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On the forcing $e^{-i\omega t}$,The other three factors have roots with real decay rate $-1/\tau$, so a real-frequency gas wave can resonate only with the undamped factor $\mathcal{D}$. The resonance condition is thereforeAt resonance, $\mathcal{D}$ annihilates the forcing, so a particular solution acquires one power of time:This is a streaming instability resonance: the gas wave's phase velocity along $\mathbf{k}$ matches the dust drift projected along the same wavevector.

A localized monochromatic cylinder emits four narrow beams, one in each quadrant, forming a St Andrew's cross. If $\alpha$ is the beam angle to the horizontal,equivalently, its angle $\theta =\pi /2-\alpha$ to the vertical obeys $\cos \theta =\omega /N$. As $\omega$ rises from zero to $N$, the beams rotate from horizontal toward vertical. For $\omega >N$ no freely propagating internal wave exists and the response is evanescent.

The wavevector and phase velocity are parallel. The dispersion relation is homogeneous of degree zero in $(k,m)$, so Euler's theorem gives $\mathbf{k}\cdot {\mathbf{c}}_g=0$: the group velocity is perpendicular to both the wavevector and phase velocity. Energy travels along the beams in the group-velocity direction.