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Study on the interaction between long and short waves

长波与短波相互作用的研究

基本信息

批准号：
06835018
负责人：
OIKAWA Masayuki
金额：
$ 0.64万
依托单位：
KYUSHU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1994
资助国家：
日本
起止时间：
1994 至 1995
项目状态：
已结题

项目摘要

(1) The resonant interaction between long and short waves is considered for two typicalscalings, that is, SH type and SK type. By applying the reductive perturbation method to capillary-gravity waves on water of finite depth, the higher-order approximations arederived for each scaling. From themodulational stability characteristics of the uniformwavetrain solutions of the higher-order equations, it has been found that while for SHtype, the higher-order terms affect significantly the stability, for SK type, they hardly do. Further, from the numerical computations for the higher-order equations, it has beenfound that (i) For the higher-order approximation of SH type, a coupled wave of longand short waves analogous to the SH type soliton is formed along with a despersive wave decaying in time from localized initial data.(ii) Under the periodic boundary condition, while for SH type scaling the higher-order terms affect significantly on the long-time evolution of solutions, for SK type scaling they do little.(2) Propagation and generation of waves are considered in a two-layr fluid with a free surface flowig with the speed V over a localized topography. When V is close to the phase speed c for the long wave limit of the internal wave mode and there is a surfacewave packet with a group velocity close to c, it is shown that the governing equationsare SH type long-short wave resonance equations the extra terms expressing the effect of the difference between V and the resonant wave speed and that of topoqraphy. This equations have the coupled soliton solutions far from the localized topography as well as various steady solutions trapped in the topography. By examining the interactions of an incident soliton and a trapped wave numerically and analytically, it has been found that according to the values ot V-c and the parameters of the incident soliton, it is transmitted, reflected and amplified, or fis ioned.

(1)对于SH型和SK型两种典型标度，考虑了长短波之间的共振相互作用。将微扰法应用于有限深度水中的毛细重力波，得到了每个尺度的高阶近似。从高阶方程的均匀波列解的调制稳定性特征可以发现，对于sh型，高阶项对稳定性有显著影响，而对于SK型，它们几乎没有影响。此外，通过对高阶方程的数值计算，发现(i)对于SH型的高阶近似，在局域初始数据中形成了一个与SH型孤子类似的长短波耦合波和随时间衰减的绝望波。（ii）在周期边界条件下，对于SH型标度，高阶项对解的长期演化有显著影响，而对于SK型标度，高阶项对解的长期演化影响不大。(2)考虑了在自由表面以速度V流过局部地形的两层流体中波的传播和产生。当内波模式的长波极限V接近相速c，且存在群速度接近c的面波包时，控制方程为SH型长短波共振方程，即表示V与共振波速之差和地形影响的附加项。该方程既有远离局域地形的耦合孤子解，也有困在局域地形中的各种稳态解。通过对入射孤子与捕获波相互作用的数值分析，发现根据V-c值和入射孤子的参数，它被传输、反射、放大或折射。