Doppler Effect Problem Solving - University of Cape Town

Doppler Effect
Problem Solving
Dr Spencer Wheaton
Dept. of Physics
University of Cape Town

What is the Doppler Effect?
Sound:
“A change in frequency heard by a
listener due to relative motion between
the sound source and the listener”
Light:
“A change in colour seen by an observer
due to relative motion between the light
source and the observer”

So the concept of relative motion is crucial!
Two objects have relative velocity if the distance
between them is changing!
Complete the table below:
A
B
C
D
E
F
G
Velocity of
source
Velocity of
observer
Doppler
effect
occurs?

A) Moving Source Only
Stationary Sound Source:
v
fL= fS v± vS
Moving Sound Source:
+ : source away
− : source towards
The wave crests bunch
up in front and spread
out behind

Question A1
(Qualitative including frequency change with time)
A music fan at a swimming pool is listening to a radio on a
diving platform. The radio is playing a constant frequency
tone when this fellow, clutching his radio, jumps off.
Describe the Doppler effect heard by a) a person left
behind on the platform, and b) a person down below
floating on a rubber raft. In each case, specify 1) whether
the observed frequency is constant, and 2) how the
observed frequency changes during the fall, if it does
change. Give your reasoning.

Question A2
(Solve for speed)
A bird is flying directly toward a stationary bird-watcher
and emits a frequency of 1250 Hz. The bird-watcher,
however, hears a frequency of 1290 Hz. What is the speed
of the bird?

Question A3
(Solve for speed and direction)
A bat locates insects by emitting ultrasonic “chirps” and
then listening for echoes from the bugs. Suppose a bat
chirp has a frequency of 25 kHz. How fast would the bat
have to fly, and in what direction, for you to just barely be
able to hear the chirp at 20 kHz?

Question A4
(Two simultaneous equations -f s and v s unknown)
Standing on a pavement, you hear a frequency of 560 Hz
from the siren of an approaching ambulance. After the
ambulance passes, the observed frequency of the siren is
480 Hz. Determine the ambulance’s speed from these
observations.

Question A5
(Interpretation of graph)
You are standing at x = 0 m, listening to a sound that is
emitted at frequency f 0 . The graph above shows the
frequency you hear during a 4-second interval. Which of the
following describes the sound source? Explain your choice.
It moves from left to right and passes you at t = 2s.
It moves from right to left and passes you at t = 2s.
It moves toward you but doesn’t reach you. It then reverses
direction at t =2 s.
It moves away from you until t = 2 s. It then reverses
direction and moves toward you but doesn’t reach you.

B) Moving Listener
f f
v L t
v L
v± v
L
L= S ( )
v
The wavecrests do not
bunch up or spread out!
+ : listener towards
− : listener away

Question B1
(Simple “plug and chug”)
The frequency of a certain police car’s siren is 1550 Hz
when at rest. What frequency do you detect if you move
with a speed of 30.0 m/s a) toward the car, and b) away
from the car?

Question B2
(Simple qualitative)
A large church has part of the organ in the front of the
church and part in the back. A person walking rapidly down
the aisle while both segments are playing at once reports
that the two segments sound out of tune. Why?

Question B3
(Ties together previous concepts of waves)
A source S generates circular waves on the surface of a
lake; the pattern of wave crests is shown in the figure
below. The speed of the waves is 5.5 m/s, and the crestto-crest
separation is 2.3 m. You are in a small boat
heading directly toward S at a constant speed of 3.3 m/s
with respect to the shore. What frequency of waves do you
observe?
S

Question B4
(Compares moving source with moving listener)
Suppose that George blows a whistle and Gloria hears it.
She will hear an increased frequency whether she is
running toward George or George is running toward her.
Are the increases in frequency the same in each case?
Assume the same running speeds.

C: Reflections Involving Two-step
Application of Equations with Only One of
Source or Listener Moving at a Time
Key Idea: When a wave reflects off a surface
the surface acts as a source of waves of
frequency equal to that frequency that a
listener moving in the same way as the surface
would hear!

