  Minor ion (such as $He^{2+}$) heating via nonresonant interaction with
spectra of linearly and circularly polarized Alfv\'{e}n waves (LPAWs and CPAWs
hereafter) is studied. The obtained analytic solutions are in good agreement
with the simulation results, indicating that newborn ions are heated by
low-frequency Alfv\'{e}n waves with finite amplitude in low-beta plasmas such
as the solar corona. The analytic solutions also reproduce the preferential
heating of heavy ions in the solar wind. In the presence of parallel
propagating Alfv\'{e}n waves, turbulence-induced particle motion is clearly
observed in the wave (magnetic field) polarized directions. After the waves
diminish, the newborn ions are heated, which is caused by the phase difference
(randomization) between ions due to their different parallel thermal motions.
The heating is dominant in the direction perpendicular to the ambient magnetic
field. The perpendicular heating,
$\eta=(T_{i\perp}^R-T_{i0\perp}^R)/T_{i0\perp}^R$ (where $T_{i0\perp}^R$ and
$T_{i\perp}^R$ are the perpendicular temperature of species $i$ before and
after genuine heating, respectively), in the spectrum of CPAWs is a factor of
two stronger than that of LPAWs. Moreover, we also study the effect of
field-aligned differential flow speed of species $i$ relative to $H^+$, $\delta
v_{ip} = (\mathbf{v_i}-\mathbf{v_p}) \cdot \mathbf{B} / |\mathbf{B}|$ (where
$\mathbf{v_i}$ and $\mathbf{v_p}$ denote vector velocities of the $H^+$ and
species $i$, respectively), on the perpendicular heating. It reveals that large
drift speed, $v_d=\delta v_{ip}$, has an effect on reducing the efficiency of
perpendicular heating, which is consistent with observations.
