# Standing Waves

AlephZero
Homework Helper
I was really talking about the definition of fundamental frequency itself. I can only see it being a useful concept if it is defined so that all frequencies produced by the instrument are truly integer multiples of the fundamental frequency. I can't think of any other useful definition.
To repeat, I think you are getting confused between the modes of (free) vibration of an object, and the components of a Fourier series for a periodic motion.

If you personally want to define "fundamental frequency" your way, that's fine - so long as ypu realize that by your definition, a guitar and a piano are NOT "musical instruments", to give just two examples.

If you disagree about the physics, sorry, but what you learned about vibrating strings in a high school level course on sound isn't the whole truth.

See http://www.simonhendry.co.uk/wp/wp-content/uploads/2012/08/inharmonicity.pdf for some real world data, on pianos.

BruceW
Homework Helper
this is a good link, thanks. So, as he says, he introduces higher terms into the wave equation:
$$T\frac{\partial^2 y}{\partial x^2} - QSK^2 \frac{\partial^4 y}{\partial x^4} = \sigma \frac{\partial^2 y}{\partial x^2}$$
and he defines the fundamental frequency as ##1/(2L) \sqrt{T/\sigma}##. And due to the higher terms in the equation, the frequencies will not necessarily be integer multiples of the fundamental frequency.

So in this definition, is the fundamental frequency useful at all? I guess maybe it is, in the case that the higher-order terms are only small, so we can make a perturbation of some kind. OK, so I have changed my mind a bit. But I would still argue that if the instrument is not approximately harmonic, then the 'fundamental frequency' as he defines it, has no meaning.

sophiecentaur