with the classical velocity deduced from static moduli. Higher values of ω/ω * result in much smaller coherent signals and a strong localization of the energy near the source. Longitudinal waves are accompanied by disordered rotations, travelling with another velocity, induced at all times during propagation. Transverse waves contain both rotational and translational components. It is speculated that reduced Cosserat theories could be relevant for such materials on the continuum scale.