Infinite One-Dimensional Periodic Systems—Characteristic Impedance

  • Ingram Bloch


This chapter is a continuation of the last one, in which the periodic systems are infinitely long—strictly speaking, semi-infinite, having a beginning but no end. We can get an indication of what to expect by letting our chain of masses, or the equivalent ladder network with series L and shunt C, become infinitely long by addition of more and more sections like those already there. Thus we let N → ∞, without m or k (or L or C) changing. Then the spectrum of normal frequencies more and more densely fills the interval from 0 to 2ω0, while ω0 remains constant at (k/m)1/2 or (LC)−1/2. It seems reasonable to suppose that a semi-infinite system can oscillate at any frequency in this range, that the discrete spectrum becomes continuous in the limit. We shall see that this expectation is correct. Similarly, the series-C-shunt-L network, when infinitely long, permits any frequency above ω0/2, as might be expected from Equation 11.48.


Phase Velocity Group Velocity Standing Wave Input Voltage Pass Band 
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  1. 1.
    Brillouin, L. Wave Propagation in Periodic Structures. New York: McGraw Hill, 1946; Dover Reprint, 1953.Google Scholar
  2. 2.
    See also: French, A.P. Vibrations and Waves. New York: Norton, 1971; and Zeines, B. Introduction to Network Analysis Englewood Cliffs, NJ: Prentice-Hall, 1967.Google Scholar

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© Springer Science+Business Media New York 1997

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  • Ingram Bloch

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