1. The treatment given in this paper is somewhat intuitive (and consequently simple) in its approach, and therefore not as rigorous as might be desired. A more extensive, rigorous treatment may be found in V. Kelvin Neil, “A study of some coherent electromagnetic effects in high-current particle accelerators,” (thesis) Lawrence Radiation Laboratory Report UCRL-9124 (April 26, 1960).
2. The possible existence of resonance is discussed in the Appendixes; see also reference 3.
3. C. E. Nielsen, A. M. Sessler, and K. R. Symon,Proceedings of the International Conference on High-Energy Accelerators and Instrumentation(CERN, Geneva, Switzerland, 1959), pp. 239–252, especially p. 246.
4. W. Gröbner and N. Hofreiter,Integraltafel(Springer-Verlag, Vienna, 1950), Part II, Sec. 335, Eq. (11a), p. 136.
5. The result for the case w∕h→∞ may also be obtained directly by reference to a corresponding electrostatic problem treated by William R. Smythe in hisStatic and Dynamic Electricity(McGraw-Hill Book Company, Inc., New York, 1950), 2nd ed. Sec. 4.20, p. 85, for a line charge centrally located between a pair of parallel conducting plates. From Smythe’s result, the current density in the boundary surfaces becomes Isurf = −1πn Inddxtan−1tanhπx2hcosn(θ−ω0t) = −12hn Insechπxhcosn(θ−ω0t), as found in our Eq. (2.3). This result also follows from the analysis by W. K. H. Panofsky and M. Phillips in theirClassical Electricity and Magnetism(Addison Wesley Publishing Company, Inc., Reading, Massachusetts, 1955), Chap. 3, Sec. 6, p. 45 ff.