Previous work has developed the theory of linearized gravitational wave (GW) interactions with matter using the Bondi–Sachs formalism, but with the perturbations restricted to be quadrupolar, i.e., the angular dependence is spherical harmonic with ℓ = 2. Here, the theory is extended to the case of GWs on a Minkowski background with general ℓ. Formulas for the GW damping and heating effects are obtained for arbitrary ℓ ≥ 2. It is found that the effects are, generally, enhanced, and this suggests that it is unlikely that higher ℓ-modes will be seen in GW observations of the post-merger signal of a binary neutron star merger, or of a core collapse supernova.