General formulas are given for the masses and spring constants of one-dimensional finite chains with linear dispersion relations, examples of which were given by Herrmann and Schmälzle in 1981 in their discussion of a well-known collision apparatus. The mathematical similarity to the problem of a Boson in a constant magnetic field is shown. The explicit formulas make a study of the continuum limit possible. This is shown to be related to the system of uniform rods studied by Bayman in 1976. Examples are given of chains with quadratic dispersion relations. Resonances that give singularities in the interaction time are discovered in certain chains of elastic spheres.
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Matthias Reinsch (1994) studied this question.
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