ABSTRACT
Maxwell’s equations are vector PDEs coupling the electric and magnetic £elds (read Chapter 1 for a quick review of Maxwell’s equations). The independent variables are the three spatial coordinates and the time variable. For time-harmonic £elds where we assume the time variation as harmonic (cosinusoidal), one can use the phasor concepts and replace ∂/∂t in the time domain with multiplication (by jω) in the phasor domain. Thus, one can reduce the dimensionality of the problem from four independent variables to three, thus reducing the mathematical complexity of the problem. Further reduction in the dimensionality of the problem is possible for certain applications. Let us review the waveguide problem (see Chapter 3) from this viewpoint. If we consider the TM modes of a rectangular waveguide ( , ) E Hz z≠ =0 0 , the longitudinal component satis£es the scalar Helmholtz equation
∂ ∂ +
∂ ∂ +
∂ ∂
⎡
⎣
⎢
⎤
⎦
⎥
+ = 2
x y z E k Ez z ,
(15.1)
= ω με. (15.2)
Since we are interested in studying the propagation or nonpropagation of the wave, we can assume that
E x y z F x yz z( , , ) ( , )= −e jβ (15.3)
and study whether and under what circumstances β is real. Substituting Equation 15.3 into Equation 15.1, we get a two-dimensional PDE in F:
∂ ∂ +
∂ ∂
⎡
⎣
⎢
⎤
⎦
⎥
+ = 2
2 0 F x
F y
k Fc2 ,
(15.4)
where
k kc 2
= − 2 2β . (15.5)
Equation 15.4 can be written as
−∇ = + =t 2
c 2F k F Fλ . (15.6)
In Equation 15.6, � t2 is the Laplacian operator in the transverse plane and λ is the eigenvalue. The waveguide problem is the eigenvalue problem involving a differential operator
and when a numerical technique is employed, the problem is converted into the familiar form of
AX X= λ , (15.7) where A is a square matrix and λ’s are the eigenvalue and X ’s are the corresponding eigenvectors. Equation 15.6 is a scalar second-order PDE and we have seen that for a rectangular region 0 < x < a and 0 < y < b. One can solve it analytically by using separation of variable technique and on imposing the PEC boundary conditions (Figure 15.1), one obtains
… …E E
m x a
n y b
= = ∞ = ∞ −sin sin e
pi pi βj 1, 2, 1, 2,, , ,
(15.8)
where
λ pi pimn mn x yk k k
m a
n b
= = + = ⎛
⎝
⎜
⎞
⎠
⎟
+ ⎛
⎝
⎜
⎞
⎠
⎟
(15.9)
Recognize in Equations 15.8 and 15.9 that the problem has double in£nity number of eigenvalues λmn and the corresponding eigenvectors are proportional to sin(mπx/a) sin (nπy/b).
