Electromagnetic Waves, Antennas, and RF: Electromagnetic Wave Equation - Exercises and Complete Worked Solutions

EM16 derived the source-free electromagnetic wave equations from Maxwell’s Equations and then specialized them to plane waves. This companion article turns those derivations into a sequence of worked problems. The emphasis is on checking candidate waves rather than merely recognizing formulas: differentiating traveling profiles, recovering one field from the other, enforcing Maxwell’s divergence constraints, determining propagation direction, building Standing Waves, extending the derivation to nonzero sources, and connecting the time-domain wave equation to the frequency-domain Helmholtz equation.

The central vacuum relations are

(1)

(2)

with

(3)

For a monochromatic source-free plane wave,

(4)

and

(5)

These relations should be viewed as consequences of the Maxwell system, not independent empirical rules [1, 2, 3, 4, 5].

How to use this problem set

Attempt all exercises in Part I before consulting Part II. For each wave problem, use the following order whenever possible:

1. identify the proposed propagation direction from the phase;
2. test the wave equation or the relevant Maxwell equation;
3. use the divergence equations to check transversality;
4. use Faraday and Ampere–Maxwell to determine relative orientation, amplitude, and phase;
5. only then substitute numerical values.

This order helps separate geometry from arithmetic and makes sign errors easier to detect.

Part I: Exercises

Exercise 1: verify the general traveling profile F(z − ct)

Let

(6)

where F is any twice-differentiable function.

Show directly from the chain rule that

(7)

Then repeat the argument for

(8)

State the propagation direction of each profile and explain why its shape does not change as it moves.

Exercise 2: sinusoidal trial solution and the dispersion relation

Consider

(9)

Substitute this field into the one-dimensional wave equation

(10)

Derive the condition relating ω and k. Then show that

(11)

Finally, state the direction of propagation for the phase kz − ωt.

Exercise 3: recover B from a specified electric plane wave

In source-free vacuum let

(12)

Assume the corresponding magnetic field has the form

(13)

Use Faraday’s law,

(14)

to determine B₀ in terms of E₀, k, and ω. Then impose the vacuum dispersion relation and show that

(15)

Use the field directions to verify that E × B points in the direction of propagation.

Exercise 4: radio-frequency wave from magnetic amplitude

A plane electromagnetic wave in vacuum has frequency

(16)

and magnetic-field amplitude

(17)

The wave propagates in the +z direction and the magnetic field points in the +y direction when the cosine factor is positive.

Find:

1. the wavelength λ;
2. the angular frequency ω;
3. the Wavenumber k;
4. the electric-field amplitude E₀;
5. the direction of the electric field.

Use c = 299792458 m/s.

Exercise 5: determine propagation direction from field orientation

At a certain phase of a plane wave,

(18)

Determine the propagation direction. Then write one possible phase factor for a monochromatic wave with positive k and positive ω.

Exercise 6: reject a longitudinal candidate using Gauss’s law

Someone proposes the source-free vacuum field

(19)

The field satisfies the scalar wave equation if ω = ck.

Compute ∇⋅ E and determine whether the field can nevertheless be a source-free electromagnetic plane wave. Explain why satisfying the wave equation alone is not enough to guarantee that a field satisfies the complete Maxwell system.

Exercise 7: full Maxwell consistency with unknown amplitude and phase

Consider

(20)

and

(21)

Use Faraday’s law and the vacuum Ampere–Maxwell law separately to determine:

1. the allowed relative phase δ;
2. the amplitude ratio E₀∕B₀;
3. the dispersion relation.

Show that both curl equations are required for the complete result.

Exercise 8: arbitrary linear polarization in the transverse plane

A wave propagates in the +z direction with electric field

(22)

Use

(23)

to obtain the magnetic field explicitly. Then prove that

(24)

and that

(25)

where

(26)

Exercise 9: build a standing electromagnetic wave from two traveling waves

Two equal-amplitude plane waves propagate in opposite directions:

(27)

(28)

1. Find the corresponding magnetic fields B₁ and B₂.
2. Add the electric fields and simplify using trigonometric identities.
3. Add the magnetic fields and simplify.
4. Find the positions of the electric-field nodes.
5. Find the positions of the magnetic-field nodes.
6. Show that an electric node is displaced by λ∕4 from the nearest magnetic node.

Exercise 10: a Gaussian electromagnetic pulse

Let

(29)

1. Verify explicitly that E_x satisfies the one-dimensional vacuum wave equation.
2. Find the position of the pulse maximum as a function of time.
3. Find the magnetic field for a +z-propagating source-free pulse.
4. Explain why the pulse is not monochromatic even though every point in the pulse propagates at the same speed c in vacuum.

