Wavelength of Stationary Waves (Cambridge (CIE) AS Physics)

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

  

• The second harmonic has three nodes and two antinodes

  • So the wavelength is equal to the length of the string

• The third harmonic has four nodes and three antinodes

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

• The nth wavelength has a wavelength of 

• The wavelengths and frequencies of the first three harmonics can be summarised as follows:

Wavelength and frequencies of different harmonics

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

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