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.
Stationary waves occur by resonance only at the natural frequencies of vibration of a medium.

2. 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.
Nodes
Antinodes λ

3. Properties of a stationary wave (2)
All particles between two adjacent nodes are in phase.
The waveform in a stationary wave does not move through medium; energy is not carried away from the source (i.e. the energy is localized).
The amplitude of a stationary wave varies from zero at a node to maximum at an antinode, and depends on position along the wave.

4. Applications of Stationary Wave
Vibration in strings
Vibration in air columns

5. 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.

6. 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.

7. 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
where T = tension
 = mass per unit length
L = length of string

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

9. 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.

10. 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

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

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

13. The quality of sound (Timbre) 1
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.
resultant
Fundamental
1st overtone
2nd overtone
3rd overtone

14. The quality of sound (Timbre) 2
the sound wave for the note middle C played on a flute, a piano, and a trumpet

15. 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
Wave source
Detector

16. Chladni’s Plate Chladni’s plate is an example of resonance in a 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.