A TM01 wave in a cylindrical wave guide can exert an alternating gradient focusing action on a beam of charged particles within the guide. The relativistic equations of motion of the particles admit three exact integrals by means of which a single nonlinear second-order total differential equation governing the motion is derived, without approximations. The linear approximation is a Mathieu equation whose solution indicates the magnitude of the focusing forces involved and shows that these are stronger when the wave is propagated in a direction opposite to that in which the particles move. It is shown that a wave providing fields of the order of several thousand volts/cm would be required to counteract the space charge defocusing effects in a typical electron beam.
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