Cambridge (CIE) AS Physics Your notes
Stationary Waves
Contents
The Principle of Superposition
Stationary Waves
Wavelength of Stationary Waves
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The Principle of Superposition
Your notes
The principle of superposition
When two or more waves arrive at the same point and overlap, their amplitudes combine
This is called superposition
The principle of superposition states that:
When two or more waves overlap at a point, the displacement at that point is equal to
the sum of the displacements of the individual waves
The superposition of surface water waves shows the effect of this overlap
There are areas of zero displacement, where the water is flat
There are areas of increased displacement, where the water waves are higher
The dogs make waves in the water which superimpose to give areas of both zero and
increased displacement.
It is possible to analyse superposition clearly when the waves are drawn on a vertical
displacement (amplitude)-displacement graph
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Your notes
Waves can superimpose so their amplitudes are added together often creating a larger
resultant amplitude
Interference is the effect of this overlap
This is explained in the next Interference & coherence
Individual wave displacements may be positive or negative and are combined in the
same way as other vector quantities
It is possible to analyse superposition clearly when the waves are drawn on a
displacement-time graph
Superposition can also be demonstrated with two pulses
When the pulses meet, the resultant displacement is also the algebraic sum of the
displacement of the individual pulses
After the pulses have interacted, they then carry on as normal
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Your notes
When two pulses overlap their displacements combine to form a resultant displacement
Worked Example
Two overlapping waves of the same types travel in the same direction. The variation
with x and y displacement of the wave is shown in the figure below.
Use the principle of superposition to sketch the resultant wave.
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Answer:
The graph of the superposition of both waves is in black: Your notes
To plot the correct amplitude at each point, sum the amplitude of both graphs at
that point
E.g. A point A, each graph has a value of 0.7. Therefore, the same point with
the resultant superposition is 2 × 0.7 = 1.4
Each square on the y-axis represents 0.2
Examiner Tips and Tricks
The best way to draw the superposition of two waves is to find where the
superimposed wave has its maximum and minimum amplitudes. It is then a case of
joining them up to form the wave. Where the waves intersect determines how much
constructive or destructive interference will occur.
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Stationary Waves
Your notes
Stationary waves
Stationary waves, or standing waves, are produced by the superposition of two waves
of the same frequency and amplitude travelling in opposite directions
This is usually achieved by a travelling wave and its reflection. The superposition
produces a wave pattern where the peaks and troughs do not move
Formation of a stationary wave
Formation of a stationary wave on a stretched spring fixed at one end
Stretched strings
Vibrations caused by stationary waves on a stretched string produce sound
This is how stringed instruments, such as guitars or violins, work
This can be demonstrated by a length of string under tension fixed at one end and
forced to vibrate due to an oscillator:
Standing wave experiment
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Your notes
Stationary wave on a stretched string kept taut by a mass and pulley system
As the frequency of the oscillator changes, standing waves with different numbers of
minima (nodes) and maxima (antinodes) form
Microwaves
A microwave source is placed in line with a reflecting plate and a small detector between
the two
The reflector can be moved to and from the source to vary the stationary wave pattern
formed
By moving the detector, it can pick up the minima (nodes) and maxima (antinodes) of the
stationary wave pattern
Stationary microwaves
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Using microwaves to demonstrate stationary waves
Air Columns Your notes
The formation of stationary waves inside an air column can be produced by sound waves
This is how musical instruments, such as clarinets and organs, work
This can be demonstrated by placing a loud speaker at the open end of an air column
with fine powder inside
At certain frequencies, the powder forms evenly spaced heaps along the tube, showing
where there is zero disturbance as a result of the nodes of the stationary wave
Stationary waves in an air column
Stationary waves can be seen in air columns using dry power
In order to produce a stationary wave, there must be a minima (node) at one end and a
maxima (antinode) at the end with the loudspeaker
Examiner Tips and Tricks
Always refer back to the experiment or scenario in an exam question e.g. the wave
produced by a loudspeaker reflects at the end of a tube. This reflected wave, with the
same frequency, overlaps the initial wave to create a stationary wave.
Formation of stationary waves
A stationary wave is made up of nodes and antinodes
Nodes are where there is no vibration
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Antinodes are where the vibrations are at their maximum amplitude
The nodes and antinodes do not move along the string.
Your notes
Nodes are fixed and antinodes only move in the vertical direction
Between nodes, all points on the stationary wave are in phase
The image below shows the nodes and antinodes on a snapshot of a stationary wave at a
point in time
Nodes and antinodes on a stationary wave
Nodes are points of zero amplitude, anti-nodes are points of maximum amplitude
L is the length of the string
1 wavelength λ is only a portion of the length of the string
Changing the frequency of the stationary wave produced will change the number of
nodes and antinodes produced and consequently the wavelength
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Your notes
Stationary waves are produced at varying frequencies
Worked Example
A stretched string is used to demonstrate a stationary wave, as shown in the diagram.
