A recent paper (Roden and Gedney, 2000) proposed the application of a Convolutional Perfectly Matched Layer (CPML) to approximate free-field conditions for Finite-Difference Time-Domain (FDTD) modelling in computational electromagnetics. On page 3 of this paper, the authors take the x-projection of Ampere's Law and show how the CPML can be nicely applied as a recursive convolution in the time domain.

I would like to apply this convolution to the wave equation

$ \frac{{\partial ^2 p}}{{\partial x^2 }} + \frac{{\partial ^2 p}}{{\partial y^2 }} = A\frac{{\partial ^2 p}}{{\partial t^2 }} + B\frac{{\partial p}}{{\partial t}} $

However, the wave equation must be re-written in terms of the x-projection and the y-projection. This leads me to my question:

In a similar fashion to the x-projection equation for Ampere's Law given in the paper by Roden and Gedney, is it possible to write an equation for the x-projection, and an equation for the y-projection of the wave equation given above? The terms "x-projection" and "y-projection" are used here in a similar fashion to the paper by Roden and Gedney.