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Okay then we have two waves that we are adding up ...
... which as you know is the wave that goes towards the right ...
... that's the wave that goes left
now k and ω is going to be the same because it is the same rope under the same tension
and the amplitude we are going to say is the same, partly for simplicity, partly we are going to say that the reflection was perfect
and so, the overall what you see on the rope is the sum of these two
so expanding y1 and y2 ...
...which of course you can expand using sum of angles
...
...
...
...
...
... these terms cancels out
and we have
simply left with ...
...
... notice how no longer have the kx-ωt combined phase, so that's why the wave this not traveling
instead what we have is we have some kind of spatial periodic movement
which then, within it, has the cos time dependence, so
it bobs up and down within the spatial envelope
funny thing to notice, one wavelength is
2 humps, not 1 hump but 2 humps
because of the + or - nature of the cos(ωt)
so remember that when we started to do calculations with these things
but once again
because we're adding up 2 waves that comes together in opposite directions
we instead of having a phase that is in the form of kx-ωt which is a travelling wave
we end up with a product of a spatial variation and in time it also changes over time
but seperately, so that's how we end up with the standing wave
as a quick aside, just to show you how
we can also do this using complex exponential...
...
...
we end up with
...
...
... which then of course as you recognize as cos(theta)...
... so this must be
...
...
so very similarly, we can also use complex exponentials to show you
that we once again end up with
a spatial oscillation that defines the cos or sin term that
which is then modulating a cos temporal variation once again
gives you that
goes down
moves
poor sketch there, but
do take note that
one wavelength looks like that, so it takes 2 humps to make 1 wavelength. Now, let's do a quick example