Tip:
Highlight text to annotate it
X
The two things that you do with the waves are add and multiply them.
When adding waves in this case we are adding waves of the same frequency they both
start at the same spot and end at the same spot.
You just put one on top of the other and add the height of one to the height of the
other. Here's the height of one, here's the height of the other and here's the result of adding those two together.
And notice the positive is still positive, the negative is still negative and you just get
a larger amplitude or a higher amplitude to the wave.
Okay. When you multiply you get a little bit different result. Anything less than the
height of one
will be a fraction times a fraction. That'll change the shape that the wave.
And also you're talking about a negative times a negative to convert it to a positive.
And for fourier analysis that's one of the main purposes of this multiplication process.
Here we go you get a positive and a positive. And
here's the original wave note that the area under the wave, the area between the,
between the wave, and the
x-axis is different.
And the shape is a bit different. Wavelength, you want to keep, bear in mind that whenever
a wave travels along the length of the string it reflects back at the end.
And in the case of the even
number waves, this would be the second partial.
You're going to have to invert the wave like that
to get a continuous wave. Here's two cycles. One
up to here, the string length and twice the string length gives you two cycles, two
complete cycles.
Here is the fundamental this is an odd-numbered wave. One,
three, five would be odd-numbered waves. This is one, and when you
move that along twice the string length you get one cycle
occurring at twice the string length. So when you add them,
here we go. First wave, second wave,
there, add the two waves and you get a new wave,
looks like that. Add this wave you get a third wave.
Something like this. You get, thats the shape,
and that's basically the position. When you pluck it, it goes back, reverts, inverts
and
reflects back on the opposite side. You get kind of a rectangular kind of shape
out of the whole thing. But you start with the triangular shape on one side.
It reflects and inverts on the other side. Looks
something like this, the rectangular shape there
is a measure of the energy that you put in when you pluck the string. And,
the distribution of that is what you do with Fourier analysis. That'll, that'll
come a bit later.
This is the basic complex wave I've been using for about eight or nine years now.
Basically it results from adding the harmonic series of waves
of various amplitudes. When you add them together
you get this basic wave and it turns out to be sufficient, to give a pretty good
approximation, the
first three partials, a pretty good approximation of,
plucking the string at one-fifth of its length.