Exercise 11: verify the curl–curl identity for a concrete vector field

Consider

(30)

Compute both sides of

(31)

explicitly and verify that they agree.

This is the identity used in EM16 to turn the two first-order curl equations into second-order wave equations.

Exercise 12: derive the electric wave equation with sources

Do not assume ρ = 0 or J = 0. Start from

(32)

(33)

and

(34)

Take one additional curl and derive

(35)

Identify the physical source terms on the right-hand side.

Exercise 13: derive the magnetic wave equation with sources

Starting from the general Ampere–Maxwell equation, take its curl and use

(36)

and Faraday’s law to derive

(37)

Explain why the magnetic source equation contains the curl of current density rather than a magnetic-charge-density term.

Exercise 14: from the wave equation to the Helmholtz equation

Suppose a source-free electric field is time harmonic and is represented by

(38)

Starting from

(39)

show that the complex spatial amplitude obeys

(40)

where

(41)

Explain why this conversion is useful in RF, antenna, optics, and steady-state sinusoidal problems.

Exercise 15: numerical finite-difference check of the wave equation

Let

(42)

Use centered second differences,

(43)

and

(44)

Define the numerical residual

(45)

1. Write a Julia program that evaluates R at a fixed point.
2. Choose Δt = 0.8Δz∕c and refine Δz by factors of two.
3. Verify that the normalized residual decreases by approximately a factor of four when the spacing is halved.
4. Explain why that factor is expected for centered second differences.

Part II: Complete Worked Solutions

Solution 1: verify the general traveling profile F(z − ct)

Define

(46)

Then

(47)

Differentiate with respect to z:

	= F′(ξ)	(48)
	= F′(ξ).	(49)

Differentiating once more,

(50)

Now differentiate with respect to time:

	= F′(ξ)	(51)
	= −cF′(ξ).	(52)

Therefore,

	= −cF′(ξ)	(53)
	= −cF′′(ξ)(−c)	(54)
	= c2F′′(ξ).	(55)

Substitute into the wave equation:

−	= F′′(ξ) −c2F′′(ξ)	(56)
	= 0.	(57)

Thus every twice-differentiable profile F(z − ct) satisfies the one-dimensional wave equation.

For

(58)

the same calculation gives

(59)

so G(z + ct) is also a solution.

To identify the direction, hold the profile argument constant. For F(z − ct),

(60)

so

(61)

As time increases, the same feature moves toward increasing z. Therefore F(z − ct) propagates in the +z direction.

For G(z + ct),

(62)

so the profile propagates in the −z direction.

The shape is preserved because the function itself is not changing; only its argument is translated in space.

Solution 2: sinusoidal trial solution and the dispersion relation

Start with

(63)

The second spatial derivative is

(64)

The second time derivative is

(65)

Substitute into the wave equation:

− k2E 0 cos(kz − ωt) −

= 0.

(66)

For a nontrivial wave,

(67)

Therefore,

(68)

For positive frequency and positive wavenumber,

(69)

Hence the phase velocity is

(70)

Using

(71)

we obtain

(72)

so

(73)

Finally, a surface of constant phase satisfies

(74)

which gives

(75)

Thus kz − ωt describes propagation in the +z direction.

Solution 3: recover B from a specified electric plane wave

The electric field is

(76)

Because only E_x is nonzero and it depends only on z,

(77)

Since

(78)

we have

(79)

The proposed magnetic field is

(80)

Its time derivative is

(81)

Faraday’s law requires

(82)

Therefore,

(83)

For a vacuum wave,

(84)

so

(85)

The cross product is

(86)

Thus the field orientation is consistent with propagation in the +z direction.

Solution 4: radio-frequency wave from magnetic amplitude

The frequency is

(87)

The wavelength is

λ	=	(88)
	= m	(89)
	≈ 0.244210 m.	(90)

Therefore,

(91)

The angular frequency is

ω	= 2πf	(92)
	= 2π(1.22760 × 109)	(93)
	≈ 7.71324 × 109 rad/s.	(94)

Thus,

(95)

The wavenumber is

k	=	(96)
	≈ 25.7286 rad/m.	(97)

Therefore,

(98)

For a vacuum plane wave,

(99)

With

(100)

we obtain

E0	= (299792458)(2.50 × 10−9)	(101)
	≈ 0.74948 V/m.	(102)

Hence,

(103)

The wave travels in +z and B points in +y. We need

(104)

Since

(105)

the electric field points in the +x direction when the cosine factor is positive.

