半空间内二维圆柱声散射理论解析解

Analytical Solution of Sound Scattering from a Two-dimensional Cylinder in Half-space

采用理论解析方法研究半空间内二维非紧致圆柱的声散射。齐次Helmholtz方程的解在柱坐标系内用自由空间格林函数的级数展开式表示。基于镜像源方法,利用刚性半空间边界反射圆柱散射声波来解决半空间边界和圆柱之间的多重散射。结合等效源原理,处理半空间边界质量型阻抗特性和刚度型阻抗特性对声传播的影响,推导单位强度简谐单极子点声源产生声场的理论表达式。总声场可以表示为四个分量的总和:入射声场、反射波以及圆柱和镜像圆柱的散射声场。采用边界积分方法对声散射进行计算,以验证理论公式的正确性。点声源模型的理论解析值与边界积分方法数值解在研究的波数和观察点角度范围内一致。

The sound scattering from a non-compact two-dimensional cylinder in half-space is studied theoretically.The analytical solution of the homogeneous Helmholtz equation is expressed by a series expansion of the free space Green’s function in a cylindrical coordinate system.Following the image source method,the multi-scattering between the cylinder and the rigid half-space surface can be tackled tactfully by using the reflected waves from the image cylinder.Incorporating the equivalent source principle,the effects of spring-like and mass-like impedance characteristics of the half-space boundary are processed respectively.The analytical solution is derived for the prediction of multiple sound scattering by the cylinder placed above the half-space surface of finite impedance.The sound field can be expressed as a sum of four components:the incident field,the reflected wave,and the scattered fields from the cylinder and its image.Furthermore,numerical solutions on the basis of the boundary integral method are conducted to validate the theoretical formulations.The analytical solutions according to the point source model are in a good agreement with the numerical results for all observer angles and wave numbers.