Standing Waves – AS Physics Revision

So… Standing Waves

Standing Wave

Standing Wave (Photo credit: Wikipedia)

 

  • Progressive / travelling waves are waves that go somewhere (it looks like the peaks are moving)
  • Standing waves are waves that just go up and down on the spot
  • They have parts that don’t move, called nodes…
  • …and parts that go up and down the whole amplitude, called antinodes
  • At nodes, the displacement from equilibrium is always zero
  • At antinodes, the average displacement is zero, but that’s only because it goes right  to the top and right to the bottom, cancelling out
  • Standing waves are formed when two travelling waves going in opposite directions superpose

In pipes and on strings

  • Standing waves can be formed on a string (such as a guitar string)
  • The fixed ends of the string become nodes
  • There’s no non-fixed end on a guitar, but if there was, it would be an antinode
  • Similarly, sound waves can be generated inside pipes (such as in musical instruments)
  • Sound waves are longitudinal, but for ease of drawing diagrams, you can get away with pretending they’re transverse, in this case
  • Open ends of the pipe will become antinodes, and closed ends will become nodes
  • By drawing a diagram of the string/pipe, labelling where the nodes and antinodes would form, and drawing a wave over it, you can work out the number of wavelengths of standing wave that fit
  • If you know the length, and the number of wavelengths in that length, you can work out the wavelength of that wave 

Fundamentals and Harmonics

  • The longest (wavelength) standing wave you can fit on a string or in a pipe (bearing in mind the rules above) is called the fundamental
  • There will be further harmonics that can fit – double, triple, quadruple the fundamental, and so on…
  • The fundamental on a guitar string is half a wavelength, because it has a node at each end and an antinode in the middle. λ=2L, where L is the length of the string.
  • The first harmonic on a guitar string is a whole wavelength, because it has a node at each end, a node in the middle and antinodes between the nodes, just like one full cycle of a sine wave. λ=L
  • The second harmonic has nodes at each end, two nodes between those, and antinodes between the nodes. λ=2L/3
  • A pipe open at both ends would have a fundamental with two antinodes, and a node in the middle, making it half a wavelength, so λ=2L
  • Its first harmonic would have antinodes at either end, and in the middle, with nodes between the antinodes… you get the picture
  • A pipe open at one end and closed at the other (or a string with one fixed end, and a loose end, I guess) could have a fundamental with only one node and one antinode, making it a quarter of a wavelength, so λ=4L
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About Matt

I like writing, filmmaking, programming and gaming, and prefer creating media to consuming it. On the topic of consumption, I'm also a big fan of eating.
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