  The acoustic scattering operator on the real line is mapped to a
Schr\"odinger operator under the Liouville transformation. The potentials in
the image are characterized precisely in terms of their scattering data, and
the inverse transformation is obtained as a simple, linear quadrature. An
existence theorem for the associated Harry Dym flows is proved, using the
scattering method. The scattering problem associated with the Camassa-Holm
flows on the real line is solved explicitly for a special case, which is used
to reduce a general class of such problems to scattering problems on finite
intervals.
