“Bilingual Course”,ZHENG Jinhai March 2009,Coastal Hydrodynamics,HOHAI UNIVERSITY,1. Linearization of basic equations,Chapter 2,2.3 Small Amplitude Wave Theory,2. Solution of the linearized equations,3. Dynamic & kinetic characteristics of small amplitude waves,4. Standing waves,2/37,Chapter 2,3. Dynamic & kinetic characteristics,Water particle velocity components Water particle trajectory Pressure field Energy and energy propagation,3/37,Chapter 2,The velocity components can be found by substituting the solution of velocity potential into the definition of potential function.,Velocity components,4/37,Chapter 2,Velocity components are harmonic functions of x and t.,The horizontal velocity component has the same phase as the elevation of the free surface.,The horizontal and vertical components are 90 out of phase.,Velocity components decrease exponentially with depth.,5/37,Chapter 2,The displacement of the water particle can be found by integrating the velocity with respect to time.,Water particle trajectory,Squaring and adding yields the water particle path as,6/37,Chapter 2,The water particles travel in an elliptical path.The elliptical motion becomes flatter with water depth.,In deep water, the orbits become true circles.,In shallow water, the major diameters of ellipses are constant.,7/37,Chapter 2,The pressure field associated with a progressive wave is determined from the unsteady Bernoulli equation.,Pressure field,The pressure equation contains two terms: the hydrostatic pressure (静水压强) & the dynamic pressure (动水压强),8/37,Chapter 2,The dynamic pressure is in phase with the water surface elevation. It is positive where the free surface is above the SWL, and is negative where the free surface is below the SWL.,In deep water dynamic pressure is very small at the bottom, while in shallow water it approaches unity.,Dynamic pressure,9/37,Chapter 2,The maximum value or the minimum one appears when wave crest or trough reaches a given point respectively.,Hydrostatic & dynamic pressure at various phases,10/37,Chapter 2,The term Kz is referred to as the “pressure response factor”(压力响应系数).,The “pressure response factor” has a maximum of unity at the mean water level and a minimum at the bottom. Below the mean water surface, it is always less than unity.,11/37,Chapter 2,A commonly used method to measure waves in either the laboratory or field by sensing the pressure fluctuations is stated as follows.,If the dynamic pressure is isolated by subtracting out the mean hydrostatic pressure, then the free surface displacement is,12/37,Chapter 2,The total energy consists of two kinds: the potential energy (势能), resulting from the displacement of the free surface； the kinetic energy (动能), due to the orbital motion of the water particles.,Wave energy,13/37,Chapter 2,The potential energy per unit crest width over one wave length is,The kinetic energy per unit crest width of a wave is,The total energy per wave per unit width is,14/37,Chapter 2,For Airy waves the potential energy is equal to the kinetic energy, which is characteristic of conservative (non-dissipative) systems.,It is worthwhile emphasizing that neither the average potential nor kinetic energy per unit area depends on water depth or wave length, but each is simply proportional to the square of the wave height.,15/37,Chapter 2,The rate at which the energy is transferred in the direction of wave propagation is called the energy flux (波能流), and it is the rate at which work is being done by the fluid on one side of a vertical section on the fluid on the other side.,Energy flux,The relationship for the energy flux is,16/37,Chapter 2,Energy flux has the units of power, and for thatreason it is denoted by P; it is commonly referredto as the wave power(波功率).,In deep water, the energy is transmitted at onlyhalf the speed of the wave profile (n=1/2), and in the shallow water, the profile and energy travel at the same speed (n=1).,17/37,Chapter 2,Conservation of the energy flux will be used later to examine the wave height variations in shoaling waves and to relate the height of breaking waves to the deep-water wave conditions.The rate of sand transport along beaches is commonly correlated with the “longshore component of the energy flux”.,18/37,Chapter 2,Group velocity (群速),If there are two trains of waves of the same height propagating in the same direction with a slightly different frequencies and wave numbers, the resulting profile, is modulated by an envelop that propagates with speed of group velocity.,19/37,Chapter 2,It is clear that no energy can propagate past a node as the wave height is zero there.Therefore, the energy must travel with the speed of the group of waves.,Characteristics of a group of waves,20/37,Chapter 2,The average rate of energy propagation per unit crest width over one wave period is seen to be the average energy per unit surface area progressing with the group velocity.,