Wave propagation through periodic lattice with defects

The discrete periodic lattice of masses and springs with line and point defects is considered. The matrix integral equations of a special form are solved explicitly to obtain the Floquet–Bloch dispersion spectra for propagative, guided and localised waves. Explicit form of the dispersion equations makes possible detailed analysis of the position and other characteristics of the spectra. For example in the case of the uniform lattice with one line inclusion along with one single defect we obtain the sharp explicit upper bound \(\frac{3}{4}-\frac{1}{2\pi }\) for the mass of single defect for which there exist localised waves in the spectral gaps. The developed method can be applied to various problems in optics, solid-state physics, or electronics in which lattice defects play a major role.