Normal modes in a monatomic periodic square chain are simulated. You can adjust the number of atoms, the ratio between the two "spring constants", the two components of the wave vector (red arrow) in the first Brillouin zone (red square), and the two components of the polarization . Sound waves (small ) and stationary waves (within the boundary of the Brillouin zone) can be easily identified. Longitudinal ( parallel to ) and transverse ( perpendicular to ) waves are also clearly reproduced.