The normal mode solution involves solving a one-dimensional equation which is very similar to that of a vibrating string. In this analogy, the ``frequencies'' of vibration give the horizontal wavenumbers associated with the modal propagation and a varying string thickness corresponds to a change in sound speed with depth. The position where the string is plucked corresponds to the source depth and the relative excitation of the ocean modes depends on the source depth just as the harmonic balance of a note is affected by the position where the string is plucked.
We will begin with a simple derivation of the normal mode equations based on separation of variables. This provides a quick means of introducing the gross features of the normal mode approach in Sect. 2.2. In Sect. 2.3, we present a generalized derivation which starts with the spectral representation of the acoustic field. This derivation is necessary for treating the more complex problems involving interfacial roughness, homogeneous halfspaces, or other more complicated boundary conditions which are treated in Sects. 2.5--2.7. Normal modes are normally thought of principally in the context of range-independent problems, however, they can be extended in various ways to both range-dependent problems and fully three-dimensional problems. These extensions are discussed in Sects. 2.8 and 2.9 respectively.