= Solution
The <Debye model> treats a three-dimensional solid as an isotropic elastic continuum with three independent acoustic <phonon polarizations>. It assumes harmonic <normal modes>, a linear <dispersion relation> $\omega=vq$, and equilibrium <Bose-Einstein statistics> with zero <phonon> <chemical potential>. Real anisotropy and nonlinear dispersion are replaced by one effective <Debye sound speed>. A cutoff restores the finite number of atomic vibrational degrees of freedom; the continuum itself would otherwise have infinitely many modes.
For sample <volume> $V$, the <wavevector> state density is $V/(2\pi)^3$ per polarization. Counting a sphere gives
$$
\mathcal N(q)=3\frac{V}{(2\pi)^3}\frac{4\pi q^3}{3}
=\frac{Vq^3}{2\pi^2}.
$$
There are $3N$ modes for $N$ atoms. Hence
$$
q_D^3=6\pi^2\frac NV,
\qquad\boxed{\omega_D=v\left(6\pi^2n_a\right)^{1/3},\quad n_a=N/V.}
$$
The associated <phonon density of states> is
$$
g(\omega)=\frac{3V\omega^2}{2\pi^2v^3}
=\frac{9N\omega^2}{\omega_D^3},\qquad0<\omega<\omega_D.
$$
If longitudinal and transverse <sound speeds> differ, use $3/v^3=1/v_L^3+2/v_T^3$ for the low-frequency count. The <Debye frequency> is an effective mode-counting cutoff, not a claim that every actual branch terminates at exactly that frequency.
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