This paper is devoted to the spectral theory (more precisely, to the variational theory of the spectrum) of guided waves in an elastic half cylinder. We use variational methods to investigate several aspects of propagating waves, including localization (see Figure 1), existence criteria and the formulas to find them. We approach the problem using two complementary methods: The variational methods for non-overdamped operator pencils to describe eigenvalues in definite spectral zones, and Ljusternik-Schnirelman critical point theory to investigate eigenvalues in the mixed spectral zone where the classical variational theory of operator pencils is not applicable.
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