In a chain of identical atoms the vibration frequency w depends on wave number k as w = w max sin (ka/2), where w max is the maximum vibration frequency, k = 2π/λ, is the wave number corresponding to frequency w 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 w, i.e. dN/dw, if the length of the chain is l. Having obtained dN/dw, find the total number N of possible longitudinal vibrations of the chain.