1Introduction to wave theory - Main Contents1.Wave Equations and Characteristics2.Wave Refraction, Diffraction and ReflectionReferences–Basic coastal engineering by Robert M. Sorensen (1978)–Hydraulics in Civil and Environmental Engineering, Andrew Chadwick and John Morfett, 2nd Ed, E & FN Spon, 1995.

21.Linear wave theory is the simplest theory for waves;2.Wave is classified by water depth;3.Wave refraction, reflection; wave break, wave run-up Threshold conceptsLecture’s expectation1.Know the terminologies used in wave theories;2.Know how to use the dispersion equation3.Know how to quantify the water surface profile, particle velocity in waves and classification of water based on water depth;4.Know how to predict wave breaking and wave runup, 5.Know how to get the wave refraction coefficients and refraction diagram;

31. Wave Equations and Characteristics•Why study waves?–Major sources of forces on offshore and coastal structures–Navigation, military and recreational purposes•Methods of study?–Experimentation (lab and field)•Empirical formulae–Observations (Lab and field)•Empirical formulae–Wave theories•Wave theories:–Linear (small amplitude) theory (Airy 1845)–Non-linear (finite amplitude) theoryDefinition of Water Waves: Water waves are fluctuations of the water level, accompanied by local currents, accelerations and pressure fluctuations.

4Types of Surface Waves•Capillary waves: surface tension is important in determining wave motions (very small in wave length and height (in mm)•Ship waves: more important in harbours•Wind waves: of greatest interests to coastal and offshore engineers•Swell, tide and tsunamis are also important to coastal and offshore engineeringFigure 1. Estimated relative ocean wave energy and primary generating forces

5Important TerminologiesWave period T: time for a particle to pass a fixed pointWave length L: distance between 2 crestsWave celerity C = L/TWave number k(=2/L): number of cycle per mFigure 2Wave Amplitude AWave Height HAngular frequency(=2/T): number of cycle per unit timeWave Height:H Wave amplitude:AWater depth:dSurface elevation: Wave steepness: H/L

61.1 Small Amplitude Wave Theory•A 2-dimensional, simplest and most useful wave theory,•A periodic velocity potential is sought to satisfy irrotational flow,•The velocity potential is used to derive equations for various wave characteristics, e.g. wave celerity, particle velocity, acceleration and pressure •Assumptions–The water is homogeneous, incompressible and surface tension forces are negligible–Flow is irrotationalvelocity potential exists & satisfy Laplace Eq–The bottom is fixed, impermeable and horizontal–Constant surface pressure–Wave height is small compared with wave length and water depth–Waves are periodic and sinusoidal

71.1.1 Governing equation and solution0//2222yx(10.1)Boundary Condition:At the bottom (y=d): (impermeable seabed)(10.2)/0vy At free surface (y=0): p=0 (atmospheric) and tv/At lateral boundaries: