Vibrations of crystals with diatomic basis

From a classical standpoint:
The vibrational modes of the atoms are considered sound waves
From a modern standpoint:
The vibrational modes of the atoms are considered phonons
The speed with which a longitudinal wave propagation through a medium of elastic material (for example solid) of density ? given by:
v_(0 )=??=?( ?_s/?)?^½ …… 1
Where ?_s is adiabatic elastic bulk modulus, or stiffness coefficient.
4-1 Vibrations of crystals with monatomic basis
Consider a one-dimensional (linear) lattice with a period (a) and with identical atoms of mass m, vibrating around each lattice point(see Fig.26) .Each atom is indexed by an integer (n), and its displacement from its equilibrium position is denoted (u). The atoms are taken to oscillate in the same direction as the lattice.
Now we consider a one-dimensional lattice with two non-equivalent per primitive basis of masses m and M with the distance between two neighboring atoms Now we consider a one-dimensional lattice with two non-equivalent per primitive basis of masses m and M with the distance between two neighboring atoms