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

本研究的目的是发展一种计算稳态声场中亥姆霍兹方程近似本征值问题的方法。在该方法中，使用以三维极坐标表示的解析解。这些解在径向上由球面贝塞尔函数组成，在天顶角方向由勒让德函数组成，在方位角方向由三角函数组成。利用配点法近似满足边界条件。这种方法可应用于轴对称空间域问题，作为可获得精确解的例子，用这种方法计算了长方体和圆柱的特征值。计算结果与精确值相符。此外，还计算了一个圆边方形区域和一个截棱锥体。对平截头棱锥进行了实验研究。计算的固有频率和振型与实验结果吻合较好，说明本文方法是有效的。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.