CIE AS Physics (9702) exams from 2022

Revision Notes

8.1.2 Stationary Waves

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

 

Stationary wave formation, downloadable AS & A Level Physics revision notes

Formation of a stationary wave on a stretched spring fixed at one end

 

  • In this section we will look at some few experiments that demonstrate stationary waves in everyday life

 

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 vibrations made by an oscillator:

Stationary wave string, downloadable AS & A Level Physics revision notes

Stationary wave on a stretched string

 

  • 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 wave microwave, downloadable AS & A Level Physics revision notes

Using microwaves to demonstrate stationary waves

 

Air columns

  • 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 fine powder inside the air column and a loudspeaker at the open end
  • 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

Air column stationary waves, downloadable AS & A Level Physics revision notes

Stationary wave in an air column

 

  • 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

 

Exam Tip

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 nodes and antinodes
    • Nodes are where there is no vibration
    • Antinodes are where the vibrations are at their maximum amplitude
  • The nodes and antinodes do not move along the string. Nodes are fixed and antinodes only move in the vertical direction
  • Between nodes, all points on the stationary 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, downloadable AS & A Level Physics revision notes

 

  • L is the length of the string
  • 1 wavelength λ is only a portion of the length of the string

 

Worked example

Worked example - nodes and antinodes (1), downloadable AS & A Level Physics revision notes

Worked example - nodes and antinodes (2), downloadable AS & A Level Physics revision notes

Exam Tip

The lengths of the strings will only be in whole or ½ wavelengths. For example, a wavelength could be made up of 3 nodes and 2 antinodes or 2 nodes and 3 antinodes.

Measuring Wavelength

  • 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

Fixed end wavelengths and harmonics, downloadable AS & A Level Physics revision notes

Diagram showing the first three modes of vibration of a stretched string with corresponding frequencies

 

  • The nth harmonic has n antinodes and n + 1 nodes

 

One or two open ends in air column

  • When a stationary wave is formed in an air column with one or two open ends, we see slightly different wave patterns in each

 

Closed and open ends fundamental, downloadable AS & A Level Physics revision notes

Diagram showing modes of vibration in pipes with one end closed and the other open or both ends open

 

  • In Image 1: only one end of the air column is open, so, the fundamental mode is now made up of a quarter of a wavelength with one node and one antinode
    • Every harmonic after that adds on an extra node or antinode
  • In Image 2: the column is open on both ends, so, the fundamental mode is made up one node and two antinodes
  • In summary, a column length L for a wave with wavelength λ and resonant frequency f for stationary waves to appear is as follows:

Table of length and corresponding resonant frequency, downloadable AS & A Level Physics revision notes

Worked example

Worked example - measuring wavelength, downloadable AS & A Level Physics revision notes

Exam Tip

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!

Author: Ashika

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.
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