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Summary

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.

A general form of Green's theorem is used to derive relations between expected costs for an optimisation problem and the distribution of state variable. A characterisation of the optimal stopping set and an alternative proof of the Howard improvement lemma emerge as non-trivial consequences.

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Chapter Four - Gravity Waves on Water

Summary

Identities relating forward and backward treatments of optimisation

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2 results

Chapter Four - Gravity Waves on Water

Summary

Identities relating forward and backward treatments of optimisation