Question C1
(Simple - led through steps)
A toy rocket moves at a speed of 242 m/s directly toward
a stationary pole (through stationary air) while emitting
sound waves at frequency f = 1250 Hz.
a) What frequency f’ is sensed by a detector that is
attached to the pole?
b) Some of the sound reaching the pole reflects back to
the rocket, which has an onboard detector. What
frequency f’’ does it detect?

Question C2
(Harder - not led in steps)
A stationary motion detector sends sound waves of
0.150 MHz toward a truck approaching at a speed of
45.0 m/s. What is the frequency of the waves reflected
back to the detector?

Question C3
(2 simultaneous equations)
A 2.00 MHz sound wave travels through a pregnant
woman’s abdomen and is reflected from the fetal heart
wall of her unborn baby. The heart wall is moving toward
the sound receiver as the heart beats. The reflected
sound is then detected by the detector and has a
frequency that differs from that emitted by 85 Hz. The
speed of sound in body tissue is 1540 m/s. Calculate the
speed of the fetal heart wall at the instant this
measurement is made?

D: Both Source and Listener Moving
moving source moving listener
v
fL= fS v± vS
NB: applies only in frame
where medium is at rest!
v± v
f = f L
L S
v± vS
f = f
L S
v± vL
( )
v
+ listener towards
− listener away
+ source away
− source towards

Question D1
(Simple plug and chug)
A railroad train is travelling at 30.0 m/s in still air. The
frequency of the note emitted by the train whistle is
262 Hz. What frequency is heard by a passenger on a
train moving in the opposite direction to the first at
18.0 m/s and a) approaching the first? b) receding from
the first?

Question D2
(Solve for v - re-arrange equation)
An ambulance with a siren emitting a whine at 1600 Hz
overtakes and passes a cyclist pedalling a bike at 2 m/s.
After being passed, the cyclist hears a frequency of
1590 Hz. How fast is the ambulance moving?

Question D3
(Solve for v - re-arrange equation)
Two trucks travel at the same speed. They are far apart
on adjacent lanes and approach each other essentially
head-on. One driver hears the horn of the other truck at
a frequency that is 1.14 times the frequency he hears
when the trucks are stationary. At what speed is each
truck moving?

E) Doppler for light:
Question E1
(Qualitative)
An astronomer measures the Doppler change in frequency
for the light reaching the earth from a distant star. From
this measurement, explain how the astronomer can deduce
that the star is receding from the earth.

Question E2
(Qualitative)
The drawing shows three situations A, B and C in which an
observer and a source of electromagnetic waves are moving
along the same line. In each case the source emits a wave
of the same frequency. The arrows in each situation denote
velocity vectors relative to the ground and have the
indicated magnitudes, either v or 2v. Rank the frequencies
of the observed waves in descending order (largest first)
according to magnitude. Explain your reasoning.

F: Shock-waves:
Sound source moving
at the speed of sound
Sound source moving faster
than the speed of sound

A shock wave is produced continuously by any object that
moves through the air at supersonic speed, not only at the
instant that it “breaks the sound barrier”
A shock wave follows the object as long as it is
travelling at supersonic speed

The sound waves that combine to form the shock wave are
created by the motion of the object itself, not by any sound
source that the object may carry
A supersonic jet airplane may have very loud engines, but
these do not cause the shock wave

Question F1
(Qualitative)
A jet airplane is flying at a constant altitude at a steady
speed v S greater than the speed of sound. Describe what
is being heard by observers at points A, B, and C at the
instant shown below, when the shock wave has just
reached point B.

Question F2
(Application of Mach cone equation)
A jet plane passes overhead at a height of 5000 m and a
speed of Mach 1.5. a) Find the Mach cone angle. b) How
long after the jet has passed directly overhead will the
shock wave reach the ground?