Debye supposed that, in spite of its periodic nature, the crystal lattice can be represented by a continuum. By assuming suitable boundary conditions, he was able to show that there can only be certain modes of vibration for a given body; the situation is much the same as that which limits the number of stationary waveforms that are possible in a string that is stretched between two fixed points. The total number of possible modes in a continuum is really infinite but Debye considered only the 3N modes of lowest frequency, N being the number of atoms, since this leads to agreement with the classical expression for the specific heat at high temperatures. (Law of Dulong and Petit).
See Also
Debye length
Debye length in a plasma
Debye length in an electrolyte
Law of Dulong and Petit
Mode