The Application of Fourier Analysis to the Uncoupling of Lattices
We shall apply here Fourier analysis and other vector space and symmetry concepts introduced in Chapter 1 to the study of certain coupled systems which have simple mechanical realizations as one-dimensional crystalline lattices. In Section 2.1 we define the constituents of such systems and their corresponding solution so that we may pose the problem of uncoupling lattices of such elements in Section 2.2. A rather detailed study of the basic solutions is made in Section 2.3 for a simple lattice and in Section 2.4 for more complicated ones which can be described by second-or farther-neighbor interaction in crystals or as molecular or diatomic chains. We go to a more general setting in studying energy and other phase-space concepts which belong properly to analytical mechanics. Sections 2.5 and 2.6 can be read after the first two sections if the reader so prefers. Although examples drawn from Sections 2.3 and 2.4 are used to illustrate examples of the theory, the reader should be able to follow the general presentation easily.
KeywordsNormal Mode Fundamental Solution Transverse Vibration Interaction Operator Simple Lattice
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