How Do Nodes Form on a Vibrating String for the nth Harmonic?

[MyNewPony]

Prove that there are n-1 nodes on a string fixed at both ends for the nth harmonic.

It is simple to show this using a diagram.

[PLAIN]http://www.space-matters.info/img/nodesandmodes.jpg

However, is there a way to show this mathematically?

 

[Borek]

You probably have to go through the harmonic frequencies and harmonic wavelengths - once you have wavelength and you know how it depends on the initial length of the string, the rest should be obvious.

 

[Thaakisfox]

Think of the properties of the sine function (which is the shape of the string with the given boundary conditions.)

 

[MyNewPony]

You probably have to go through the harmonic frequencies and harmonic wavelengths - once you have wavelength and you know how it depends on the initial length of the string, the rest should be obvious.

Wavelength = (2/n)*length of string

Can I get a hint on what to do next?