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定常声场Helmholtzs方程近似本征值问题的方法。在这种方法中，使用了以三维极坐标表示的解析解。解由径向的球贝塞尔函数、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。采用配置法近似满足边界条件。该方法适用于轴对称空间域问题，并以矩形六面体和圆柱体为例进行了计算。计算结果与精确值相吻合。此外，还对具有圆边和棱锥体的方体区域进行了计算。对金字塔的锥体进行了实验研究。计算得到的固有频率和固有振型与实验结果吻合较好，说明该方法是有效的。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.