1、Coastal Hydrodynamics,1. Linearization of basic equations,Chapter 2,2.3 Small Amplitude Wave Theory,2. Solution of the linearized equations,3. Dynamic it is commonly referred to as the wave power(波功率).,In deep water, the energy is transmitted at only half the speed of the wave profile (n=1/2), and i

2、n the shallow water, the pro 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 alon

3、g beaches is commonly correlated with the “l(fā)ongshore 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

4、 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 rat

5、e 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.,The speed at which the energy is transmitted is equal to the group velocity.,21/37,Chapter 2,The group velocity is defined as,This derivative

6、can be evaluated from the dispersion relationship,22/37,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 heig

7、ht, thus giving a standing wave in front of the wall.,Standing waves are also called clapotis (駐波).,23/37,Chapter 2,The surface elevation of standing waves can be expressed as,It is seen that the height of the standing wave is twice the height of each of the two progressive waves forming the standin

8、g wave.,24/37,Chapter 2,Water surface displacement associated with a standing wave,25/37,Antinode(波腹),Node( 波節(jié)),Chapter 2,The velocity components of standing waves are,It is of interest that the horizontal and vertical components of velocity under a standing wave are in phase.,26/37,Chapter 2,The ex

9、treme values of u and w in space occur under the nodes and antinodes of the water surface profile, and they are equal to zero under the antinodes and nodes.,27/37,Chapter 2,A standing wave could exit within a basin with two walls situated at two antinodes. Why?,The lateral boundary condition at the

10、vertical wall would be one of no flow through the wall. Inspection of the equation for the horizontal velocity shows that at locations of antinodes the no-flow condition is satisfied.,28/37,Chapter 2,The potential and kinetic energies of standing waves averaged over one wave length per unit crest wi

11、dth are,Thus both the potential and kinetic energies of standing waves are twice those of progressive waves.,29/37,Chapter 2,At certain times, the velocity is zero everywhere in the standing wave system. It is therefore evident that at some times all the energy is potential and at other times all th

12、e energy is kinetic. That is to say, the energy changes form periodically from kinetic to potential energy, and vice versa.,30/37,Chapter 2,The displacement of a water particle under a standing wave is,The water particle path under a standing wave is a straight line.,31/37,Chapter 2,The pressure at

13、any depth under a standing wave is,Note that under the nodes, the pressure is solely hydrostatic. The dynamic pressure is in phase with the water surface elevation, and as before it is a combined result of the local surface displacement and the vertical acceleration.,32/37,Chapter 2,If the wave heig

14、hts of the incident wave and the reflected wave are different, the superposition creates a partial standing wave.,The surface elevation of a partial standing wave is,33/37,Chapter 2,It is realized that the successive antinodes and nodes appear at the intervals of L/4.,34/37,Chapter 2,The reflection coefficient (反射系數(shù)) based on the linear wave theory can be determined by measuring the amplitudes at the antinode and node of the composite wave train.,35/37,Homework,Chapter 2,Pleas