Harmonics (AQA A Level Physics): Revision Note

Harmonics

• Stationary waves can have different wave patterns, known as harmonics

  • These depend on the frequency of the vibration and the situation in which they are created

• These harmonics can be observed on a string with two fixed ends

• As the frequency is increased, more harmonics begin to appear

Harmonics on a string

• 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 first harmonic or fundamental frequency

• The particular frequencies (i.e. resonant frequencies) of stationary waves formed depend on the length of the string L and the wave speed v

• 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

• Therefore, the frequency is equal to:

$f_1=\frac{v}{\lambda }=\frac{v}{2L}$

• The second harmonic has three nodes and two antinodes

• The wavelength is L and the frequency is equal to:

$f_2=\frac{v}{\lambda }=\frac{v}{L}$

$f_2=2f_1$

• The third harmonic has four nodes and three antinodes

• The wavelength is 2L / 3 and the frequency is equal to:

$f_3=\frac{v}{\lambda }=\frac{3v}{2L}$

$f_3=3f_1$

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

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

• Comparing the equations for the first, second and third harmonics shows that for the nth harmonic:

$\mathrm{frequency}\mathrm{of}n\mathrm{th}\mathrm{harmonic}=n\times \mathrm{frequency}\mathrm{of}\mathrm{first}\mathrm{harmonic}$

Frequency of the first harmonic

• The speed of a wave travelling along a string with two fixed ends is given by:

$v=\sqrt{\frac{T}{\mu }}$

• Where:

  • $T$ = tension in the string (N)

  • $\mu$ = mass per unit length of the string (kg m^–1)

• For the first harmonic of a stationary wave of length L, the wavelength, λ = 2L

• Therefore, according to the wave equation, the speed of the stationary wave is:

$v=f\lambda =f\times 2L$

• Combining these two equations leads to the frequency of the first harmonic:

$f=\frac{1}{2L}\sqrt{\frac{T}{\mu }}$

• Where:

  • $f$ = frequency (Hz)

  • $L$ = the length of the string (m)