Page 107 - Electromagnetics

2.10.4 Scalar wave equation for a conducting medium
In many applications, particularly those involving planar boundary surfaces, it is
convenient to decompose the vector wave equation into cartesian components. Using
2
2
2
2
∇ V = ˆ x∇ V x + ˆ y∇ V y + ˆ z∇ V z in (2.325) and in (2.326), we find that the rectangular
components of E and H must obey the scalar wave equation
2
∂ψ(r, t) ∂ ψ(r, t)
2
∇ ψ(r, t) − µσ − µ
= s(r, t). (2.327)
∂t ∂t 2
For the electric field wave equation we have
∂ J α i 1
ˆ
ψ = E α , s = µ + α ·∇ρ,
∂t
where α = x, y, z. For the magnetic field wave equations we have
i
ψ = H α , s = ˆα · (−∇ × J ).
2.10.5 Fields determined byMaxwell’s equations vs. fields deter-
mined bythe wave equation
Although we derive the wave equations directly from Maxwell’s equations, we may
wonder whether the solutions to second-order differential equations such as (2.314)–
(2.315) are necessarily the same as the solutions to the first-order Maxwell equations.
Hansen and Yaghjian [81] show that if all information about the fields is supplied by the
sources J(r, t) and ρ(r, t), rather than by specification of field values on boundaries, the
solutions to Maxwell’s equations and the wave equations are equivalent as long as the
second derivatives of the quantities

∇· E(r, t) − ρ(r, t)/
, ∇· H(r, t),
are continuous functions of r and t. If boundary values are supplied in an attempt to
guarantee uniqueness, then solutions to the wave equation and to Maxwell’s equations
may differ. This is particularly important when comparing numerical solutions obtained
directly from Maxwell’s equations (using the FDTD method, say) to solutions obtained
from the wave equation. “Spurious” solutions having no physical significance are a con-
tinual plague for engineers who employ numerical techniques. The interested reader
should see Jiang [94].
We note that these conclusions do not hold for static fields. The conditions for equiv-
alence of the first-order and second-order static field equations are considered in § 3.2.4.

2.10.6 Transient uniform plane waves in a conducting medium
We can learn a great deal about the wave nature of the electromagnetic field by solving
the wave equation (2.325) under simple circumstances. In Chapter 5 we shall solve for
the field produced by an arbitrary distribution of impressed sources, but here we seek a
simple solution to the homogeneous form of the equation. This allows us to study the
phenomenology of wave propagation without worrying about the consequences of specific
source functions. We shall also assume a high degree of symmetry so that we are not
bogged down in details about the vector directions of the field components.
We seek a solution of the wave equation in which the fields are invariant over a chosen
planar surface. The resulting fields are said to comprise a uniform plane wave. Although

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