A note on equivalence between the standard and nearly perfectly matched layers for the second-order wave equation

By either rigorously or approximately transforming the mapped wave equation from the complex stretched coordinate system to the real one, the standard perfectly matched layer (PML) or the nearly PML (NPML) can be obtained. The two are equivalent for the first-order wave equation but not for the second-order one. Over the past two decades, two parallel literatures have emerged, one about the PML and the other about the NPML. Although the PML and NPML are theoretically equivalent regarding absorption efficiency, the suggestion has been that the NPML is superior regarding (i) implementation complexity, (ii) computational efficiency, and particularly (iii) memory efficiency. The present paper restructures the PML formulation to halve its memory efficiencies. Compared with the current formulation, we show that in fact the PML and the NPML are almost equivalent regarding these three aspects. Moreover, when the classical coordinate stretching function is used, the PML is superior to the NPML.