  We study the accretion of a Schwarzschild black hole due to spherically
symmetric perturbations sourced by a minimally coupled massless scalar field.
The backreaction of the black hole to low-frequency ingoing scalar waves is
computed analytically as a second-order perturbative effect, using matched
asymptotic expansions to relate the behaviour of the scalar field in the
vicinity of the horizon and at null infinity. As an application of our results,
we compute the mass increase due to (i) ingoing wave packets with an arbitrary
profile and (ii) incoherent radiation. Our results could serve as a model for
the backreaction of environmental scalar fields on black holes.
