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Hi, welcome to another set of lectures
this time is on standing wave and resonance
Hopefully, by now, you are not dreading these yet
especially in light of last week
but now that we have done all that hard work: we have looked at and come up with the wave equation
we have also dealt with the superposition principle
as well as the reflection at an interface. We can now put that all together
to talk about the phenomenon of standing waves, and ultimately about resonance
Now, resonance here,
You may have heard of it in your 221 course, but here we have a slightly different
meaning for it in a slightly different context. Very much related
but in this context, it's a little bit different and I would argue it's even more important than the resonance that you would have seen
it is much more prevalent because it deals with travelling waves and how it emerges into standing waves
So let's get started with that, let's look at how standing waves comes about
Standing waves is thus called because of the way it ends up looking
and we will see how it looks like. It is actually a very apt description
where it comes from, it comes from the fact that a wave
crashes on an interface, reflects and interferes with itself
as a quick reminder, if I have
a wave that comes in
give it a little bit of asymmetry, and it is travelling this way
it hits a certain interface
when it comes out
if you remember, not only does it traveled the other way
it also gets flipped upside down because this is an infinitely hard wall, so this is for a fixed end
just to be very very clear
now for a pulse like that
you have one pulse coming in
and the other pulse coming out
and they only overlap in the very short instance where the pulse is passing through the interface, but most of the time
you have a good looking pulse going in, a good looking pulse going out. However,
if we consider plane waves
so friendly reminder, the plane wave we can describe as a cos curve, and unlike the pulse, which starts and ends
the cos curve is kind of spreaded out through all space. It keeps coming. There is more and more of it; it goes on forever
so what happens is when the wave hits the interface
and travels in that way
there is a part of it that would "go through" the interface
and that's going to give us the reflection
which is also going that way but gets reflected
right behind it, there is another way
that looks just like it
so that's going to interact with our reflected wave
so how does this way reflect?
so let's put this as red
as we know
because it reflected, it gets flipped horizontally, so it goes the other way
but because we have a fixed end, we are going to flip it vertical as well
once again, with the pi phase shift for the plane wave or just the negative sign
and when that comes back, well, that just adds up onto that
and so ultimately we will end up with
is a wave that's
twice as high
so there's your 0 point
and you end up with a wave that is twice as high when they add up -- superposition
OK, little later on
the wave has moved forward a little bit more
and you have that
same thing
this part is going to get reflected
once again, we flip
horizontal to go backwards, vertical because of the phase shift
in as it lines up, it's going to give you
nothing actually, everything cancels out
and then later on, it moves again
and this part of the wave looks like that
this parts gets reflected once again
so we will try and do this fairly quickly
horizontal and vertical, like that, not super scaled drawing, but you can see how
as it comes back it's going to add up the other way so it'll give you something that looks like this
and then later on
last little bit of the cycle
which is going to happen very much the same as
the second scenario here
so that becomes
something like this, and you end up with
once again, a flat line
so
what's actually happening is you have a wave travelling to the right adding up to a wave travelling to the left
but effectively what you see is a wave that does not seem to move
it just have the same shape but it bobs up and down, up and down, up and down
and that's why we call these "standing waves", because it does not look like the wave is going anywhere,
it just stays in place and stands still
and moves up and down like that
there is a little bit of vocabulary, you see that there are these points where these red dots are, the string does not seem to move
we call those nodes, these stationary points where the wave does not seem to be
doing anything to the string. Those are called nodes. And then in between the nodes, we have
places where it moves the most, and we call those the anti-nodes
and so this is the basic look of the standing wave and we can look at how that comes about using our equations