The dispersion relation for a one-dimensional monoatomic lattice chain is given by the equation \[\omega = \frac{2}{a}\,\vartheta_s\left|\sin\!\left(\frac{ka}{2}\right)\right|,\] where \(a\) is the interatomic spacing, \(k = \dfrac{2\pi}{\lambda}\), and \(\vartheta_s\) has the dimension of velocity. The relation between the phase velocity \(V_p\) and group velocity \(V_g\) in the long wavelength limit is given by: