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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