The change of energy and phase of a particle gyrating in a homogeneous magnetic field and a perpendicularly propagating electrostatic wave is calculated analytically for resonant and nonresonant frequencies in the regime where the wavelength2&pgr;k⊥−1is of the order of the gyroradiusa. This new result provides a quantitative basis for a simple wave‐particle interaction model and resolves certain discrepancies between previously obtained results and numerical calculations of the ranges ofk⊥afor which instability is possible. The long‐time behavior of the particle motion is discussed using a stroboscopic orbit theory. Significant modifications of the particle orbits by a single wave, which necessitate the use of a “strong turbulence” theory, are shown to occur ife&dgr;&psgr;/KT⊥∼(2&pgr;)1/2(K⊥a)−1/2(&ohgr;−l&OHgr;) /&ohgr;, wheree&dgr;&psgr;/KT⊥is the ratio of wave potential energy to perpendicular particle kinetic energy andlis the harmonic number.