Lets try a different example.
We use that first example sort of in a lengthy way
to introduce all the different terminology
and how we think about these examples.
But for a different example
we could look at something with two open ends.
So instead of any closed ends,
we now require that at both ends of this pipe
which has two open ends,
there must be atmosphere pressure
for the reasons we talked about.
So now we need to find the wavelength
and frequency of the waves in this pipe.
We would do the same sort of analysis.
We first require again
that we have nodes at both open ends.
We know that that is the length of the pipe.
And we could also draw out the rest of our wave
and see what its length is,
compared to the length of the pipe and so just by looking at this,
we can find with the longest wavelength is that can fit in our pipe.
Since I couldn't make it possibly any longer than this
and really think about that.
Try to imagine stretching this wave out even more than it is now.
There's no way you can match the requirement
that it has to be zero at either end of the open-ended pipe.
So this really is the longest wavelength that would fit
inside this particular pipe with two open ends.
In this case the length of the pipe is now only half
of the wavelength of the sound and we can rearrange to see
that the wavelength is twice the length of the pipe.
We could keep doing this in the same way that we described
making shorter and shorter waves
and as we make shorter and shorter waves,
we would still maintain that we have two nodes,
two zero points at either end of our open-ended pipe.
And we'll again do the same thing as indexing each one of these
by the number n, as we described.
We'll use this number n slightly differently
but again the best way to do this is as we already shown
which is where you write them out
and you see what integers you need and we substitute those with the number n,
just using the same sort of logic we talked about before.
In this case, one more time we can solve the equations just by using our velocity equation
and once we've done that,
we now have v over 2L times n.
Let's do one more example.
We have two closed ends now.
So we have a completely closed sort of pipe
and we can ask how this sound has to behave in this pipe
and this will be the same sort of thing.
Except that the two ends, we require both ends be at maxima or minima.
So this is where we have to be careful.
We have on one end the maximum and the other end a minimum.
So why is this allowed?
Well, remember our requirements at the end of the pipe
where that there will be antinodes, the exact opposite,
the maximum pressure or the minimum pressure.
But both of those are allowed,
both the maximum and the minimum at the closed end of a pipe.
It doesn't have to be always the maximum pressure
because again as we allow the sound to evolve in time.
They'll be trading off.
The sound will be fluctuating
and these maximum points will go to minimum and maximum, etc.
We can now write the entirety of this wave
and do the exact same sort of analysis. We have a length of pipe.
We have a length of the wave, the wavelength.
And we can compare the two and find the length of the pipe.
Again, it's going to be half the wavelength of your wave.
We can find the wavelenght just by rearranging
and then we can do again our last step
by writing both the frequency using our velocity equation
and then generalizing all the different frequencies we could fit in our pipe
by continually compressing our wave,
maintaining our requirement that we have either maxima or minima at the ends
and then seeing what patterns we notice.
Whether it's always an odd number or even number
and then using the integer n to make even numbers with 2 times n
where odd numbers with 2 times n plus 1.
And again, this is one of the more potentially confusing topics
and how we create standing waves.
How we satisfy these boundary conditions.
How we create more and more waves.
How we describe those waves using an integer n,
but it is also very common.
It's a very common sort of question to ask,
so be sure to go over this a few times
to make sure it makes a lot of sense to you.
The basic logic that we're using as we go through
these examples especially the first one.