Modified By Sangita Holkar Phy Dept

1. Stationary Waves
Stationary waves are produced by superposition
of two progressive waves of equal amplitude and
frequency, travelling with the same speed in
opposite directions.
http://www2.biglobe.ne.jp/~norimari/science/JavaEd/e-wave4.html

2. Production of Stationary Waves
A stationary wave would be set up by
causing the string to oscillate rapidly at a
particular frequency.
If the signal frequency is increased further,
overtone patterns appear.

3. Properties of a stationary wave (1)
Stationary waves have nodes where there is no
displacement at any time.
In between the nodes are positions called antinodes,
where the displacement has maximum amplitude.
λ
A vibrating loop
N A N A N
VibratorVibrator

4. Properties of a stationary wave (2)
The waveform in a stationary wave does not move
through medium; energy is not carried away from the
source.
The amplitude of a stationary wave varies from zero
at a node to maximum at an antinode, and depends
on position along the wave.

5. Vibrations of particles in a
stationary wave
At t = 0, all particles are at rest because
the particles reach their maximum
displacements.
At t = ¼T,
Particles a, e, and i are at rest because
they are the nodes.
Particles b, c and d are moving
downward.
They vibrate in phase but with different
amplitude.
Particles f, g and h are moving upward.
They vibrate in phase but with different
amplitude.
t = 0
t = ¼T
t = ¼T
t = ⅜T
t = ½T
a
b
c
d
e
f
g
h
ii
a
b c d
e
f g h
ii

6. Properties of a stationary wave (2)
All particles between two adjacent nodes
(within one vibrating loop) are in phase.
Video
1. Stationary waves (string)
2. Stationary waves (sound)

7. Standing waves in a string fixed at both ends.
Normal modes of a string
...3,2,1,n
2
==
n
L
nλ
...3,2,1,n
2
===
L
v
n
v
f
n
n
λ
Wavelength:
Frequency:
1
...3,2,1,n
T
2
fnf
L
n
f
n
n
⋅=
==
µ
µ
Tv =:Using
frequencylfundamentathecalledis
T
2
1
1
µL
f =

8. Standing waves in a string fixed at both ends.
f1 is called the fundamental frequency
The higher frequencies fn are integer
multiples of the fundamental frequency
These normal modes are called
harmonics.
f1 is the first harmonic, f2 is the second
harmonic and so on…

9. Modes of vibration of strings
Picture of Standing Wave Name Structure
1st Harmonic
or
Fundamental
1 Antinode
2 Nodes
2nd Harmonic
or
1st Overtone
2 Antinodes
3 Nodes
3rd Harmonic
or
2nd Overtone
3 Antinodes
4 Nodes
4th Harmonic
or
3rd Overtone
4 Antinodes
5 Nodes
5th Harmonic
or
4th Overtone
5 Antinodes
6 Nodes
L = ½λ1
f1 = v/2L
L = λ2
f2 = v/L
L = 1½λ3
f3 = 3v/2L
L = 2λ4
f4 = 2v/L
L = 2½λ5
f5 = 5v/2L
L

10. Investigating stationary waves using
sound waves and microwaves
Moving the detector along the line between the wave
source and the reflector enables alternating points of
high and low signal intensity to be found. These are the
antinodes and nodes of the stationary waves.
The distance between successive nodes or antinodes
can be measured, and corresponds to half the
wavelength λ.
If the frequency f of the source is known, the speed of
the two progressive waves which produce the stationary
wave can be obtained. Reflector
Detector
Wave source

11. Resonant Frequencies of a Vibrating
String
From the experiment, we find that
 There is a number of resonant frequencies
in a vibrating string,
 The lowest resonant frequency is called
the fundamental frequency (1st
harmonic),
 The other frequencies are called overtones
(2nd
harmonic, 3rd
harmonic etc.),
 Each of the overtones has a frequency
which is a whole-number multiple of the
frequency of the fundamental.

