Analysis of nonlinear water waves due to Coupled Vibration Equations and Application to wave breaking

最近的研究指出，Nochino的耦合振动方程(CVE)适用于孤立波等非线性水波和波浪破碎。该项目的目的是通过数值计算来研究任意水深构型下的非线性水波的连续波传播能力，特别是波浪在坡面上的变形和波浪的制动。在项目的第一年(1996)，为了满足底部边界条件，我们应用了具有偶数阶和奇数次级数的勒让德多项式，推导出了任意水深配置的连续波传播方程。用新的CVE进行了数值计算，计算结果表明波底不稳定，这与波浪破碎的不稳定性有明显的区别。在第二年(1997年)，用变分算法考察了新CVE数值计算非线性水波的不稳定性。对CFL数和微分方程组的数值不稳定性问题进行了仔细的测试和讨论。但计算结果仍不稳定。经过仔细的研究，发现波前的对数波对CFL数问题有一定的刺激作用。数值计算从静水开始，第一年波从入射边界入射。计算区域中的波组成了行进波尖处的波前。因此，在目前的计算方法中，相对水深小于1/20的区域内的波浪是稳定的。这个区域隐含着浅水波，也与Boussinesq方程中的一个相同。

In recent reserches, it is pointed out that the coupled vibration equation (CVE) by Nochino is appicable for nonlinear water waves such as Solitary wave and for wave breaking. The goal of the project is to investigate the capability of CVE for nonlinear water waves on arbitarary configuration of depth, espesially the wave deformation on slope pf bottom and wave braking by the numerical calculation.On first year of project (1996), the CVE for the orbitirary config ation of depth is derived by applying the Legendre polyniominals with both the even and odd oder series, in order to satisfy the baoundary condition at bottom. The numerical calculation are excuted with the new CVE.The results of the calculation show unstable at the wave bottom, which is clearly different from the unstablity of wave breaking.On the second year (1997), the unstability of numerical calculation for nolinear water waves by the new CVE investigated by trying varius algorism of calculation. The problems of CFL number and the numerical instability of differenrial equations was carefully tested and ovewhlemed. the alculation results, however, sill show unstable. After careful research, it is found that the log wave at wave front stimulate the problem of CFL number. The numerical calculation is started from the water at still, and waves incident through the incident baoundary on the first year developed. The waves in the calculation region make up the wave front at the tip of waves progressing. Consiquently, waves in the region of its relative depth less than 1/20 are stable in the present calculation method. This region implies the shallow water waves and also is the same as one of the Boussinesq Equation is applicable.