In 1958, Courant and Snyder analyzed the problem of alternating-gradient beam transport and treated a model without focusing gaps or space charge. Recently we revisited their work and found the exact solution for matched-beam envelopes in a linear quadrupole lattice [O.A. Anderson and L.L. LoDestro, Phys. Rev. ST Accel. Beams, 2009]. We extend that work here to include the effect of gaps. We derive the exact envelopes and show results for various field strengths, occupancies eta, and gap-length ratios. We find the peak envelope excursion. It has a broad minimum as a function of the phase advance sigma (typically around 34o) over which it varies less than +-1percent. The phase-advance numbers also change little over the full range of gap ratios. However, the required field strengths vary appreciably. In the second stable band, the higher field strength necessitated by the lower occupancy accentuates the remarkable compression effect predicted for the FD (gapless) model.

The Kapchinskij-Vladimirskij equations are widely used to study the evolution of the beam envelopes in a periodic system of quadrupole focusing cells. In this paper, we analyze the case of a matched beam. Our model is analogous to that used by Courant and Snyder [E.D. Courant and H.S. Snyder, Ann. Phys. 3, 1 (1958)] in obtaining a first-order approximate solution for a synchrotron. Here, we treat a linear machine and obtain an exact solution. The model uses a full occupancy, piecewise-constant focusing function and neglects space charge. There are solutions in an infinite number of bands as the focus strength is increased. We show that all these bands are stable. Our explicit results for the phase advance sigma and the envelope a(z) are exact for all phase advances except multiples of 180o, where the behavior is singular. We find that the peak envelope size is minimized at sigma = 90o. Actual operation in the higher bands would require very large, very accurate field strengths and would produce significantly larger envelope excursions.