Coastal Hydrodynamics,Coastal Hydrodynamics,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),Chapter 2The pressure field a,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.,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.,Wave energy,Chapter 2The total energy con,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,Chapter 2The rate at which th,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.,2/36,Chapter 2Group velocity (群速)I,Chapter 2,The group velocity is defined as,This derivative can be evaluated from the,dispersion relationship,3/36,Chapter 2The group velocity i,Chapter 2,4. Standing waves,Standing waves,(,立波),often occur when incoming,waves are completely reflected by vertical walls.,If a progressive wave were normally incident on,a vertical wall, it would be reflected backward,without a change in height, thus giving a,standing wave in front of the wall.,Standing waves are also called clapotis,(,驻波),.,4/36,Chapter 24. Standing wavesSta,Chapter 2,If the wave heights of the incident wave and the,reflected wave are different, the superposition,creates,a partial standing wave,.,5/36,Chapter 2If the wave heights,Chapter 2,The,reflection coefficient,(,反射系数),based on,the linear wave theory can be determined by,measuring the amplitudes at the antinode,and node of the composite wave train.,6/36,Chapter 2The reflection coeff,1.,Stokes wave theory,Chapter 2,2.4,Finite Amplitude Wave Theory,2.,Trochoidal wave theory,3.,Cnoidal wave theory,4.,Solitary wave theory,7/36,1. Stokes wave theory Chapter,Chapter 2,In 1847, Stokes considered waves of small but,finite height progressing over still water of,finite depth and presented a second-order,theory.,1.,Stokes waves,The method of Stokes has been extended,to higher orders of approximation.,The tabulated solutions to third and fifth,orders are very useful in applications.,8/36,Chapter 2In 1847, Stokes cons,Chapter 2,The solution depends on the presumed small,quantity,ka, which will be defined as,.,Therefore we will decompose all quantities,into a power series in,.,Perturbation approach,(,摄动法),9/36,Chapter 2The solution depends,Chapter 2,Velocity potential is,Second-order solution,For finite height waves there is an additional,term added onto the equation obtained in,the linear wave theory.,10/36,Chapter 2Velocity potential i,Chapter 2,Water surface displacement is,Second-order solution,For finite height waves there is an additional,term added onto the basic sinusoidal shape.,11/36,Chapter 2Water surface displa,Chapter 2,The added term enhances the crest amplitude,and detracts from the trough amplitude, so,that the Stokes wave profile has steeper crests,separated by flatter troughs than does the,sinusoidal wave.,12/36,Chapter 2The added term enhan,Chapter 2,The dispersion equation remains the same,that is,Second-order solution,It is noted that a correction occurs to the,dispersion equation at the third order,which would result in a slight increase in,wave celerity.,13/36,Chapter 2The dispersion equat,Chapter 2,Velocity components are,Second-order solution,14/36,Chapter 2Velocity components,Chapter 2,The effect of the added term is to increase the,magnitude but shorten the duration of the,velocity under the crest and decrease the,magnitude but lengthen the duration of the,velocity under the trough. For the horizontal,velocity, the velocities are greater under the,crest but are reduced under the trough when,compared with those of the linear wave.,15/36,Chapter 2The effect of the ad,Chapter 2,Comparison of bottom orbital velocity under Stokes wave with that of linear wave of the same height and length,(H=4m, h=12m, T=12sec),16/36,Chapter 2Comparison of bottom,Chapter 2,An interesting departure of the Stokes wave,from the Airy wave is that the particle orbits,are not closed. The crest position after one,wave cycle progresses in the direction of,wave propagation.,Second-order solution,17/36,Chapter 2An interesting depar,Chapter 2,The net horizontal displacement within one,wave period is,Thus there results a so-call “mass transport”,(,质量输移),with an associated velocity ,18/36,Chapter 2The net horizontal d,Chapter 2,Wave energy,Second-order solution,19/36,Chapter 2Wave energy Second-o,Chapter 2,Wave pressure,Second-order solution,20/36,Chapter 2Wave pressure Second,Chapter 2,The first solution for periodic waves of finite,height was developed by Gerstner in 1802.,From the developed equations, he concluded,that the surface profile is