In this chapter we discuss free linear waves on a sphere, again for an ideal fluid. Waves are a basic mechanism by which a fluid adjusts to changes and propagates momentum and energy. Many of the approximations that are applied to the basic equations of oceanic motions and that form the remainder of this book can be characterized by the wave types that they eliminate. Free linear waves, as opposed to forced waves, are solutions of homogeneous linear equations. They can be superimposed. Mathematically, these homogeneous linear equations form an eigenvalue problem. The eigenfunctions determine the wave form, the eigenvalues the dispersion relation. Specifically, we consider linear waves on a stably stratified motionless background state. The linearized equations have separable eigensolutions. The vertical eigenvalue is characterized by the speed of sound, gravitation, and stratification. The horizontal eigenvalue problem is characterized by rotation and the spherical geometry of the earth. We then classify the various wave solutions into:

• sound (or acoustic) waves;

• surface and internal gravity waves; and

• barotropic and baroclinic Rossby waves;

and consider the limit of short-wave solutions. This chapter owes much to Kamenkovich (1977) where more details and proofs can be found.

Basic kinematic concepts of waves, such as propagating and standing waves, and the geometric optics approximation are covered in Appendix E.