# 7: Longitudinal Oscillations and Sound

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Transverse oscillations of a continuous system are easy to visualize because you can see directly the function that describes the displacement. The mathematics of longitudinal oscillations of a continuous linear space translation invariant system is the same. It must be, because it is completely determined by the space translation invariance. But the physics is different.

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In this chapter, we introduce two physical systems with longitudinal oscillations: massive springs and organ pipes.

1. We describe the massive spring as the continuum limit of a system of masses connected by massless springs and study its normal modes for various boundary conditions.
2. We discuss in some detail the system of a mass at the end of a massive spring. When the spring is “light,” this is an important example of physics with two different “scales.”
3. We discuss the physics of sound waves in a tube, by analogy with the oscillations of the massive spring. We also introduce the “Helmholtz” approximation for the lowest mode of a bottle.

This page titled 7: Longitudinal Oscillations and Sound 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.