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Chapter 21 – Mechanical Waves
A PowerPoint Presentation by
Paul E. Tippens, Professor of
Physics
Southern Polytechnic State
University
© 2007
Mechanical Waves
A mechanical wave is a physical
disturbance in an elastic medium.
Consider a stone dropped into a 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, T, is the time
for one complete
oscillation. (seconds,s)
Frequency, f, is the
number of complete
oscillations per
second. Hertz (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: T = 0.500 s
1
1
f  
T 0.500 s
Frequency: f = 2.00 Hz
A Transverse Wave
In a transverse wave, the vibration of the
individual particles of the medium is
perpendicular to the direction of wave
propagation.
Motion of
particles
Motion of
wave
Longitudinal Waves
In a longitudinal wave, the vibration of the
individual particles is parallel to the
direction of wave propagation.
v
Motion of
particles
Motion of
wave
Water Waves
An ocean wave is a combination of transverse and
longitudinal.
The individual particles
move in ellipses as the
wave disturbance moves
toward the shore.
Wave speed in a string.
The wave speed v in
a vibrating string is
determined by the
tension F and the
linear density m, or
mass per unit length.
v
F
m

FL
m
L
m = m/L
v = speed of the transverse wave (m/s)
F = tension on the string (N)
m or m/L = mass per unit length (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
v = 28.0 m/s
200 g
Note: Be careful to use consistent units. The
tension F must be in newtons, the mass m in
kilograms, and the length L in 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 l as illustrated.
l
A
B
Wavelength l is distance between two
particles that are in phase.
Velocity and Wave Frequency.
The period T is the time to move a distance of
one wavelength. Therefore, the wave speed is:
v
l
T
1
but T 
f
so
v fl
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 fl
Production of a Longitudinal Wave
l
l
• 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
l
Frequency f = waves
per second (Hz)
v
s
t
Velocity v (m/s)
Wavelength l (m)
v fl
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
f = 120 Hz
The distance moved during
a time of one cycle is the
wavelength; therefore:
l = 0.20 m
v = fl
v = (120 Hz)(0.2 m)
v = 24.0 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 = m/L
v
E
 2 2 f 2 A2 m
L
P  2 2 f 2 A2 m v
Example 3. A 2-m string has a mass of 300 g and
vibrates with a frequency of 20 Hz and an amplitude o
50 mm. If the tension in the rope is 48 N, how much
power must be delivered to the string?
m 0.30 kg
m 
 0.150 kg/m
L
2m
v
F
(48 N)

 17.9 m/s
m
0.15 kg/m
P  2 2 f 2 A2 m v
P = 22(20 Hz)2(0.05 m)2(0.15 kg/m)(17.9 m/s)
P = 53.0 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
ln 
n
n  1, 2, 3, . . .
Possible Frequencies f = v/l :
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
m

Characteristic
frequencies:
FL
m
and
n
fn 
2L
nv
f 
2L
F
m
; 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
m
; 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
m

v fl
FL
m
n
fn 
2L
E
 2 2 f 2 A2 m
L
F
m
1
f 
T
; n  1, 2, 3, . . .
P  2 2 f 2 A2 m v