1) Standing waves are formed by the superposition of two waves traveling in opposite directions. They have points called nodes where the disturbance is always zero and amplitudes vary along the waveform. 2) Standing waves on a string or open pipe have harmonic frequencies where the wavelength is related to the length of the string or pipe by integer multiples. Closed pipes have a different harmonic series. 3) The fundamental frequency and harmonics determine the quality of the sound from different instruments. Resonance occurs at the natural harmonic frequencies of the system.

1. Topic 4.5 Standing Waves

2. Standing Waves
The Formation

3. Wave Reflection

4. Reflection from fixed or free end

5. When 2 waves of
the same speed and wavelength
and equal or almost equal amplitudes
travelling in opposite directions
meet, a standing wave is formed
Standing Waves

6. Standing Waves
The standing wave is the result of the
superposition of the two waves travelling
in opposite directions
The main difference between a standing
and a travelling wave is that in the case of
the standing wave no energy or
momentum is transferred down the string

7. Standing Waves
A standing wave is characterised by the
fact that there exists a number of points
at which the disturbance is always zero.
These points are called Nodes and the
amplitudes along the waveform is
different
In a travelling wave there are no points
where the disturbance is always zero, all
the points have the same amplitude.

8. At the correct frequency
a standing wave is formed
The frequency is increased
until a different standing
wave is formed
Standing Waves

9. Standing wave

11. Standing wave

13. Nodes and antinodes
λ
Hyperlink

15. • If you take a wire and stretch it between two
points then you can set up a standing wave
• The travelling waves are reflected to and fro
between the two ends of the wire and
interfere to produce the standing wave
•This has a node at both ends
• and an antinode in the middle
•– it is called the fundamental

16. • With this wave the length of the string is
equal to half the wave length
• L = ½ 
•   = 2L
• As v = f 
• Then f = v / 
•  f = v / 2L
• This is the fundamental frequency of the
string (the 1st harmonic)

17. • This is not the only standing wave that can
exist on the string
• The next standing wave is
This is called the 2nd Harmonic
L = 

18. • With this wave the length of the string is
equal to the wave length
• L = 
•   = L
• As v = f 
• Then f = v / 
•  f = v / L
• This is the 2nd Harmonic frequency of the
string
• Notice it is twice the fundamental frequency

19. • The next standing wave is
L = 3/2 
This is called the 3rd Harmonic

20. • With this wave the length of the string is equal to
3/2 of the wave length
• L =3/2 
•   = 2/3L
• As v = f 
• Then f = v / 
•  f = v / 2/3L
•  f = 3v / 2L
• This is the 3rd Harmonic frequency of the string
• Notice it is three times the fundamental frequency

21. • Notice that the only
constraint is that the ends
of the string are nodes.
• In general we find that the
wavelengths satisfy
 = 2L
n
Where n = 1,2,3,4……

22. • This is the harmonic series
• The fundamental is the dominant vibration
and will be the one that the ear will hear
above all the others
• The harmonics effect the quality of the note
• It is for this reason that different musical
instruments sounding a note of the same
frequency sound different
• (it is not the only way though)

23. Resonance and Standing Waves
2. Standing Waves in Pipes

24. • Sound standing waves are also formed
in pipes
• Exactly the same results apply
• There are two types of pipes
–1. Open ended
–2. Closed at one end
• Nodes exist at closed ends
• Antinodes exists at open ends

25. a) Open Ended
• Fundamental Frequency (1st Harmonic)
L =  /2
•  = 2L
•As v = f 
•Then f = v / 
• f = v / 2L

26. • 2nd Harmonic
•  = L
•As v = f 
•Then f = v / 
• f = v / L
L = 

27. • 3rd Harmonic
•  = 2/3L
•As v = f 
•Then f = v / 
• f = v / 2/3L
• f = 3v / 2L
L = 3/2

28. • The harmonics are in the same series
as the string series
• If the fundamental frequency = f
• Then the 2nd harmonic is 2f, 3rd is 3f and
the 4th is 4f… etc

29. b) Closed at one End
• Fundamental Frequency (1st Harmonic)
L =  /4
•  = 4L
•As v = f 
•Then f = v / 
• f = v / 4L

30. • Next Harmonic
•  = 4/3L
•As v = f 
•Then f = v / 
• f = v / 4/3L
• f = 3v / 4L
L = 3/4

31. • And the next harmonic
•  = 4/5L
•As v = f 
•Then f = v / 
• f = v / 4/5L
• f = 5v / 4L
L = 5/4

32. • The harmonics are DIFFERENT to the
string and open pipe series
• If the fundamental frequency = f
• Then there is no 2nd harmonic
• The 3rd is 3f
• There is no 4th harmonic
• The 5th is 5f

33. Modes of vibration in strings
λ
λ
Hyperlink

34. Harmonics

35. bench
1
2
3
5
7
8
9
Frequency Adjust
1
10
100
1000
10
100
1000
Frequency range
Outputs
A
power
10Hz 100kHz
1kHz 10kHz100Hz
Frequency
55 Hz
Wave
2 m
15 cm
vibration
generator
signal generator
rubber cord (4 mm2
)
Set up this experiment and
produce the first 8 standing
waves. Record the wavelength
for each one. Record the
frequency for each resonant
standing wave. Plot a suitable
graph to determine the
relationship between frequency
and wavelength.

36. A 0.3 m section of discarded garden hose will produce a trumpet sound when
blown as one blows a trumpet. Changing the length will change the pitch of the
"trumpet".

38. Measurement of velocity of sound
1. Measure difference in length between
2 successive resonances.
2. Use this distance to calculate the
wavelength.
3. Use this value and the frequency of
the tuning fork to calculate the speed
of sound.

39. Hyperlink
Travelling vs Standing Waves

40. Stationary wave Traveling wave
Amplitude
Frequency
Wavelength
Phase
Energy
Complete the table

41. Stationary wave Traveling wave
Amplitude All points have different amplitudes.
Maximum at the antinodes.
All points have the same amplitude.
Frequency Same for all points on the wave. Same for all points on the wave.
Wavelength Double the distance between 2 nodes. Distance between 2 successive points in
phase.
Phase All points between 2 nodes are in phase. All points along 1 wavelength have a
different phase.
Energy Energy is not transmitted by the wave, but
contained within it.
Energy is transmitted by the wave.