standing waves

• standing waves are a result of interference, -> Two waves travelling in opposite directions that have same frequency, amplitude, and velocity (in same medium) - identical in every aspect but direction.

  
• A standing wave is formed from the interference of two waves with identical amplitude, frequency, and speed that travel in opposite directions
• A node is a point on the (resultant) standing wave where there is always destructive interference. The two waves are perfectly out of phase, so this point always has zero displacement.

  An antinode is a point on the standing wave where there is always constructive interference. The two waves are perfectly in phase, so this point reaches maximum displacement.
• Explain how a standing wave is established along the guitar string?


  ends of the strings are tied down. string plucked -> a wave starts to travel in the string. When part of the wave hits one of the tied down ends of string -> reflected back (because the end is fixed) -) creates two waves in the string: identical in every way but their direction
• Each end is a node, because each end is fixed -> No vibration at the ends.

  Two nodes, one at each end of string. Between these two nodes must be an antinode. So this is the simplest standing wave. (1st harmonic = fundamental frequency)

  
• standing waves fixed ends

  
• Two closed ends or Two open ends

  1st harmonic (fundamental) = half a wave so λ = 2 x L

  f1 = v/2 x L

  L = λ/2
• Two closed ends or Two open ends

  2nd harmonic = whole wave so λ= L

  f2 = v/L or f = 2fo (fundamental frequency)

  L = λ
• Two closed ends or Two open ends


  3rd harmonic = λ = (2/3) x L

  f3 = v/(2/3) x L f = 3fo

  L = 3 x λ/2
• Two closed ends or Two open ends

  4th harmonic = two whole wavelengths so λ=(1/2) x L

  f4 = v/(1/2) x L or f=4fo

  L = 2λ
• standing waves fixed ends

  
• standing waves open ends

  
• standing waves one fixed one open ends

  
• One open and One closed end

  1st harmonic (fundamental) = quarter of a wavelength so λ = 4 x L

  f1 = v/4 x L

  L = λ/4 or (1/4) x λ
• One open and One closed end


  3rd harmonic = λ = (4/3) x L

  f3 = v/(4/3) x L or f=3fo

  L = 3λ/4
• One open and One closed end

  5th harmonic = λ = (5/4) x L

  f5 = v/(5/4) x L or f=5fo

  L = 5λ/4