  In conventional acoustic scattering theory, a large-distance asymptotic
approximation is employed. In this approximation, a far-field pattern, an
asymptotic approximation of the exact result, is used to describe a scattering
process. The information of the distance between the target and the observer,
however, is lost in the large-distance asymptotic approximation. In this paper,
we provide a rigorous theory of acoustic scattering without the large-distance
asymptotic approximation. The acoustic scattering treatment developed in this
paper provides an improved description for the acoustic wave outside the
target. Moreover, as examples, we consider acoustic scattering on a rigid
sphere and on a nonrigid sphere. We also illustrate the influence of the near
target effect on the angular distribution of outgoing waves. It is shown that
for long wavelength acoustic scattering, the near target effect must be
reckoned in.
