It is known that any {\em real coordinate transformation} (RCT) to compress waves in an unbounded domain into a bounded domain results in infinite oscillations that cannot be resolved by any grid-based method. In this paper, we intend to show that it is viable if the outgoing waves are compressed along the radial direction and the resulting oscillatory pattern is extracted explicitly. We therefore construct a perfectly matched layer (PML)-type technique for domain reduction of wave scattering problems using RCT, termed as real compressed layer (RCL). Different from all existing approaches, the RCL technique has two features: (i) the RCL-equation only involves real-valued coefficients, which is more desirable for computation and analysis; and (ii) the layer is not ``artificial'' in the sense that the computed field in the layer can recover the outgoing wave of the original scattering problem in the unbounded domain. Here we demonstrate the essential idea and performance of the RCL for the two-dimensional Helmholtz problem with a bounded scatterer, but this technique can be extended to three dimensions in a similar setting.

翻译：已知任何通过实坐标变换将无界域中的波压缩到有界域的方法都会产生无法通过任何网格方法解析的无限振荡。本文旨在证明：若沿径向方向压缩出射波并显式提取由此产生的振荡模式，则该方法是可行的。为此，我们利用实坐标变换构建了一种用于波散射问题区域缩减的理想匹配层类技术，称为实压缩层（RCL）。与现有方法不同，RCL技术具有两个特征：（i）RCL方程仅包含实值系数，更利于计算与分析；（ii）该层并非"人工"的，因为层内计算场能恢复无界域中原始散射问题的出射波。本文以有界散射体的二维亥姆霍兹问题为例，展示了RCL的基本思想与性能，但该技术可类似推广至三维情形。