12. Factors that determine the fundamental
frequency of a vibrating string
The frequency of vibration depends on
 the mass per unit length of the string,
 the tension in the string and,
 the length of the string.
The fundamental frequency is given by
µ
T
L
fo
2
1
= where T = tension
µ = mass per unit length
L = length of string

13. Vibrations in Air Column
When a loudspeaker producing sound is
placed near the end of a hollow tube, the tube
resonates with sound at certain frequencies.
Stationary waves are set up inside the tube
because of the superposition of the incident
wave and the reflected wave travelling in
opposite directions.
http://www.walter-fendt.de/ph11e/stlwaves.htm

14. Factors that determine the fundamental
frequency of a vibrating air column
The natural frequency of a wind
instrument is dependent upon
 The type of the air column,
 The length of the air column of the instrument.
Open tube Closed tube

15. Standing Waves in a String,
3
This is the first normal mode
that is consistent with the
boundary conditions.
There are nodes at both ends.
There is one antinode in the
middle.
This is the longest wavelength
mode:
 ½λ1 = L so λ1 = 2L
The section of the standing
wave between nodes is called a
loop.
In the first normal mode, the
string vibrates in one loop. Section 18.3

16. Standing Waves in a String, 4
Consecutive normal modes add a loop at each step.
 The section of the standing wave from one node to the next is called a loop.
The second mode (c) corresponds to to λ = L.
The third mode (d) corresponds to λ = 2L/3.
Section 18.3

17. /4/4

18. Name
Modes of vibration for an open tube
Picture of Standing Wave Structure
1st Harmonic
or
Fundamental
2 Antinodes
1 Node
2nd Harmonic
or
1st Overtone
3 Antinodes
2 Nodes
3rd Harmonic
or
2nd Overtone
4 Antinodes
3 Nodes
4th Harmonic
or
3rd Overtone
5 Antinodes
4 Nodes
5th Harmonic
or
4th Overtone
6 Antinodes
5 Nodes
L = ½λ1
f1 = v/2L
L = λ2
f2 = v/L
L = 1½λ3
f3 = 3v/2L
L = 2λ4
f4 =2v/L
L = 2½λ5
f5 = 5v/2L

19. Modes of vibration for a closed tube
Picture of Standing Wave Name Structure
1st Harmonic
or
Fundamental
1 Antinode
1 Node
3rd Harmonic
or
1st Overtone
2 Antinodes
2 Nodes
5th Harmonic
or
2nd Overtone
3 Antinodes
3 Nodes
7th Harmonic
or
3rd Overtone
4 Antinodes
4 Nodes
9th Harmonic
or
4th Overtone
5 Antinodes
5 Nodes
L = ¼λ1
f1 = v/4L
L = ¾λ3
f3 =3v/4L
L = 1¼λ5
f5 =5v/4L
L = 1¾λ7
f7 = 7v/4L
L = 2¼λ9
f9 =9v/4L

20. The quality of sound (Timbre)
The quality of sound is determined by the
following factors:
 The particular harmonics present in addition to the
fundamental vibration,
 The relative amplitude of each harmonic,
 The transient sounds produced when the vibration is
started.
1st
overtone Fundamental
2nd
overtone
3rd
overtone
resultant
http://surendranath.tripod.com/Harmonics/Harmonics.html

21. Chladni’s Plate
Chladni’s plate is an example of resonance in
a plate.
There are a number of frequencies at which
the plate resonate. Each gives a different
pattern.

22. Resonance.
 All elastic objects have natural frequencies of
vibration that are determined by the materials they
are made of and their shapes.
 When energy is transferred at the natural
frequencies, there is a dramatic increase of
amplitude called resonance.
 The natural frequencies are also called resonant
frequencies.

23. When the frequency of an applied force, including
the force of a sound wave, matches the natural
frequency of an object, energy is transferred very
efficiently. The condition is called resonance.