Discretization of the Wave Equation

Let us consider the general form of the wave equation that governs the evolution of any electric or magnetic field component f. Its homogeneous version in a Cartesian coordinate system (x, y, z) is given by 

In this case and according to the theoretical aspects of [2, 3], there are two Laplacian difference operators for the discretization of (3.4). The first one is expressed as 

Having determined the basic attributes of (3.7), analysis will now focus on the development of a representation for V2e/kr with r = xx + yy and x, y the respective unit vectors. Since in the one-dimensional case, application of (3.5) gives d2 = 2 [cos kAh) — 1] ejkx, the 

Inspection of (3.11) leads to the direct conclusion that, despite our primary goal, p depends on angle. Such a relation is fairly weak, implying that a p k Ah, 9), which exactly satisfies (3.11) at a specific value 9 = 9ex, still remains a sensible approximation for 9 9ex. An indicative ex for minimizing the deviation ofL fe r]/g-/k 1 from2 [cos(7A7) — 1] maybe derived via cos4 9ex = 1/2. Under these considerations and through the assistance of (3.10), expression (3.11) becomes 

The solution of (3.4) in a 3-D domain is straightforward, since the derivation process preserves the preceding abstractions and only operator L2[.] has to be appropriately redefined. 

---