Propagation and kinetic roughening of wave fronts in disordered lattices

The dynamics of a wave front propagating in diluted square lattices of elastic beams is analyzed. We concentrate on the propagation of the first maximum of a semi-infinite wave train. Two different limits are found for the velocity depending on the bending stiffness of the beams. If it vanishes, a one-dimensional chain model is derived for the velocity and the amplitude is found to decrease exponentially. The first maximum is localized and the average width of the wave front is always finite. For very stiff beams an effective-medium model gives the correct velocity and the amplitude of the first maximum decays according to a power law. No localization of the first maximum is observed in the simulations. In this limit scaling arguments based on Huygen’s principle suggest a growth exponent of $1/2,$ and a roughness exponent of $2/3.$ The growth exponent fits the simulation data well, but a considerably lower roughness exponent (0.5) is obtained. There is a crossover region for the bending stiffness, wherein the wave-front behavior cannot be explained by these limiting cases.