Harmonics

• What are stationary waves known to have?

  Different wave patterns called harmonics
• What do harmonics depend on?

  Frequency of vibration and creation situation
• How can harmonics be observed?

  On a string with two fixed ends
• What happens as the frequency of a stationary wave increases?

  More harmonics begin to appear
• What is the simplest wave pattern of a vibrating string fixed at both ends?

  A single loop with two nodes and an antinode

  
• What is the first harmonic also known as?

  Fundamental frequency
• What do the resonant frequencies of stationary waves depend on?

  Length of the string L and wave speed v
• How is the wavelength of the lowest harmonic related to the string length L?

  Wavelength is $2L$
• Why is the wavelength of the lowest harmonic $2L$?


  There is only one loop, which is a half wavelength
• How many nodes and antinodes does the second harmonic have?

  Three nodes and two antinodes

  
• What is the wavelength of the second harmonic?

  The wavelength is $L$
• How many nodes and antinodes does the third harmonic have?

  Four nodes and three antinodes
• What is the wavelength of the third harmonic?

  The wavelength is $\frac{2L}{3}$

  
• How is the nth harmonic defined in terms of nodes and antinodes?

  Has n antinodes and n + 1 nodes
• How can the wavelengths and frequencies of the first three harmonics be summarized?

  • First harmonic: Wavelength = $2L$, Frequency = $f_1$
  • Second harmonic: Wavelength = $L$, Frequency = $2f_1$
  • Third harmonic: Wavelength = $\frac{2L}{3}$, Frequency = $3f_1$

  
• What is the relationship between the frequency of the nth harmonic and the first harmonic?

  Frequency = n × frequency of first harmonic

  
• If a stationary wave in the third harmonic has a frequency of 150 Hz, what is the frequency of the first harmonic?

  50 Hz

  
• How do you calculate the frequency of the fifth harmonic from the first harmonic?

  Multiply the first harmonic frequency by 5

  
• What is the formula for the speed of a wave along a string?

  v = $\sqrt{\frac{T}{\mu}}$
• What is the wavelength for the first harmonic of a stationary wave of length L?

  Wavelength is $2L$

  
• How is the speed of the stationary wave calculated using the wave equation?

  v = f × $2L$
• How do you derive the frequency of the first harmonic from the wave speed equation?

  Combine wave speed and wavelength equations

  
• What should you remember about matching wavelengths with harmonics?

  • First harmonic: half or quarter wavelength
  • Second harmonic: full wavelength

  
• What is the frequency of the first harmonic produced by a guitar string of mass 3.2 g and length 90 cm under a tension of 65 N?


  Calculate using tension and length