圆形薄板二维驻波的研究

Research for two-dimensional standing waves on circular plate

在理论上和实验上对圆形薄板二维驻波波节图形(克拉尼图形)进行了研究,通过在极坐标下对圆形薄板的竖向小振动方程进行分离变量,求解出圆形薄板小振动方程在自由边条件下的解析解的简正模式,并进行了数值模拟.发现通过调节圆形薄板上点振动源的位置和频率,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种情形,且波节线的数量可严格控制.并在此基础上计算了圆形薄板上二维振动的波矢和波速,以及弹性模量等物理量,且理论结果跟实验符合得很好.

The analytical solution of Chladni figures on a thin mental board is a hard nut to crack in theoretical acoustics.When the boundary of a board is clamped or simply supported,the question can be solved quite completely,but in the condition of free boundary,the solution becomes rather difficult.In this paper,the two-dimensional standing waves of a circular plate（ Chladni figures） in various frequencies are investigated experimentally and theoretically.It is found that the Chladni figures can be precisely controlled by adjusting the frequency and position of the vibration source.The patterns of three kinds have been observed,only having circular nodal lines,radial nodal lines and the combination of the two kinds of nodal lines,respectively.Furthermore,the wave vectors,phase velocities and the elastic modulus of the plate are also obtained.The results of experiments well correspond with the analytical solutions.