环形薄板二维驻波的研究

在理论上和实验上对环形薄板二维驻波波节图形(克拉尼图形) 进行了研究. 通过在极坐标下对垂直板面方向小振动方程进行分离变量, 求解出环形薄板小振动方程在外边界悬空时分别在两种内边界条件, 即内边界悬空和内边界简支下的解析解的简正模式, 并计算了在第一种边条件下几种共振模式的径向波速近似值, 以及两种边条件下的圆形驻波波节线的半径和薄板的弹性模量. 发现通过调节环形薄板上点振动源的频率, 可精确控制薄板上出现的克拉尼图形. 实验上观察到了仅有圆形波节线, 仅有辐射状波节线, 以及两种波节线同时存在3 种简正模式的情形, 且波节线的数量可严格控制. 理论结果跟实验符合得很好. 

THE RESEARCH OF 2-DIMENSIONAL STANDING WAVES ON ANNULAR PLATE

The analytical solution of Chladni figures on a thin metal plate is a difficult problem in theoretical acoustics. The problem can be solved quite completely if the plate is circular or rectangular and the vibration frequency is low, but the solution becomes rather difficult if the plate is annular and the frequency is high. In this paper, the two-dimensional standing waves on an annular plate (Chladni figures) are investigated both experimentally and theoretically at two kinds of boundary conditions, the free outer edge with simply supported inner edge and both free edges , respectively. It is found that the Chladni figures can be precisely controlled by adjusting the vibrating frequency and position of the vibration source. Three kinds of patterns have been observed, only having circular nodal lines, only having radial nodal lines, and having both of the two kinds of nodal lines, distinguishedly. Furthermore, the approximate radial wave velocities, the radii of standing wave nodal circles in different frequencies, and the elastic modulus of the plate are also obtained. The results of experiments well correspond with those from the analytical solutions.