当前位置: X-MOL 学术Q. J. Mech. Appl. Math. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Image Conditions for Elliptical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly Reflecting Flat Boundaries: Application to in Situ Tunable Noise Barriers
The Quarterly Journal of Mechanics and Applied Mathematics ( IF 0.9 ) Pub Date : 2020-03-02 , DOI: 10.1093/qjmam/hbaa005
Ho-Chul Shin 1
Affiliation  

Two-dimensional time-harmonic multiple scattering problems are addressed for a finite number of elliptical objects placed in wedge-shaped acoustic domains including half-plane and right-angled corners. The method of separation of variables in conjunction with the addition theorems is employed in the elliptical coordinates. The wavefunctions are represented in terms of radial and angular Mathieu functions. The method of images is applied to consider the effect of the infinitely long flat boundaries which are perfectly reflecting: either rigid or pressure release. The wedge angle is |$\pi/n$| rad with integer |$n$|⁠; image ellipses must be appropriately rotated to realise the mirror reflection. Then, the ‘image conditions’ are developed to reduce the number of unknowns by expressing the unknown expansion coefficients of image scattered fields in terms of real counterparts. Use of image conditions, therefore, leads to the |$4n^2$|-fold reduction in the size of a matrix for direct solvers and |$2n$|-times faster computation in building the system of linear equations than the approach without using them. Multiple scattering models using image conditions are formulated for rigid, pressure release and fluid ellipses under either plane- or cylindrical-wave incidence, and are numerically validated by the boundary element method. Furthermore, potential applications are presented: arrays of elliptically shaped scatterers make in situ tunable noise barriers by rotating scatterers. Finally, polar-coordinate image conditions (for circular objects) are also discussed when coordinates local to circles are also rotated. In Appendix, analytic formulae are provided, which permits the elliptical-coordinate addition theorems used in this article to be calculated by summation instead of numerical integration.

中文翻译:

具有完美反射平面边界的椭圆坐标变量分离声多重散射模型的图像条件:在现场可调节噪声屏障中的应用

对于放置在楔形声域(包括半平面角和直角角)中的有限数量的椭圆对象,解决了二维时谐多重散射问题。椭圆坐标中采用了与加法定理结合的变量分离方法。波函数以径向和角度Mathieu函数表示。应用图像方法来考虑无限长的平坦边界的影响,这些边界完美地反映了:刚性释放或压力释放。楔角为| $ \ pi / n $ | rad与整数| $ n $ |⁠; 图像椭圆必须适当旋转才能实现镜面反射。然后,开发“图像条件”以通过以真实对应物表示图像散射场的未知膨胀系数来减少未知数。因此,使用图像条件将导致| $ 4n ^ 2 $ |。直接求解器和| $ 2n $ |的矩阵大小的两倍减小与不使用线性方程组的方法相比,构建线性方程组的时间快了两倍。针对平面,圆柱波入射下的刚性,压力释放和椭圆形,使用图像条件制定了多种散射模型,并通过边界元方法进行了数值验证。此外,还提出了潜在的应用:椭圆形散射体的阵列通过旋转散射体在原位形成可调谐的噪声屏障。最后,当圆局部坐标也旋转时,还将讨论极坐标图像条件(用于圆形对象)。在附录中,提供了解析公式,该公式允许通过累加而不是数值积分来计算本文中使用的椭圆坐标加法定理。
更新日期:2020-03-02
down
wechat
bug