1) Standing waves are produced by the superposition of two waves of equal amplitude and wavelength travelling in opposite directions. The displacements of the two waves add vectorially, resulting in areas of constructive and destructive interference. 2) On a string fixed at both ends, the fundamental mode has a node at each end and an antinode in the middle. Higher harmonics have additional nodes and antinodes appearing at regular intervals, with frequencies that are odd integer multiples of the fundamental frequency. 3) In pipes closed at both ends, the patterns of nodes and antinodes are similar to those on strings, with the fundamental mode having a node at each end and an antinode in the middle. Higher harmon

1. presents
a production
Standing
Waves
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Image: Chladni design

2. Superposition
The displacements
of two waves
travelling at the
plus same place add
vectorially to
produce a
resultant.
equals
What will be the
resultant when the
waves have each
travelled 1/4 of a
wavelength further?

3. Waves in 1D
Wave from left
Standing wave Wave from right
where they cross
The black dotted line shows the vector addition of
the red and blue waves. Identify the positions where
displacement is max and where it is zero.

4. Waves on a string
fundamental mode
N A N f = v/2L
Distance L
N A N A N
2nd harmonic f = v/L
3rd harmonic
N A N A N A N
f = 3v/2L
N = node, A = antinode,
position of zero position of max. Draw the next
amplitude amplitude 2 modes

5. Waves in a pipe
fundamental mode
N Distance L A
f = v/4L
3rd harmonic
N A N A
f = 3v/4L
5th harmonic
N A N A N A
f = 5v/4L
Here, the node is a position of zero Draw the next
amplitude but also maximum pressure 2 modes

6. Amplitude and phase
For this standing wave on a string, each point
has a different amplitude to its neighbour.
All the points between two adjacent
nodes are in phase with each other.
The points on the left side are π rad out of
phase with the points on the right side.

7. Standing vs travelling
Standing wave Travelling wave
The wave shape The wave shape
doesn’t move progresses
Neighbouring points Neighbouring points
have different amplitudes have the same amplitude
Neighbouring points Neighbouring points
have the same phase have different phase
It stores energy It transmits energy

8. a production
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