矩形单元声场波函数构造及其应用

在波叠加法中，结构外部声场是在离散边界上对Green函数进行积分并叠加得到，但数值积分的计算效率较低。而等效源法虽然提高了计算效率，但其面源简化为点源的过程中存在较大的积分近似误差。针对上述两种方法的缺陷，构造了一种波函数以替代离散单元关于Green函数积分的声场。首先，利用球坐标系下Helmholtz方程的解，推导了替代矩形单元积分的一般形式波函数及效率更高的内推波函数。其次，当离散单元为正方形时，将其近似成圆形域，进一步简化了内推波函数的表达式。最后，将所构造的波函数应用于声场计算。数值结果表明，在计算单个矩形单元外部辐射声场时，构造的波函数不仅保证了计算精度，而且相比于直接积分大幅度提高了计算效率。其中，矩形域一般形式和内推形式的波函数计算效率是直接积分的5～6倍，圆形域内推波函数计算效率达到了直接积分的12～13倍。在简支板声源和立方箱体辐射声源数值算例中，圆形域内推波函数在整个计算频段的声场计算精度均高于等效源法。

Construction and application of wave function of sound field of rectangular element

In the wave superposition method, the sound field outside the structure is obtained by integrating and superimposing the Green function on the discrete boundary, but the computational efficiency of the numerical integration is low. In order to avoid numerical integration and improve computational efficiency, a wave function is constructed to replace the sound field integrated by Green function of discrete element. Firstly, by using the solution of Helmholtz equation in spherical coordinate system, the general wave function and the more efficient internal wave function are derived instead of the rectangular element integral. Secondly, when the discrete element is square, it is approximated to a circular domain, which further simplifies the expression of the extrapolated wave function. Finally, the constructed wave function is applied to the acoustic field calculation. The numerical results show that the wave function constructed not only ensures the accuracy of calculation, but also improves the efficiency of calculation compared with direct integration. Among them, the computational efficiency of the wave function in the general form and the extrapolation form in the rectangular domain is 5-6 times that of the direct integration, and the computational efficiency of the extrapolation wave function in the circular domain is 12-13 times that of the direct integration. In the calculation of simply supported plate sound source and cubic box radiation sound source, the calculation accuracy of the acoustic field of the push wave function in the circular domain is higher than that of the equivalent source method in the whole calculation frequency band.