  In this paper, the interaction of an incident finite amplitude longitudinal
wave with a localized region of nonlinearity is considered. This interaction
produces a secondary field represented by a superposition of first-, second-,
and third-harmonic components. The secondary field is solely a result of the
quadratic and cubic elastic nonlinearity present within the region of the
inclusion. The second-harmonic scattering amplitude depends on the quadratic
nonlinearity parameter $\beta$, while the first- and third-harmonic amplitudes
depend on the cubic nonlinearity parameter $\gamma$. The special cases of
forward and backward scattering amplitudes were analyzed. For each harmonic,
the forward-scattering amplitude is always greater than or equal to the
backward scattering amplitude in which the equality is only realized in the
Rayleigh scattering limit. Lastly, the amplitudes of the scattered harmonic
waves are compared to predicted harmonic amplitudes derived from a plane wave
model.
