# Standing Waves

Please could somebody explain why the fundamental frequency always has the biggest amplitude in comparison to the rest of the harmonics?

sophiecentaur
Gold Member
2020 Award
The reason that the fundamental usually dominates will be because of the shape of the string when plucked - the Fourier Transform of the initial shape will normally have a maximum at the fundamental mode. But it doesn't necessarily. You can excite a string with asymmetrical plucking at 1/4 and 3/4 along its length and get the first overtone dominating with no significant fundamental OR by placing a dampening finger, half way along a guitar string (or other places) you can get higher overtones (not exact harmonics unless the medium is ideal - but referred to by guitarists as harmonics - but they're not Physicists so it's OK for them)

1 person
AlephZero
Homework Helper
The fundamental doesn't always have the biggest amplitude.

For example see Fig 2 (page 6 of the PDF) in http://www.haskins.yale.edu/sr/SR111/sr111_23.pdf

Several counter-examples of organ reed pipes here: http://www.pykett.org.uk/reedpipetones.htm - e.g Fig 9 the 5th harmonic is loudest, followed by the 3rd.

The reason why lower frequency harmonics usually have bigger amplitudes than higher, is because of the basic equation for the frequency of a vibrating object, ##f = (1/2\pi)\sqrt{K/M}## where K and M measure the stiffness and mass of whatever is vibrating. Often, the amount of vibrating mass is about the same for all the harmonics, and the stiffness increases as the frequency increases. For a fixed amount of forrce, you get a bigger amplitude of vibration from a low stiffness (because displacement = stiffness / force), and so the low frequency harmonics tend to have bigger amplitudes.

1 person
Don't they die out??

Two waves traveling in opposite direction should die out according to super position principle (Zero amplitude). Then how standing waves are produced? Is this not a contradiction? Please help. thank you.

sophiecentaur