Department of Electronics and Telecommunications,
Faculty of Information Technology, Mathematics and Electrical Engineering,
NTNU
Norwegian University of Science and Technology
TFE4130 Electromagnetic and Acoustic Waves
Problem Set 2
Problem 1
Consider a linear, isotropic, and homogeneous medium without sources (i.e., ρ = 0 and J = 0).
a) State the Maxwell equations in this medium, expressed with E and H. State the
equations twice: (i) For the physical, time-varying fields (time domaim), and (ii) for the
corresponding phasors (frequency domain).
b) Consider the frequency domain version of Maxwell’s equations. Prove that the two
divergence equations (Gauss’ law and the corresponding law for the magnetic field) are
redundant.
c) Use Maxwell’s equations in the time domain to obtain the wave equation
n2 ∂ 2 E
= 0,
c2 ∂t2
and identify the parameter n from the permittivity and permeability.
∇2 E −
(1)
d) Use Maxwell’s equations in the frequency domain to obtain the Helmholtz equation
∇2 E + k 2 E = 0.
(2)
Problem 2
In this problem we still consider a linear, isotropic, and homogeneous medium without sources.
a) If a waveform is described by f (t, z) = A cos(kz − ωt), in what direction does the wave
travel?
Set k = ω = 1 and draw the waveform for a few values of t on the range z ∈ [−2π, 2π].
Do the same for the waveform f (t, z) = A cos(kz + ωt). In what direction does this wave
travel?
b) Write the general solution to the one-dimensional wave equation
2
∂
1 ∂2
−
f (z, t) = 0.
∂z 2 c2 ∂t2
1
(3)
c) Write the Helmholtz equation for the electric field component in the x-direction, Ex .
Assume that Ex is only dependent on z. Find the general solution to the equation.
d) One solution to the Helmholtz equation is given as
Ex (z) = E0+ exp(−jkz).
(4)
What is the physical electric field (in the time domain)?
e) A plane wave is of the form
E(r) = E0 e−jk·r ,
(5a)
−jk·r
(5b)
H(r) = H0 e
,
with constant vector amplitudes E0 and H0 . Show that the electric and magnetic field
of the plane wave are both perpendicular to k. I.e., Eqs. (5) describe a transverse
electromagnetic wave.
Hint: Calculate the divergences ∇ · E and ∇ · H using the vector formula
∇ · (V A) = V ∇ · A + A · ∇V .
f ) Prove that E and H are perpendicular. Writing E = E x̂, H = H ŷ and k = kẑ, prove
that the wave impedance
E
(6)
η=
H
is
r
µ
η=
.
(7)
Hint: Consider one of the Maxwell curl equations, and use the vector formula
∇ × (V A) = (∇V ) × A + V ∇ × A.
Problem 3
a) The velocity of a locomotive is 100 km/t along a straight railway. If its whistle has
frequency f = 440 Hz, what is the frequency as heard by a stationary observer close to
the railway? Assume that the locomotive approaches the observer (and that the distance
between the railway and the observer is much less than the distance between the
locomotive and the observer).
The speed of sound in air is taken to be 340m/s.
b) The Doppler radar used by police to check the speed of a car, emits an electromagnetic
wave of frequency f . We assume that the road is straight, and that the radar is located
close to the road. After the radar signal is reflected by an approaching car, what will be
the detected frequency as seen by the radar? You may certainly assume that the velocity
is much less than the speed of light, to simplify the expression.
Hint: Since the car reflects the radar wave, you may imagine that it first detects the
wave (and thereby observes the frequency), and then emits a wave with this frequency.
2