Abstract
This chapter is a review of the longest-period vibrations of the planet called free oscillations. We start our examination of mode-theory for a sphere using the simple but effectively analogous problem of a vibrating string. Then we discuss the solutions for a a uniform sphere in terms of spherical harmonics, with particular focus on eigenfrequencies. We use a homogeneous earth model as a guide to explore long-period observations of the Earth using seismic spectra that clearly illuminate the modal nature of the planet's long-period vibrations. We discuss observations of Earth's natural periods of oscillations and connect them with model-based calculations of their corresponding eigenfunctions. We also discuss mode splitting as a result of Earth's rotation and heterogeneity. Finally, we discuss mode attenuation and the process by which we can use these fundamental observations to build seismic models of the planet.
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