On the possibility of refraction of dust acoustic waves

We theoretically investigate conditions for the refraction of long-wavelength dust acoustic waves by arrays of periodic cylinders in a dusty plasma. This is based on a recent analysis of the refraction of shallow water waves by periodic cylinder arrays (Hu and Chan, Phys. Rev. Lett. 95 (2005), 154501). In the dusty plasma case, however, the boundary conditions involve the formation of voids around the cylinders. Possible experimental conditions are discussed. Dust acoustic waves (DAWs) are the very ultra low-frequency waves, where the inertia is provided by massive highly charged dust grains, while the restoring force comes from the pressures of Boltzmann distributed inertia less electrons and ions [1]. Recently, the diffraction of DAWs by a circular cylinder placed in a dusty plasma was studied experimentally by Kim et al. [2], who pointed out the similarity between the equations describing DAWs and those describing sound waves in a gas. That means, the dynamics of the long wavelength (in comparison with the dusty plasma Debye radius [3]) waves can be described by the same type of linearized continuity and momentum equations as sound waves [2]. For the DAWs, the governing equations are ∂n1 ∂t + n0∇ · u1 = 0, (1a) and ∂u1 ∂t + cd n0 ∇n1 = 0, (1b) where n1 is the small dust number density perturbation in the equilibrium dust number density n0, u1 is the dust fluid velocity, and the dust acoustic speed (for cold dust) is cd = ωpdλD , where ωpd = (4πZ en0/m) 1/2 is the dust plasma frequency, with Z and m being the charge state and the mass of a dust grain, respectively. The dusty plasma Debye radius λD is given by [3] 1 λD = 1 λDe + 1 λDi , where λDj = (Tj/4πnje ), with j = e, i denoting the electrons and ions, respectively. Assuming that all perturbed quantities vary with time as exp(−iωt), where ω is the 232 M. Rosenberg et al. frequency, we can Fourier transform (1a) and (1b) and combine the resultant equations to obtain