A range of phenomena important in space plasmas calls for simulation and can be approximated by the two-dimensional geometry in which the magnetic field lines are straight and parallel and all gradients are perpendicular to the field. The design of simulations itself motivates analytical theory, in this case linear theory for a velocity distribution. A general formulation of linear theory with the unperturbed state stratified uses the method of Robertson et al. [1981>, the constants of the motion being energy W and canonical momentum Py. This is developed using the periodicity of any particle trajectory, leading to a simple condition determining the direction of energy exchange between resonant particles in an element dWdPy, and a wave, which is a generalization of the energization of a wave by an overtaking beam. A Larmor radius expansion is performed including the second harmonic of the trajectory frequency, but no higher harmonics, and including the gradient of the electric field, which would be essential for Kelvin-Helmholtz instability. The objective is to determine the dependence of the perturbations on gyrophase. The important result is that the third Fourier component is of second order in the Larmor radius and the formulae are obtained for the Fourier components up to the second. When the Larmor radius expansion is valid, fine structure in the velocity distribution arises only from resonances. A less rigorous discussion of the opposite case, treating the magnetic field as weak, shows that particles faster than the phase speed pass through resonance in the sense of stationary phase. This generally dominates the perturbations, and generally these vary rapidly with gyrophase.