Problem 6.189

In a chain of identical atoms the vibration frequency \(\omega\) depends on wave number \(k\) as \(\omega=\omega_{\max } \sin (k a / 2),\) where \(\omega_{m u x}\) is the maximum vibration frequency, \(k=2 \pi / \lambda\) is the wave number corresponding to frequency \(\omega, a\) is the distance between neighbouring atoms. Making use of this dispersion relation, find the dependence of the number of longitudinal vibrations per unit frequency interval on \(\omega,\) i.e. \(d N / d \omega,\) if the length of the chain is \(l .\) Having obtained \(d N / d \omega,\) find the total number \(N\) of possible longitudinal vibrations of the chain.