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Most introductory physics students are familiar with standing wave patterns on a string. These types of vibrations are called extended modes, for the obvious reason that they extend over the entire system. However, if the symmetry of the string is broken, say by splicing in a lighter section, then another type of vibration, a localized mode, can occur. Surprisingly, the lighter section vibrates with a large amplitude that decreases exponentially with distance from the string defect. In a landmark 1958 paper, Philip Anderson introduced the notion of localization in the context of electronic wave function in media such as amorphous semiconductors and disordered insulators. For many physicists, the concept of localization is an interference phenomenon that occurs for all types of waves, including light, sound, water, and seismic waves. It occurs for longitudinal and transverse waves and in classical as well as quantum systems, and can appear in systems that are merely nonperiodic and not necessarily disordered. Despite the fundamental nature of localization, its ubiquity, and the amount of research devoted to it, localization is not a topic commonly presented to undergraduates. However, the phenomenon is easy enough for students to explore in a homework assignment, and rich enough to be the basis of a research project. We believe that localization should be a standard topic in a course on waves. Localization also can be easily incorporated into a classical mechanics course in conjunction with the study of normal modes in a system of coupled oscillators.
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