Chapter 21 – Mechanical Waves
A PowerPoint Presentation by
Paul E. Tippens, Professor of Physics
Southern Polytechnic State University
©
2007
Objectives: After completion of this
module, you should be able to:
• Demonstrate your understanding of transverse and
longitudinal waves.
• Define, relate and apply the concepts of frequency,
wavelength, and wave speed.
• Solve problems involving mass, length, tension, and
wave velocity for transverse waves.
• Write and apply an expression for determining the
characteristic frequencies for a vibrating string with
fixed endpoints.
Mechanical Waves
A mechanical wave is a physical
disturbance in an elastic medium.
Consider a stone dropped into a lake.
lake
Energy is transferred from stone to floating log, but
only the disturbance travels.
Actual motion of any individual water particle is small.
Energy propagation via such a disturbance is known
as mechanical wave motion.
Periodic Motion
Simple periodic motion is that motion in which a
body moves back and forth over a fixed path,
returning to each position and velocity after a
definite interval of time.
1
f 
T
Amplitude
A
Period,
Period
Period,T,
T,isisthe
thetime
time
for
forone
onecomplete
complete
oscillation.
oscillation.(seconds,s)
(seconds,s)
Frequency,
Frequency
Frequency,f,f,isisthe
the
number
numberof
ofcomplete
complete
oscillations
oscillationsper
per
-1
second.
second.Hertz
Hertz(s(s-1))
Review of Simple
Harmonic Motion
x
F
It might be helpful for
you to review Chapter 14
on Simple Harmonic
Motion. Many of the same
terms are used in this
chapter.
Example: The suspended mass makes 30
complete oscillations in 15 s. What is the
period and frequency of the motion?
15 s
T
 0.50 s
30 cylces
x
F
Period:
Period: TT == 0.500
0.500 ss
1
1
f  
T 0.500 s
Frequency:
Frequency: ff == 2.00
2.00 Hz
Hz
A Transverse Wave
In
In aa transverse
transverse wave,
wave, the
the vibration
vibration of
of the
the
individual
individual particles
particles of
of the
the medium
medium isis
perpendicular
perpendicular to
to the
the direction
direction of
of wave
wave
propagation.
propagation.
Motion of
particles
Motion of
wave
Longitudinal Waves
In
In aa longitudinal
longitudinal wave
wave,, the
the vibration
vibration of
of the
the
individual
individual particles
particles isis parallel
parallel to
to the
the
direction
direction of
of wave
wave propagation.
propagation.
v
Motion of
particles
Motion of
wave
Water Waves
An
An ocean
ocean wave
wave isis aa combicombination
nation of
of transverse
transverse and
and
longitudinal.
longitudinal.
The
The individual
individual particles
particles
move
move in
in ellipses
ellipses as
as the
the
wave
wave disturbance
disturbance moves
moves
toward
toward the
the shore.
shore.
Wave speed in a string.
The
The wave
wave speed
speed vv in
in
aa vibrating
vibrating string
string isis
determined
determinedby
bythe
the
tension
tension FF and
and the
the
linear
linear density
density,, or
or
mass
massper
perunit
unitlength.
length.
v
F


FL
m
L
 = m/L
vv==speed
speedof
ofthe
thetransverse
transversewave
wave(m/s)
(m/s)
FF==tension
tensionon
onthe
thestring
string(N)
(N)
or
orm/L
m/L==mass
massper
perunit
unitlength
length(kg/m)
(kg/m)
Example 1: A 5-g section of string has a
length of 2 M from the wall to the top of a
pulley. A 200-g mass hangs at the end.
What is the speed of a wave in this string?
F = (0.20 kg)(9.8 m/s2) = 1.96 N
v
FL
(1.96 N)(2 m)

m
0.005 kg
vv == 28.0
28.0 m/s
m/s
200 g
Note:
Be careful
careful to
to use
use consistent
consistent units.
units.
Note: Be
The
, the
The tension
tension FF must
must be
be in
in newtons
newtons,
the mass
mass
m
, and
.
m in
in kilograms
kilograms,
and the
the length
length LL in
in meters
meters.
Periodic Wave Motion
A vibrating metal plate produces a
transverse continuous wave as shown.
For one complete vibration, the wave moves
a distance of one wavelength  as illustrated.

