Theory and application of acoustical multipoles with complex singularities

具有复杂源位置的源是数学结构，用于描述光学和电动力学等领域的波传播，但在声学中并不为人所知。该项目的目的是了解这些被称为“复杂多极”的元素的数学和物理特性，并应用它们来描述声学中的波动现象。在项目的第一期，进行了复杂多极的分析。在等效声源方法中引入复单极子，并证明了复单极子与真实单极子相结合的优点，可以较好地模拟高频低声源条件下的挡板驱动声辐射。复杂单极子的聚焦辐射是重建驱动器方向性的关键。复单极子、偶极子和四极子被用来描述三维半空间格林函数及其在局部反应地面上的导数。该表示比现有的其他形式具有更好的收敛率和适用性。利用聚焦辐射对特殊声元和衍射产生散射。根据第一个项目阶段取得的结果，将继续对这些来源进行理论和数值调查，并在第二个项目期间完成。提出了一种基于复杂多极的非局部反应地面上的三维半空间格林函数。将复杂多极的适用性扩展到二维问题。特别是，具有复线源的二维半空间格林函数的表示可能可以通过使用与三维相同的方法来获得，并且应该表现出同样有利的分析性质。最后，重点将从频域转移到时域。除了具有复杂空间坐标外，具有复杂时间坐标的源也会产生脉冲光束。特别选定的问题的解应该用梁函数的展开来有效地表示。

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.