• 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

   

• 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:

• The second harmonic has?three nodes?and?two antinodes
• The wavelength is?L?and the frequency is equal to:

• The third harmonic has?four nodes?and?three antinodes
• The wavelength is 2L / 3 and the frequency is equal to:

• The nth harmonic has n antinodes and n + 1 nodes
• The wavelengths and frequencies of the first three harmonics can be summarised as follows:

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

Step 1: Calculate the frequency of the first harmonic

f3?= 3 f1

f1?= f3?÷ 3 = 150 ÷ 3 = 50 Hz

Step 2: Calculate the frequency of the fifth harmonic

f5?= 5 f1

f5?= 5 × 50 = 250 Hz

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

• Where:
  • T?= tension in the string (N)
  • μ = 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 = fλ = f × 2L

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

• Where:
  • f?= frequency (Hz)
  • L?= the length of the string (m)