Solution 5: determine propagation direction from field orientation

For a source-free plane wave, the propagation direction follows the direction of

(106)

Here

(107)

Therefore,

E × B	∥y × (−z)	(108)
	= −x.	(109)

Thus the wave propagates in the negative x direction:

(110)

A convenient phase for propagation in the −x direction is

(111)

Indeed, holding phase constant gives

(112)

or

(113)

The constant-phase surface therefore moves toward decreasing x.

Solution 6: reject a longitudinal candidate using Gauss’s law

The proposed field is

(114)

Its divergence is

∇⋅ E	=	(115)
	= −kE0 sin(kz − ωt).	(116)

This is generally nonzero. But in source-free vacuum, Gauss’s Law requires

(117)

Therefore the proposed longitudinal field is not an allowed source-free electromagnetic plane wave.

The important lesson is that the second-order wave equation is a consequence of Maxwell’s equations, but by itself it does not encode every Maxwell constraint. A candidate field can satisfy

(118)

while failing

(119)

Therefore a complete electromagnetic solution must satisfy both the wave equation and the relevant Maxwell constraints.

Solution 7: full Maxwell consistency with unknown amplitude and phase

Let

(120)

Then

(121)

and

(122)

Faraday’s law gives

(123)

Meanwhile,

(124)

Thus

(125)

for every z and t. The simplest nontrivial possibility is that the fields are in phase,

(126)

with

(127)

Now apply Ampere–Maxwell. Since only B_y depends on z,

(128)

For δ = 0,

(129)

so

(130)

The right-hand side is

μ0𝜖0

= μ0𝜖0ωE0 sin 𝜃x.

(131)

Therefore,

(132)

Use the Faraday result

(133)

Substitution gives

(134)

Cancel E₀ and multiply by ω:

(135)

Hence,

(136)

Thus

(137)

Finally,

(138)

or

(139)

Faraday’s law determines the relative amplitude and phase relation once ω∕k is known. Ampere–Maxwell supplies the second relation needed to determine the vacuum dispersion relation itself.

Solution 8: arbitrary linear polarization in the transverse plane

The electric field is

(140)

For propagation in +z,

(141)

Use

(142)

Therefore,

B

= cos(kz − ωt).

(143)

Thus

(144)

Now take the dot product of the amplitude vectors:

E0 ⋅ B0	= ⋅	(145)
	=	(146)
	= 0.	(147)

Therefore,

(148)

The magnetic amplitude is

B0	=	(149)
	= .	(150)

Hence,

(151)

This result does not depend on which transverse direction the linear polarization chooses.

Solution 9: build a standing electromagnetic wave from two traveling waves

For the +z wave,

(152)

The associated magnetic field is

(153)

because

(154)

For the −z wave,

(155)

Now E₂ × B₂ must point in −z, so the magnetic field reverses direction:

(156)

Add the electric fields:

E

= E0 x.

(157)

Using

(158)

we obtain

(159)

Now add the magnetic fields:

B

= y.

(160)

Using

(161)

we obtain

(162)

Electric nodes occur where

(163)

Thus

(164)

or

(165)

Magnetic nodes occur where

(166)

so

(167)

and therefore

(168)

The nearest electric and magnetic nodes differ by

(169)

This spatial offset is a characteristic feature of a standing electromagnetic wave.

Solution 10: a Gaussian electromagnetic pulse

Define

(170)

Then

(171)

This is exactly the form F(z − ct) from Exercise 1, so it must satisfy the one-dimensional wave equation. We can also verify it directly.

First,

(172)

Differentiating again,

(173)

For the time derivative,

(174)

Differentiating again gives

(175)

Therefore,

−

= 0.

(176)

The pulse maximum occurs when the exponent is zero:

(177)

Thus

(178)

For a +z-propagating pulse with E ∥x,

(179)

Hence

(180)

The Gaussian pulse is not monochromatic because it is localized in space. A localized waveform requires a superposition of many wavenumbers and therefore many frequencies. In nondispersive vacuum, all of those spectral components satisfy ω = ck and therefore propagate with the same speed c, so the pulse retains its shape.

