椭圆形港湾内水波共振的解析解

AN ANALYTICAL SOLUTION FOR OSCILLATIONS WITHIN AN ELLIPTICAL HARBOR

通过坐标变换将线性长波方程转换为基于椭圆坐标系的水波运动方程,并采用分离变量法分别得到马丢方程描述的极角方向运动方程和拓展型马丢方程描述的径向运动方程。椭圆形港湾内的水波共振可以表示为马丢函数与拓展型马丢函数的乘积。由边界处自由水面法向量梯度为零求得水波共振的特征参数。椭圆形港湾内水波共振的极角方向波节点数与马丢函数的阶数相同,径向波节点数与边界条件相关。

A linear function is obtained by transforming the shallow-water wave equation from rectangular coordinates to elliptic coordinates, which gives the ordinary and the modified Mathieu equations respectively to describe oscillations in the polar and the radial directions by applying the method of separation of variables. Oscillations within an elliptical harbor can be described by appropriate products of radial and angular Mathieu functions. Eigenvalues are obtained by implementing the no-flux condition at the boundary. The oscillation is a two dimension distribution, and there are n nodes running in the polar direction, which is the same as the order of the angular Mathieu function; the nodes in radial direction are related with the boundary condition.