A
B
Wavelength is distance between two
particles that are in phase.
Velocity and Wave Frequency.
The
The period
period TT isis the
the time
time to
to move
move aa distance
distance of
of
one
one wavelength.
wavelength. Therefore,
Therefore, the
the wave
wave speed
speed is:
is:
v

T
but
1
T
f
so
v f
The frequency f is in s-1 or hertz (Hz).
The velocity of any wave is the product of
the frequency and the wavelength:
v f
Production of a Longitudinal Wave


• An oscillating pendulum produces condensations
and rarefactions that travel down the spring.
• The wave length l is the distance between
adjacent condensations or rarefactions.
Velocity, Wavelength, Speed

Frequency f = waves
per second (Hz)
v
s
t
Velocity v (m/s)
Wavelength  (m)
v f
Wave equation
Example 2: An electromagnetic vibrator
sends waves down a string. The vibrator
makes 600 complete cycles in 5 s. For one
complete vibration, the wave moves a
distance of 20 cm. What are the frequency,
wavelength, and velocity of the wave?
600 cycles
f 
;
5s
ff == 120
120 Hz
Hz
The distance moved during
a time of one cycle is the
wavelength; therefore:
 == 0.020
0.020 m
m
v = f
v = (120 Hz)(0.02 m)
vv == 2.40
2.40 m/s
m/s
Energy of a Periodic Wave
The energy of a periodic wave in a string is a
function of the linear density m , the frequency f,
the velocity v, and the amplitude A of the wave.
f
A
 = m/L
v
E
 2 2 f 2 A2 
L
P  2 2 f 2 A2  v
Example 3. A 2-m string has a mass of 300 g and
vibrates with a frequency of 20 Hz and an amplitude
of 50 mm. If the tension in the rope is 48 N, how
much power must be delivered to the string?
m 0.30 kg
 
 0.150 kg/m
L
2m
v
F
(48 N)

 17.9 m/s

0.15 kg/m
P  2 2 f 2 A2  v
P = 22(20 Hz)2(0.05 m)2(0.15 kg/m)(17.9 m/s)
PP==53.0
53.0W
W
The Superposition Principle
• When two or more waves (blue and green) exist in
the same medium, each wave moves as though the
other were absent.
• The resultant displacement of these waves at any
point is the algebraic sum (yellow) wave of the two
displacements.
Constructive Interference
Destructive Interference
Formation of a
Standing Wave:
Incident and reflected
waves traveling in
opposite directions
produce nodes N and
antinodes A.
The distance between
alternate nodes or antinodes is one wavelength.
Possible Wavelengths for Standing Waves
Fundamental, n = 1
1st overtone, n = 2
2nd overtone, n = 3
3rd overtone, n = 4
n = harmonics
2L
n 
n
n  1, 2, 3, . . .
Possible Frequencies f = v/:
Fundamental, n = 1
f = 1/2L
1st overtone, n = 2
f = 2/2L
2nd overtone, n = 3
f = 3/2L
3rd overtone, n = 4
f = 4/2L
n = harmonics
f = n/2L
nv
fn 
2L
n  1, 2, 3, . . .
Characteristic Frequencies
Now, for a string under
tension, we have:
v
F


Characteristic
frequencies:
FL
m
and
n
fn 
2L
nv
f 
2L
F

; n  1, 2, 3, . . .
Example 4. A 9-g steel wire is 2 m long
and is under a tension of 400 N. If the
string vibrates in three loops, what is the
frequency of the wave?
For three loops: n = 3
n
fn 
2L
3
f3 
2L
F

; n3
FL
3

m
2(2 m )
Third harmonic
2nd overtone
400 N
(400 N )(2 m )
0.009 kg
f3 = 224 Hz
Summary for Wave Motion:
v
F


v f
FL
m
n
fn 
2L
E
 2 2 f 2 A2 
L
F

1
f 
T
; n  1, 2, 3, . . .
P  2 2 f 2 A2  v
CONCLUSION: Chapter 21
Mechanical Waves