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sound: Marupavana from Georgia Kelly

Standing waves

Higher harmonics for 2 fixed ends


We arrived at

u(x,t) = - 2A sin(kx) cos(ωt)


The cos(ωt) part describes the oscillatory movements. It is independent of the position x. Consequently, all points of the cord oscillate in phase, right as we have observed. Only their amplitudes are different, which is described by

2A sin(kx)


At x=0 the term 2A sin(kx) vanishes, which means that the cord does not oscillate x=0. That describes the extremum at the lefthand side of the cord which has a node because it is a fixed point of the cord.
At x= we are at the extremum at righthand side of the cord. That must also be a node of the oscillatory movements. Hence, 2A sin(k) must vanish. That can be achieved for

k = π, 2π, 3π, 4π, 5π, ...


With k=2π/λ and v=fλ, one arrives at

f = v/2, 2v/2, 3v/2, 4v/2, 5v/2, ...


exactly as we had found before.
The above formula describes the relation between the lengths of cords and the tones produced, for musical instruments like a harp, a piano, a violin or a guitar.


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