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9: The Boundary at Infinity

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    Although the wave phenomena we can see in the laboratory live in finite regions of space, it is often convenient to analyze them as if the traveling waves come in from and go out to infinity. We have described traveling waves in infinite translation invariant systems. But traveling waves are more complicated and more interesting in systems in which there are boundaries that break the translation symmetry.


    In this chapter, we introduce a new kind of “boundary condition” in systems that lack a boundary! It will enable us to discuss reflection and transmission, and in general, the phenomenon of scattering.

    1. We discuss forced oscillation problems in semi-infinite systems, that extend to infinity in one direction. We show that we can impose a “boundary condition” even though there is no boundary, by specifying the amplitude of a wave traveling in one direction. We then discuss scattering problems in infinite systems, describing the amplitudes for transmission and reflection. We study the motion of a general wave with definite frequency.
    2. We discuss electromagnetic plane waves in a dielectric.
    3. We discuss reflection and transmission by a mass on a string and two masses on a string, showing how to use a “transfer matrix” to simplify the solution to the scattering problem. We analyze reflection from a boundary between regions with different wave number and show how to eliminate the reflection with a suitable “nonreflective coating.”

    This page titled 9: The Boundary at Infinity is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Howard Georgi via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.