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Well, the following video is going be on
Fourier analysis. Its very superficial, and the subject is really,
pretty advanced, more advanced than I can really competently teach so take it with a
grain of salt.
If you want to check this out for yourself you probably should have a
a year or more of calculus.
This is my best first attempt.
When the string is released two
harmonic series of waves travel in opposite directions along the
string.
They trace out a parallelogram. And return to the starting point.
It looks something like this.
The size of that parallelogram is a measure of the string' s initial energy.
The highest partials quickly transfer the energy to the guitar
And the string will lose that rectangular shape very quickly.
Here's the rectangular shape. You release it. It goes into decay.
The string changes shape continuously as the energy is transferred to the
guitar
In the order from highest partial down to lowest. The string eventually takes the shape of the
fundamental and comes to a stop.
This shows how the partials are added
and combined to form this complex or combined sound wave we have down here.
It looks something like this. And you should
keep in mind that the waves travel
the complete length of the string in one direction and then back in the other direction
in order to complete one cycle. So,
the string length would actually go from about here to about here
and you would take this whole section of the wave and
move it over this way you would get that rectangular shape
In Fourier analysis you're going to start
with the complex wave. And by a multiplication process
you're going to derive the amplitudes of each of these frequencies
these partials. The word sine is a
mistranslation the original word apparently would've referred to an
analogy to the bow and arrow. Something like this.
And a sine would have been half the original position of the bow string.
Cosine apparently would have been a measure of the
arrows energy. How far are you going to shoot the arrow. The
simple harmonic motion of the energy waves is treated mathematically
as a change in the ratios of the blue right triangle here.
This is about as simple as I've ever seen the math expressed. It comes from this website
right here.
Basically the amplitude of the complex sound wave
at any time consists of the sum of this sequence,
of harmonics. And this is how it's expressed
And this is the summation symbol. This means the amplitude is equal to the sum
of the
partials, (over the,) one over the number of the partial times the
sine of the number of partials times theta. Theta here would be a measure of time,
In this instance. The specific mathematical technique used is to
multiply two waves of the same kind and the same frequency.
You've got two sine waves. And
this one is going to be amplitude one. And this is an unknown amplitude we want to
find.
First we want to just convert this negative part to a positive.
And when you multiply
the amplitude of this wave times the amplitude of this wave at this point,
it'll be a positive times a positive. But when you do it on this
point it'll be a negative times a negative. It'll
change it to a positive. You'll have a positive and a positive.
Now once you've converted the
sine wave to all positive values, you want to get it into a rectangular shape
like this.
Visually it'll look like this. There you go.
And now you've got a rectangle shape. The actual wave will be half of that height,
so you multiply it by one-half. The amplitude is here,
multiplied by one-half. The idea is to derive the amplitudes
or relative strength each of the simple harmonic partials
that make up this complex wave. Basically take a sine wave of the same frequency.
Amplitude one. And you multiply it times,
the original complex wave.
Basically take at each regular point along the
time line here, take the amplitude of one wave and you multiply it
times the amplitude of the other wave at that point.
The result is a rather strange looking wave thats all positive.
Taking one-half of the wave put it directly over here, you've got a rectangle
The amplitude of that wave is over here.
You've got one-half the time line, one-half T here.
And the area is, amplitude times one-half T.
So you divide the area by one-half T.
And you get the amplitude of the partial. And the process continues, to get the
amplitude of the second partial. Start with a sine wave twice the frequency
of the original.
Amplitude one again. You multiply
times the original complex wave. The result
looking like this. Its got a positive component and a negative component to it.
And basically in the integration process
just increase the number squares to get something that roughly approximates
the area between the straight line and the curve. Subtract the
negative component from the positive component and then you take
one-half T, one-half the length of the period
divided into the number of squares and that gives you
the amplitude of the second partial. The process continues for all the upper partials.
As waves get more and more complex it's easier to do the math than it is to
do a visual demonstration. And this animation shows how the relative strength
of the partials
depends on where the string is plucked. Here's the
partials right here. Here's the string right here
Here's the relative strength or amplitude of the partials.
This is the string plucked midway.
One-third of its length, One-fourth of its length, One-fifth of its length,
which is where most of the time you pluck it. And one-tenth
would show a much higher strength of the higher partials.
Okay that's it for now.
It's very superficial. And I myself am going to
have to go back to study algebra which is what I was doing when the iPad came out
and I had to make the switch over to video.
Right now I think I'll just go play guitar for awhile.