Which row in the table correctly describes the length of L and the name of X and Y?
Length L Point X Point Y
A 5 wavelengths Node Antinode
B 1 Antinode Node
2 wavelengths
2
C 1 Node Antinode
2 wavelengths
2
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D 5 wavelengths Antinode Node
Your notes
Answer: C
Step 1: Determine the number of wavelengths in the length of the string
1
The string has 2 wavelengths
2
This rules out A and D
Step 2: Determine points X and Y
X is a point of 0 displacement - a node
Y is a point of maximum displacement - an antinode
Therefore, the correct row is C
Examiner Tips and Tricks
The lengths of the strings will only be in terms of whole or ½ wavelengths. For example,
a wavelength could be made up of 3 nodes and 2 antinodes or 2 nodes and 3
antinodes.
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Wavelength of Stationary Waves
Your notes
Wavelength of stationary waves
Stationary waves have different wave patterns depending on the frequency of the
vibration and the situation in which they are created
Two fixed ends
When a stationary wave, such as a vibrating string, is fixed at both ends, the simplest
wave pattern is a single loop made up of two nodes and an antinode
This is called the fundamental mode of vibration or the first harmonic
The particular frequencies (i.e. resonant frequencies) of standing waves possible in the
string depend on its length L and its speed v
As you increase the frequency, the higher harmonics begin to appear
The frequencies can be calculated from the string length and wave equation
For a string of length L, the wavelength of the lowest harmonic is 2L
This is because there is only one loop of the stationary wave, which is a half
wavelength
2L
λn = n
2L
λ1 = 1
= 2L
The second harmonic has three nodes and two antinodes
So the wavelength is equal to the length of the string
2L
λ2 = 2
=L
The third harmonic has four nodes and three antinodes
2L
λ3 = 3
The nth harmonic has n antinodes and n + 1 nodes
The nth wavelength has a wavelength of
2L
n
The wavelengths and frequencies of the first three harmonics can be summarised as
follows:
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Wavelength and frequencies of different harmonics
Your notes
Diagram showing the first three modes of vibration of a stretched string with
corresponding frequencies
One or two open ends in an air column
When a stationary wave is formed in an air column with one or two open ends, slightly
different wave patterns are seen in each
Harmonics in an air column
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Your notes
Diagram showing modes of vibration in pipes with one end closed and the other open or
both ends open
Image 1 shows stationary waves in a column which is closed at one end
At the closed end, a node forms
At the open end, an antinode forms
Therefore, the fundamental mode is made up of a quarter wavelength with one node and
one antinode
Every harmonic after that adds on an extra node and antinode
Hence, only odd harmonics form
Image 2 shows stationary waves in a column which is open at both ends
An antinode forms at each open end
Therefore, the fundamental mode is made up of a half wavelength with one node and
two antinodes
Every harmonic after that adds on an extra node and an antinode
Hence, odd and even harmonics can form
In summary, a column length L for a wave with wavelength λ and resonant frequency f for
stationary waves to appear is as follows:
Air Column Length & Frequencies Summary Table
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Your notes
Worked Example
A standing wave is set up in a column of length L when a loudspeaker placed at one
end emits a sound wave of frequency f. The column is closed at the other end. The
speed of sound is 340 m s−1.
For a column of length 7.5 m, what is the wavelength of the second lowest note
produced?
Answer:
Step 1: Determine the positions of the nodes and antinodes
One end of the column is closed, and the loudspeaker represents an open end
Hence, an antinode forms at the loudspeaker (open end) and a node forms at the
closed end
The fundamental frequency represents the lowest note - this would be 1 node
and 1 antinode
So, the second-lowest note must have 2 nodes and 2 antinodes
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Your notes
Step 2: Write an expression for the length of the sound wave in the column
3
In the column, there is a quarter wavelength and a half wavelength, or λ
4
Therefore the length of the column is:
3λ
L = 4
nλ
Note: for a column with an open and closed end, L = 4
, this would
represent the third harmonic (n = 3)
Step 3: Determine the wavelength of the second lowest note
4L 4 × 7.5
λ = 3
= 3
= 10 m
Examiner Tips and Tricks
The fundamental counts as the first harmonic or n = 1 and is the lowest frequency with
half or quarter of a wavelength. A full wavelength with both ends open or both ends
closed is the second harmonic. Make sure to match the correct wavelength with the
harmonic asked for in the question!
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