Spontaneous phonon decay selection rule: $N$ and $U$ processes

We show that the spontaneous decay of a phonon by anharmonic processes of any order into a set of phonons of higher phase velocity is impossible for both $N$ and $U$ processes. The result is valid for crystals of arbitrary symmetry and anisotropy and in the presence of frequency and angular dispersion. The implications for the propagation of large-wave vector phonons is discussed.