Wave localization in stratified square-cell lattices. The antiplane   problem
  Steady-state and transient antiplane dynamic processes in a structured solids consisting of uniform periodic square-cell lattices connected by a lattice layer of different bond stiffnesses and point masses are analyzed. A semi-infinite lattice covered by a layer is also considered. Localization phenomena that are characterized by a waveguide-like propagation along the layer direction and exponential attenuation along its normal are studied. Waveguide pass-bands and attenuation factors are obtained analytically, while transient processes developed under the action of a monochromatic local source are numerically simulated. As a result, it is shown how a two-dimensional problem is transformed with time into a quasi-one-dimensional one and how a layer traps the source energy. Special attention is paid to revealing particularities of transient waves in cases where steady-state solutions are absent: resonant waves with frequencies demarcating pass- and stop-bands at the ends of the Brillouin zone and wave transition in the vicinities of transition points in dispersion curves. In the latter case, a simultaneous onset of different localization phenomena - a spatial star-like beaming and a one-dimensional waveguide-like localization - is shown. 
Grigory Osharovich and Mark Ayzenberg-Stepanenko
cond-mat.mtrl-sci cond-mat.mes-hall cond-mat.other math-ph math.MP