Continuous Systems, Wave Equation, Lagrangian Density, Hamilton’s Principle
The presence of waves on periodic networks suggests that such networks may be related to continuous systems such as stretched strings on which also waves can be propagated. We explore this possibility by letting the segments of a periodic system become smaller and more numerous until, in the limit, the system is continuous. We concentrate our attention on the mechanical chain of masses discussed in Chapter 12, which becomes a continuous string in the limit, but shall later comment briefly on the limiting forms of other periodic systems.
KeywordsFrequency Ratio Power Flow Lagrangian Density Periodic System Continuous System
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- 1.The author is indebted to the late M.J. Moravcsik for the method of displaying frequency ratios used in Figure 14.2.Google Scholar
- 2.Many books contain discussions of vibrating strings; for example, the books, already cited: Symon K.R. Mechanics 3d ed. Reading, MA: Addison Wesley, 1971; and Morse, P.M. Vibrations and Sound 2d ed. New York: McGraw Hill, 1948.Google Scholar
- 3.The application of Hamilton’s principle to continuous systems and, in general, the mechanics of such systems, are discussed in Goldstein, and in the early chapters of various books on quantum field theory, such as Wentzel, G. Quantum Theory of Fields. New York: Interscience, 1949, and Schweber, S. Introduction to Relativistic Quantum Field Theory. Evanston; Row Peterson, 1961.Google Scholar