trochoidal in form.,His solution is limited to waves in water of,infinite depth.,2.,Trochoidal waves,In 1935, Gaillard attempted to extend the,theory to water of finite depth and introduced,an elliptic trochoidal wave theory,for shallow water waves.,21/36,Chapter 2The first solution f,Chapter 2,For an angle of rotation, the surface depression,below crest level is,22/36,Chapter 2For an angle of rota,Chapter 2,This theory has been wide application by civil,engineers and naval architects because the,solutions are exact and the equations simple,to use.,However, mass transport is not predicted and,the velocity field is rotational. Even worse,in the trochoidal wave the particles rotate in,a sense opposite to the rotational that would,be present in a wave generated by a wind,stress on the water surface.,23/36,Chapter 2This theory has been,Chapter 2,In 1895, Korteweg and de Vries developed a,shallow water wave theory which allowed periodic,waves to exits. The name cnoidal is derived from,the fact that the wave profile is expressed by the,cn(),function of Jacobis elliptic functions.,3.,Cnoidal waves,The cnoidal wave is a periodic wave that may have,widely spaced sharp crests separated by wide,troughs and so could be applicable to wave,description just outside the breaker zone.,24/36,Chapter 2In 1895, Korteweg an,Chapter 2,The first-order approximate solution is given by the following equations:,25/36,Chapter 2The first-order appr,Chapter 2,It is worthwhile mentioning that the cnoidal wave,has the unique feature of reducing to a profile,expressed in terms of cosines at one limit and,to the solitary wave theory at the other limit.,It is apparent then that we could simply apply,the cnoidal wave and ignore other theories.,Unfortunately, the mathematics of the cnoidal,wave is difficult, so that in practice it is,applied to as limited range as possible.,26/36,Chapter 2It is worthwhile men,Chapter 2,The solitary wave is a translation wave consisting,of a single crest. There is therefore no wave,period or wave length associated with,the solitary wave.,4.,Solitary waves,The solitary wave was first described by Rusell,in 1844, who produced it in a laboratory wave,tank by suddenly releasing a mass of water,at one end of the wave tank. The first theoretical,consideration was that of Boussinesq.,27/36,Chapter 2The solitary wave is,Chapter 2,The first-order approximation of the water,surface in the solitary wave theory is,28/36,Chapter 2The first-order appr,Chapter 2,Because of its similarity to real waves and its,simplicity, the solitary wave has been wide,application to nearshore studies.,It would appear that the solitary wave would,not be particularly useful in describing the,periodic wind waves we deal with in the,ocean.,29/36,Chapter 2Because of its simil,Chapter 2,2.5 Wave Theory Limits of Applicability,How does one decide which of the,wave theories is applicable to his,particular problem?,30/36,Chapter 22.5 Wave Theory Lim,Chapter 2,This validity is composed of two parts:,the mathematical validity, which is the ability,of any given wave theory to satisfy the,mathematically posed boundary value problem.,the physical validity, which refers to how well,the prediction of the various theories agrees,with actual measurements.,31/36,Chapter 2This validity is c,Chapter 2,Ranges of applications of the several wave theories as function of the rations H/gT,2,and h/gT,2,32/36,Chapter 2Ranges of applicatio,Chapter 2,In general, the linear wave theory does well in,deep water when wave steepness is small,while the Stokes wave theory proved to be,more applicable when wave steepness is large.,In the intermediate water, almost all kinds of,wave theories could be used. The cnoidal,wave theory and the solitary wave theory do,well in shallow water.,33/36,Chapter 2In general, the line,MINI-EXAMINATION,Chapter 2,34/36,As far as the water surface, the,particle velocity and the particle,orbit are concerned, what are,differences between linear waves,and Stokes II waves?,MINI-EXAMINATION Chapter 234/3,Homework,Chapter 2,Please ponder upon the following problems.,Standing waves often occur when,incoming waves are completely reflected,by vertical walls. At which phase,position would the wall be located?,What are differences between the,linear waves & Stokes waves?,How to decide the suitable wave,theory for a specific problem?,35/36,Homework Chapter 2Please ponde,THANK YOU,“,Coastal Hydrodynamics”, chapter 2,THANK YOU“Coastal Hydrodynamic,