中文摘要：

为满足偏心源和多体散射理论研究发展的需求,阐明了柱面坐标下波函数变换的基本类型,指出了现有变换公式的局限性,提出了全域变换公式,给出了应用实例并讨论了应用前景。研究表明,不同柱面坐标下波函数变换可以分为内域和外域两种基本类型,现有Graf加法公式仅适用于内域变换问题。提出的新公式突破了现有公式的限制,可完成弹性波散射问题中不同圆柱坐标系下波函数全空间域上的变换。应用新公式,构造了圆形空穴的外线源散射和内线源散射的解答,通过实例与大圆弧近似方法进行了对比,表明了新公式的优势和有效性。波函数变换基本类型的划分、全域变换概念的引入和公式构建的原理具有普适性,可以推广到其它坐标系下波函数的坐标变换公式的建立中。

英文摘要：

To supply the essential tools for the research on scattering problems by the eccentric sources and the multi-scatterer,the wave functions transform method in the cylindrical coordinate system is investigated.The basic types of transforms of wave functions in cylindrical coordinates are distinguished and the limitation of the existing transform formula is pointed out.A new complete-domain transform formula of cylindrical wave functions is presented and its application cases are demonstrated.The results indicate that the transforms of wave functions in cylindrical coordinates can be classified as two types of the interior and the exterior and the existing Graf＇s addition formula is only suitable for the interior transform.The new formula goes through the restriction of the existing formula and can perform the transform between two different cylindrical coordinates in the whole space.By using the new formula the scattering solutions by the sources outside and inside a cylindrical cavity are attained and the advantage and effectiveness of the new formula are proved by comparison with the approximate model based on a large cycle method.The division of the transform types,the concept of complete-domain and the principle of the new formula here are universal and they may be further extended to establish the coordinate transform formula of wave functions in other coordinate systems.