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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.

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What actually do you think the x and y projections are? I looked at the Roden and Gedney paper, and it seems to me that they consider Maxwell's equations as specified by two vectors $E,H$, and the x projection is just the equation for $E_x$. As far as I can tell you are dealing with a scalar wave equation, what are you trying to project? – Willie Wong Aug 3 '10 at 21:23
That's a good question, Willie. I would like to do something similar to Roden and Gedney, but all that I have is a scalar wave equation. Might there be a way to start with a similar (vector) equation and then take an x-projection and a y-projection? – Nicholas Kinar Aug 3 '10 at 21:58
First question: what do you think is the reason that Roden and Gedney needed to look at the projections? Second question: what do you think these projections gain for you? Why not just do whatever coordinate transform directly with the scalar equation? – Willie Wong Aug 4 '10 at 1:39
Using the properties of convolution, I applied the Roden and Gedney CPML directly to the scalar wave equation given in my original posting. I found that when applied to my scalar wave equation, the CPML will absorb waves traveling in the x-direction and y-direction (perpendicular to the CPML). Waves arriving at other angles of incidence were reflected. This may be the reason why the CPML in the Roden and Gedney paper was applied only to the x-projection and the y-projection equations. – Nicholas Kinar Aug 4 '10 at 2:20

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