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1、2-4 Finite Amplitude Wave Theory,A sine (or cosine) function or a superposition of such functions can describe a wave whose height is extremely small compared with its length. When the wave height becomes relatively large, it is impossible to ignore the non-linear terms which are neglected in small
2、amplitude wave theory. In such cases a finite amplitude wave theory is necessary.,2-4-1 Trochoidal Wave摆线波Theory,Among finite amplitude wave theories, the trochoidal wave theory, which was originally derived by Gerstner for deep water waves, gives the only exact solution for deep water waves.,2-4-2
3、Stokes Wave Theory,Stokes obtained an asymptotic 渐近的 solution 解 for deep water irrotational waves with a permanent wave profile.,Perturbation Method 摄动法 ： Assuming that the solution can be represented in terms of a power series 级数 expansion 展开 of some small parameter ,The index n represents the nth-
4、order for any particular quantity.,The parameter is related to the ratio of wave height to wave length, or to water depth, and is usually assumed to be small. Each of the velocity potentials must satisfy governing differential equation ( Laplaces equation) and boundary conditions.,The velocity poten
5、tial, wave elevation, celerity and wave length are: （The second order asymptotic solution),For deep water, they are:,The water particle velocities in the x- and z- directions for the second-order theory,速度不对称 正向(向岸)历时变短, 波峰时水平速度增大 负向(离岸)历时增长,波谷时水平速度减小,2-4-3 Mass Transport with Stokes Waves,The water
6、 particle path over one wave cycle obtained with the Stokes wave theory is not a closed curve as in small amplitude wave theory. A sample calculation of that path is shown in Fig. 2.4.3., which indicates that the crest position after one wave cycle progresses in the direction of wave propagation com
7、pared with the crest position at the beginning of the wave cycle.,The displacements,Thus there results a so-called mass transport 质量输移 or drift漂流.,The mass transport velocity in shallow water is:,The mass transport velocity in deep water is given by,Comparison of the First- and Second-Order Theories
8、,A comparison of first- and second-order theories is useful to obtain insight about the choice of a theory for a particular problem. It should be kept in mind that linear (or first-order) theory applies to a wave that is symmetrical about the SWL and has water particles that move in closed orbits. O
9、n the other hand, Stokes second-order theory predicts a wave form that is unsymmetrical about the SWL but still symmetrical about a vertical line through the crest and has water particle orbits that are open.,Pressure (2nd order),等号右边第一、二项是微幅波的结果，第 三、四项是非线性影响的修正项，他们均 与波陡有关。,In deep water, the effect
10、 is less and can be neglected, while in shallow water, the effect can not be neglected.,Energy,The kinetic energy 动能 per unit crest width in one wave length, Ek, is:,Energy,The potential energy 势能 per unit crest width over one wave length Ep is given by:,Kinetic Energy Potential Energy (kh) is large
11、r,Total Energy per unit crest width over one wave length is given by:,The difference is significant.,2-4-4 Critical Wave Condition (Stokes Waves),A wave will be considered to break when the water particle horizontal velocity component u at its crest波峰 is equal to or larger than the wave celerity c.
12、(u=c),The result of Michells calculation shows that the maximum steepness of deep water waves is given by:,The critical steepness of shallow water waves is:,浅水非线性波理论,当水深很浅，Stocks 波的高阶项可能变得很大，因 而不能适用，这时就应作为浅水非线性波来研究。 椭圆余弦波Cnoidal Wave Theory 是最主要的浅水 非线性波理论之一。 椭圆余弦波Cnoidal Wave Theory的一个极限情况 是当波长无穷大时，
13、趋近于孤立波solitary wave。 椭圆余弦波Cnoidal Wave Theory的另一个极限情 况是当振幅很小或相对水深（h/H)很大时，称为 浅水正弦波shallow water sinusoidal wave。,2-4-5 Cnoidal Wave Theory 椭圆余弦波,The name cnoidal is derived from the fact that the wave profile is expressed by Jacobis elliptic functions.,In the asymptotic expansion of the Stokes wave
14、theory, the parameter H/L is assumed to be small compared with unity. While for shallow water waves, the relative water depth h/L has an important influence on wave motion. Therefore H/h as well as H/L should be considered in a finite amplitude wave theory.,2-4-5 Cnoidal Wave Theory 椭圆余弦波,In order t
15、o represent the relative magnitude between the above two quantities, a parameter called Ursells parameter 参数 is introduced:,它表示非线性波理论中的二阶项和一阶项的比值。,The region U1. A wave in this region deforms with propagation. In the intermediate region where U is comparable to unity, permanent waves exist which are
16、 called cnoidal waves.,The limiting condition,The limiting condition for cnoidal wave is given by: Hmax=0.73ht,2-4-5 Cnoidal Wave Theory 椭圆余弦波,Solitary Wave孤立波 波长无穷大时,Cnoidal Wave,Shallow Water Sinusoidal Wave 振幅很小或相对水深很大时,2-4-6 Solitary Wave 孤立波 Theory,The surface profile of cnoidal waves is periodic, but tends toward that of a non-periodic wave with a single crest as k approaches unity. This limiting wave is called a solitary wave.,Listed below are the main results o
