Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

In three dimensions a component $u$ of a time-harmonic plane wave satisfies

\nabla^{2} u + k^{2} u = 0 ⟶ \frac{\partial ^{2} u}{\partial x ^{2}} + \frac{\partial ^{2} u}{\partial y ^{2}} + \frac{\partial ^{2} u}{\partial z ^{2}} + k^{2} u = 0. (6.1.13)

We solve it by the separation of variables: assume a product of three functions of one variable each, $u (x, y, z) = X (x) Y (y) Z (z)$ . Substituting and dividing by $X Y Z$ ,

- k_{x}^{2} \frac{X ^{''}}{X} + - k_{y}^{2} \frac{Y ^{''}}{Y} + - k_{z}^{2} \frac{Z ^{''}}{Z} + k^{2} = 0. (6.1.14)

The first term depends on $x$ only, the second on $y$ only, the third on $z$ only, and their sum is the constant $- k^{2}$ for all $x$ , $y$ , $z$ . That is possible only if each term is a constant by itself. Calling these constants $- k_{x}^{2}$ , $- k_{y}^{2}$ and $- k_{z}^{2}$ ,

k_{x}^{2} + k_{y}^{2} + k_{z}^{2} = k^{2}, (6.1.15)

and each factor obeys an equation of the form Eq. (6.1.4): $X^{''} + k_{x}^{2} X = 0$ with the forward solution $X = e^{- j k_{x} x}$ , and likewise $Y = e^{- j k_{y} y}$ , $Z = e^{- j k_{z} z}$ . Collecting the three numbers into the wave vector

k = k_{x} a_{x} + k_{y} a_{y} + k_{z} a_{z} = k a_{n}, (6.1.16)

and writing $r = x a_{x} + y a_{y} + z a_{z}$ for the position vector, the solution becomes

u (x, y, z) = C e^{- j k \cdot r} = C e^{- j k a_{n} \cdot r} = C e^{- α (a_{n} \cdot r)} e^{- j β (a_{n} \cdot r)} . (6.1.17)

The unit vector $a_{n}$ , also written $a_{k}$ , is the direction of propagation. Condition Eq. (6.1.15) says simply that $k \cdot k = k^{2}$ : the length of the wave vector is fixed by the medium and the frequency, and only its direction is free.

The wavefronts

The phase of $C e^{- j k \cdot r}$ is constant where

k \cdot r = k a_{n} \cdot r = const . (6.1.18)

$a_{n} \cdot r$ is the distance of the point from the origin measured along $a_{n}$ , so these surfaces are planes perpendicular to $a_{n}$ , hence the name plane wave. Neighbouring crests are $λ = 2 π / β$ apart along $a_{n}$ . Along the coordinate axes the phase repeats less often. If $a_{n}$ lies in the $x z$ -plane at the angle $θ$ from the $z$ -axis (Figure 6.1.4), then $k_{x} = k sin θ$ , $k_{z} = k cos θ$ and

λ_{x} λ_{z} = \frac{2 π}{k _{x}} = \frac{λ}{sin θ}, = \frac{2 π}{k _{z}} = \frac{λ}{cos θ}, (6.1.19)

both longer than $λ$ . In general, dividing Eq. (6.1.15) by $4 π^{2}$ ,

\frac{1}{λ _{x}^{2}} + \frac{1}{λ _{y}^{2}} + \frac{1}{λ _{z}^{2}} = \frac{1}{λ ^{2}} . (6.1.20)

A point of constant phase observed on the $z$ -axis therefore runs along it at the speed $ω / k_{z} = v_{p} / cos θ$ , faster than $v_{p}$ .

The wavefronts of \mathrm e^{-\mathrm j\vec k\cdot\vec r} are planes perpendicular to \vec k, one wavelength apart along \vec k. Measured along the axes the same fronts are farther apart: \lambda_x=\lambda/\sin\theta and \lambda_z=\lambda/\cos\theta. — **Figure 6.1.4.** The wavefronts of $e^{- j k \cdot r}$ are planes perpendicular to $k$ , one wavelength apart along $k$ . Measured along the axes the same fronts are farther apart: $λ_{x} = λ / sin θ$ and $λ_{z} = λ / cos θ$ .

No paradox

Only the intersection of a wavefront with the axis moves at $v_{p} / cos θ$ , like the point where a breaking wave meets a straight beach, which runs along the shore faster than the water moves. No energy and no signal travels that fast. The same geometry returns in waveguides, where two plane waves zig-zag between metal walls (Section 7.2), and Exercise 6.1.17 builds such a guide from two crossing plane waves.

Quick check

A $3 GHz$ wave in free space travels in the $x z$ -plane at $6 0^{\circ}$ from the $z$ -axis. Find $λ$ , $λ_{z}$ and the speed of the phase along $z$ .

$λ = c / f = 10 cm$ , $λ_{z} = λ / cos 6 0^{\circ} = 20 cm$ , and the phase runs along $z$ at $c / cos 6 0^{\circ} = 6 \times 1 0^{8} m/s$ . Along $x$ , $λ_{x} = λ / sin 6 0^{\circ} = 11.5 cm$ .

6.1.2 Plane waves in three dimensions

The wavefronts