In this part we develop the mathematical apparatus necessary for the quantification of waves and flows. As we noted in the introduction, this apparatus consists of three components. The first, kinematics, places a constraint on the possible motions of the particles comprising a continuous body in order that the body indeed remains continuous; this constraint is developed in Chapter 3. The second component, dynamics, essentially applies Newton's second law to each particle in the body. As we see in Chapter 4, this gives us a vector equation, called the momentum equation, relating the acceleration of a fluid particle to the forces acting on it. The third component of our apparatus, rheology, is a set of equations describing how the material of a body deforms or flows in response to forces. Our rheological equations include the change of volume induced by a change in pressure, as explained in Chapter 5, and the change of shape induced by changes in deviatoric stresses, as explained in Chapter 6.

The primary mathematical quantities of interest are the deformation (or its time derivative the velocity), the strain tensor (or its time derivative the rate-of-strain tensor) and the stress tensor. While stress and strain are used fairly interchangeably in colloquial usage, they have precise and distinct meanings in the present context. Stress quantifies the applied force and strain (or rate of strain) quantifies the deformation (or flow) that occurs in response to that force. More specifically, the stress tensor describes the structure of the forces acting within a body, while the strain (and rate-of-strain) tensor describes the structure of the internal deformation.

Waves and flows occur as changes to the static state of a body, so before tackling waves and flows, we need to determine the static state and develop the equations governing small deviations from this state; this is accomplished in Chapter 7. The statics and dynamics of rotating fluids are of particular importance in geophysical fluid dynamics; the basic concepts of – and equations governing – rotating fluids are considered in Chapter 8.