Theory and application of acoustical multipoles with complex singularities

具有复杂源位置的源是数学结构，其已被用于描述波在诸如光学和电动力学等领域中的传播，但在声学中并不为人所知。本项目的目的是了解这些元素的数学和物理特性，称为“复多极”，并将其应用于描述声学中的波动现象。在项目的第一阶段，对复多极进行了分析。将复单极子引入等效声源法中，并与真实的单极子结合，证明了复单极子在低声源数情况下模拟折流板驱动器高频声辐射的优越性。复合单极子的聚焦辐射是驱动器方向性重建的关键。用复单极子、复偶极子和复四极子来描述局部反应地面上的三维半空间绿色函数及其导数。这种表示法比现有的其它表示法具有更好的收敛速度和适用性。利用聚焦辐射产生特殊声学元件的散射和衍射，在第一项目阶段取得的成果的基础上，将在第二项目期间继续并完成对这些源的理论和数值研究。基于复多极子理论推导了非局部反应地面上三维半空间的绿色函数。复多极子的适用性扩展到二维问题的目的。特别是，表示的二维半空间绿色的功能与复杂的线源，可能会获得通过使用相同的方法，在三维，并应表现出相同的有利的分析性能。最后，重点将从频域转移到时域。除了复空间坐标之外，还具有复时间坐标的源产生脉冲束。特别选定的问题的解决方案，应有效地表示梁函数的扩展。

Sources with complex source positions are mathematical structures that have been used to describe propagation of waves in fields like optics and electrodynamics but are not well known in acoustics. The objective of this project is to understand the mathematical and physical properties of these elements, referred as “complex multipoles” and apply them to describe wave phenomena in acoustics.In the first period of the project, the analysis of the complex multipoles has been performed. Complex monopoles were incorporated into the equivalent source method and proved to be advantageous in combination with real monopoles to simulate the sound radiation from baffled drivers at high frequencies with a low number of sources. The focused radiation of the complex monopoles turns out to be key in the reconstruction of the driver’s directivity. Complex monopoles, dipoles and quadrupoles have been used to describe the 3D half-space Green’s function and its derivatives over a locally reacting ground. This representation has shown a better convergence ratio and applicability than other existing forms. Focused radiation was used to generate scattering from special acoustic elements and diffraction.Based on the results obtained in the first project phase, the theoretical and numerical investigations of those sources will be continued and completed in the second project period. A 3D half-space Green’s function over a non-locally reacting ground based on complex multipoles is planned to derive. An extension of the applicability of the complex multipoles to 2D problems is intended. In particular, a representation of the 2D half-space Green’s function with complex line sources can be obtained probably by using the same approach as in three dimensions and should exhibit the same favourable analytical properties. Finally, the emphasis will be shifted from the frequency domain to the time domain. Sources with a complex time coordinate in addition to the complex spatial coordinate generate pulsed beams. Solutions of particularly selected problems should be efficiently represented by an expansion in beam functions.