Solution 11: verify the curl–curl identity for a concrete vector field

The vector field is

(181)

First compute its curl:

∇× A	=	(182)
	= 0x − zy + yz.	(183)

Now curl that result:

∇× (∇× A)	= ∇×	(184)
	= 2x.	(185)

So the left-hand side is

(186)

Now compute the right-hand side. The divergence is

∇⋅ A	= + +	(187)
	= 2x + x + x	(188)
	= 4x.	(189)

Therefore,

(190)

The vector Laplacian is obtained component by component:

(191)

Thus

(192)

Hence

∇(∇⋅ A) −∇2A	= 4x − 2x	(193)
	= 2x.	(194)

Therefore,

(195)

The identity is verified for this explicit field.

Solution 12: derive the electric wave equation with sources

Begin with Faraday’s law:

(196)

Take the curl of both sides:

(197)

Use the curl–curl identity:

(198)

Gauss’s law gives

(199)

so

(200)

Ampere–Maxwell gives

(201)

Therefore,

−(∇× B)

= −μ0 − μ0𝜖0.

(202)

Substitute both results:

(203)

Rearrange:

(204)

The source terms are therefore:

• the spatial gradient of charge density, (1∕𝜖₀)∇ρ;
• the time variation of current density, μ₀∂J∕∂t.

When both vanish in the local propagation region, the equation reduces to the homogeneous vacuum wave equation derived in EM16.

Solution 13: derive the magnetic wave equation with sources

Start from Ampere–Maxwell:

(205)

Take the curl:

(206)

Use the curl–curl identity:

(207)

Gauss’s law for magnetism gives

(208)

so the left side becomes

(209)

Faraday’s law gives

(210)

Therefore,

(211)

Substitute:

(212)

Multiply by −1 and rearrange:

(213)

There is no term proportional to a magnetic charge density because Maxwell’s classical theory contains

(214)

rather than a magnetic analogue of

(215)

Thus magnetic-wave source structure enters through circulating electric current, represented by ∇× J.

Solution 14: from the wave equation to the Helmholtz equation

Represent the real physical field as

(216)

Because the wave equation is linear, it is sufficient to work with the complex field

(217)

The Laplacian acts only on the spatial amplitude:

(218)

The second time derivative is

	= (−iω)2Ee−iωt	(219)
	= −ω2Ee−iωt.	(220)

Substitute into

(221)

This gives

(222)

Since the exponential factor is never zero,

(223)

Define

(224)

Then

(225)

This is the vector Helmholtz equation in a homogeneous source-free region.

The time-domain wave equation describes general transients and arbitrary time dependence. The Helmholtz equation removes the known sinusoidal time dependence and leaves a purely spatial boundary-value problem. That makes it especially useful for steady-state RF fields, antennas, resonators, diffraction, and monochromatic optics.

Solution 15: numerical finite-difference check of the wave equation

A direct Julia implementation is:

using Printf

c = 299_792_458.0
lambda = 1.0
k = 2pi / lambda
omega = c * k
E0 = 1.0

E(z, t) = E0 * cos(k*z - omega*t + 0.37)

z0 = 0.23
t0 = 0.17 / c

println(" N normalized residual")

for N in (40, 80, 160, 320)
dz = lambda / N
dt = 0.8 * dz / c

dzz = (E(z0 + dz, t0) - 2E(z0, t0) + E(z0 - dz, t0)) / dz^2
dtt = (E(z0, t0 + dt) - 2E(z0, t0) + E(z0, t0 - dt)) / dt^2

R = dzz - dtt / c^2
Rnorm = abs(R) / (k^2 * abs(E(z0, t0)))

@printf("%4d %.6e\n", N, Rnorm)
end

The centered second-difference formulas are second-order accurate:

(226)

and

(227)

With

(228)

both truncation errors scale as (Δz)².

For the test point and phase used above, representative normalized residuals are approximately

(229)

for N = 40, 80, 160, 320, respectively.

Each time N doubles, Δz is halved. A second-order error should therefore change by approximately

(230)

That is exactly the trend in the numerical residual. The finite-difference experiment therefore checks two things at once: the analytic plane wave satisfies the continuum wave equation, and the chosen numerical approximation converges at the expected second-order rate.

1 What EM16E1 adds to the series

EM16 derived the electromagnetic wave equation and its basic plane-wave consequences. EM16E1 turns those results into working tools.

The main progression is

(231)

(232)

(233)

The most important conceptual point is that the scalar-looking wave equation is necessary but not sufficient for a physical electromagnetic field. The complete field must also satisfy the Maxwell divergence and curl equations, which enforce transversality, field orientation, phase relationships, and the amplitude relation E₀ = cB₀.

A natural next topic is electromagnetic energy and power flow: electric and magnetic energy density, the Poynting vector, intensity, and inverse-square spreading.

References