Infinite One-Dimensional Periodic Systems—Characteristic Impedance

Abstract

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.