  The linear response of two-dimensional amorphous elastic bodies to an
external delta force is determined in analogy with recent experiments on
granular aggregates. For the generated forces, stress and displacement fields,
we find strong relative fluctuations of order one close to the source, which,
however, average out readily to the classical predictions of isotropic
continuum elasticity. The stress fluctuations decay (essentially) exponentially
with distance from the source. Only beyond a surprisingly large distance, $b
\approx 30$ interatomic distances, self-averaging dominates, and the quenched
disorder becomes irrelevant for the response of an individual configuration. We
argue that this self-averaging length $b$ sets also the lower wavelength bound
for the applicability of classical eigenfrequency calculations.Particular
attention is paid to the displacements of the source, allowing a direct
measurement of the local rigidity. The algebraic correlations of these
displacements demonstrate the existence of domains of slightly different
rigidity without, however, revealing a characteristic length scale, at least
not for the system sizes we are able to probe.
