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.