Resonant-frequency primitive waveforms and star waves in lattices

For square and triangular lattices we have found a new line-localized primitive waveform (LPW) existing at a resonant frequency. In two-dimensional (2D) case, the LPW represents a line of oscillating particles, while the lattice outside this line remains at rest. We show that: (a) A single LPW does not conduct energy; however, a band consisting of two or more neighboring LPWs is a conductor with the phase-shift-dependent energy flux velocity. (b) Any canonical sinusoidal wave consists of LPWs. In turn, the LPW can be represented by a superposition of the sinusoidal waves (these two types of waves are connected by the discrete Fourier transform). (c) There are two (three) LPW orientations for the square (triangular) lattice, and this is why the sinusoidal-wave group velocity orientation is piecewise constant at this frequency; it coincides with the nearest LPW orientation. (d) LPW can also exist at a lower frequency being localized at the lattice halfplane boundary. Further, for 3D lattices plane-localized waveforms are found to exist in a frequency region. Finally, for the point harmonic excitation of 2D lattices we show that starlike waves develop with the rays in the LPW directions.