Development of a Practical Method of Simulating Three-Dimensional Acoustic Fields in Closed Space with Arbitrary Shape Using Analytic Solutions

本研究的目的是建立一种计算稳定声场的亥姆霍兹方程近似特征值问题的方法。该方法采用三维极坐标表示的解析解。其解由径向的球面贝塞尔函数、天顶角方向的勒让德关联函数和方位角方向的三角函数组成。采用配点法满足近似边界条件。该方法可用于求解轴对称空间域的问题。作为可以得到精确解的例子，用该方法计算了一个直角平行六面体和一个圆柱的特征值。计算结果与实际值吻合。在此基础上，对具有圆边的四边形区域和金字塔锥体进行了计算。对金字塔的截锥体进行了实验研究。计算得到的固有频率和固有模态与实验结果吻合较好。我们可以断定目前的方法是有效的。然而，为了扩大该方法的应用，还需要进行更多的研究。

The purpose of this research is to develop a method of calculating approximately eigenvalue problems of Helmholtzs Equation governing steady acoustic fields. In this method, analytic solutions represented in three-dimensional polar coordinates are used. The solutions consist of the spherical Bessel functions in the radial direction, the Legendre associated functions in the zenith angle direction and the trigonometric functions in the azimuth angle direction. The collocation method is utilized to satisfy approximately boundary conditions. This method can be applied to problems of axisymmetric spatial domains.As examples of the cases in which the exact solutions can be obtained, eigenvalues of a rectangular parallelepiped and a circular cylinder are calculated by this method. The results agree with the exact values. Furthermore, calculations about a boxy domain with round edges and a frustum of pyramid are done. An experiment is done about the frustum of pyramid. The calculated natural frequencies and the natural modes agree well with the experimental results.We can conclude that the present method is valid. To expand application of the method, however, more investigations are necessary.