圆柱形容器中自由表面波的三波内共振相互作用

由Luke变分原理导出了圆柱形容器中自由毛细重力表面波的基本方程.首先对速度势和自由面波高作Galerkin展开,用多重尺度法导出了控制方程前二阶摄动方程,在此基础上讨论了三个波二阶内共振的非线性相互作用,导出了三波内共振的非线性耦合作用方程和守恒律.针对非退化的作用方程,分析了相平面上平衡点的位置,研究了参数对应共振与非共振的各种情况,在不同参数情况下求出二阶作用方程的稳态解并分析了解的稳定性态,还讨论了只在有限时间内有效的解.分析表明,在非退化情况下,由于初始条件不同,3个波之间能量传递的模式不尽相同,有可能能量在3波之间周期性传递,亦可能单波的能量有衰减或增长.

Internal Resonant Interactions of Three Free Surface-Waves in a Circular Cylindrical Basin

The basic equations of free capillary_gravity surface_waves in a circular cylindrical basin were derived from Luke's principle.Taking Galerkin's expansion of the velocity potential and the free surface elevation,two_order perturbation equations were derived by use of expansion of multiple scale.The nonlinear interactions with two_order internal resonance of three free surface_waves were discussed based on the above.The results include:derivation of the couple equations of resonant interactions among three waves and the conservation laws;analysis of the positions of equilibrium points in phase plane;study of the resonant parameters and the non_resonant parameters respectively in all kinds of circumstances; derivation of the stationary solutions of second_order interaction equations corresponding to different parameters and analysis of the stability property of the solutions;discussion of the effective solutions only in the limited time range.The analysis makes it clear that the energy transformation mode among three waves differs because of the different initial conditions under nontrivial circumstance.The energy may either exchange among three waves periodically or damp or increase in single waves.