On the forcing $e^{-i\omega t}$,
$$
\mathcal D=-i(\omega-\mathbf k\mathbin\cdot\Delta\mathbf u).
$$
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 therefore
$$
\boxed{\omega=\mathbf k\mathbin\cdot\Delta\mathbf u}.
$$
At resonance, $\mathcal D$ annihilates the forcing, so a particular solution acquires one power of time:
$$
\sigma'(t)=Ct\,e^{-i\omega t}+O(1),
\qquad
\boxed{|\sigma'(t)|\sim|C|t}.
$$
This is a <streaming instability resonance>: the gas wave's <phase velocity> along $\mathbf k$ matches the dust drift projected along the same wavevector.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 2 / c / ii / Solution

A resonance condition is the relation between forcing parameters and a homogeneous mode that makes the forcing operator singular on that mode. A repeated inversion then produces algebraic amplitude growth.


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 Incoming links: Resonance condition