  This paper studies the coherent acceleration of ions interacting with two
electrostatic waves in a uniform magnetic field B0. This is a generalization of
an earlier analysis of waves propagating perpendicularly to B0 to include the
effect of wavenumbers along B0. The Lie transformation technique is used to
develop a perturbation theory describing the ion motion, and results are
compared with numerical solutions of the complete equations of motion. Coherent
energization occurs when the Doppler-shifted wave frequencies differ by nearly
an integer multiple of the ion cyclotron frequency. When the difference in the
parallel wavenumbers of the two waves is increased the coherent energization of
ions is limited to a small part of the phase space. The energization of ions
and its dependence on wave parameters is discussed.
