Deterministic and probabilistic mathematical theories have in common that they construct mathematical representations of real-world phenomena. On a basic level this can be regarded as a type of explicit problem-solving. This involves presenting the problem in ‘abstract form’ in symbols (often numbers, letters, or geometrical elements). These symbols are then manipulated in accordance with precise rules: Strings of symbols in sets of equations come to represent ideas. The construction of mathematical theories involves testing whether experimental observations fit the postulated ‘mathematical rules.’ If they do not fit then the ‘mathematical rules’ may be refined, extended, or new ones may be formulated. Newly mathematically formalized ideas are validated by testing whether they align with observations but also by examining whether they are consistent with other, previously established, mathematical rules.

Chapter 4 introduces basic differential equations and boundary conditions for gravity waves propagating along a water surface. Assuming low wave amplitudes, equations are linearised. Then a quantitative discussion is given for harmonical (sinusoidal) waves propagating either on deep water, or otherwise on water of constant depth. Phase and group velocities are introduced, and then formulas are derived for the potential energy and the kinetic energy associated with a water wave. A closely related result is an important formula for the wave-power level, which equals the wave’s group velocity multiplied by the wave’s stored – kinetic + potential – energy per unit of sea surface. An additional subject is the wave’s momentum density. A section concerns real sea waves. Further, circular waves are mathematically described. Two sections of the chapter concern mathematical tools to be applied in Chapters 5–8 of the book. A final section considers water waves analysed in the time domain.