Transcript

Sounds are made when objects, such as this violin string, vibrate. The to and fro motion of the string continuously pushes the air molecules around it together and pulls them apart again causing a wave of pressure changes in the air. These pressure changes are picked up by our ears and interpreted by our brains as sound. Many objects can generate sound, such as the vibrating cone of a speaker.

But what does sound actually look like? That may seem a bit of a strange question. Sound is something that we hear, not see. We’re about to see a sound wave as it propagates out from a speaker, using a special type of photography called Schlieren photography. This is a photographic method which actually allows us to see the movement of the air molecules as a sound wave travels through them.

This image is a recreation of an experiment conducted by Dr. Michael Hargather and Dr.Gary Settles at Penn State University where they placed a speaker emitting a single tone in front of a camera inside a Schlieren setup and filmed what happened to the air molecules around the speaker.

This black rectangle here is the speaker and travelling away from it, through the air, are waves of sound. They look very like the waves in water. As the air molecules bunch together we get the crest of a wave showing an area of high pressure. As the air molecules are pulled apart again we see a trough forming showing an area of low pressure. If indeed this was water and we were to place and imaginary ball just in front of the speaker, we could envisage how the ball would bob up and down on the waves as they pass by.

Now if we draw a line showing not only the current position of the ball, but where that ball was a moment ago and a moment before that and a moment before that and so on, we can actually draw a representation of the sound waveform.

How could we represent this line mathematically? Well it just so happens that there is a mathematical function that looks very much like this simple wave shape. That function is called a Sine function.

You may remember the Sine function from your school days as having something to do with the angles of triangles. If you take the result of the Sine of the angle of a triangle and multiply it by the length of the hypotenuse it will give you the height of the triangle’s apex. But what has this purely mathematical function got to do with the sound waves we saw before? Well just watch this:

As the angle of our triangle increases it rotates the apex of the triangle around in a circle.

Now, if we plot all the different angles through which our triangle has rotated on a separate graph, you can see that the shape it describes looks very similar to the sound wave we saw before, which is why a Sine wave is a pretty good approximation of a sound wave.

But, how can something so simple like a Sine wave be so versatile? After all we use sound to transfer information in speech or music. How can such a simple wave like this contain so much information? Well the short answer is that it can’t. This wave just sounds like a single tone. What can we do to make this Sine Wave sound a little more like something you might hear in the real world?

Until now, we’ve looked at our Sine wave as a signal that changes over time like a floating ball moving up and down on the crest of a wave. I want to look at our wave in slightly different way.

I’m going to assume that the basic shape of our Sine wave never changes, it always looks like a rise and fall over time. So what properties of my wave can I change?

There are, in fact, 3 properties I can change but for now I’m going to deal with only 2 of them, we’ll deal with the 3rd in the next lecture.

I’m going to change the frequency, or pitch, and the amplitude, or loudness, of my wave.

So if I already know what my basic Sine Wave looks like, I don’t really need to see the wave itself. What interests me is its frequency and its amplitude. So I’m going to plot my Sine Wave on a new graph.

We can now see our single Sine Wave represented as a peak. As the frequency increases, the peak moves to the right, and as it decreases again the peak moves back to the left.

As the amplitude decreases, the peak gets smaller and as the amplitude increases again, the peak gets larger.

This new way of looking at my Sine Wave is known as looking at it in the Frequency Domain, plotting the amplitude of my Sine Wave against its Frequency as opposed to the way we looked at it before in the Time Domain when we plotted how the amplitude of our Sine Wave changed over time. This distinction is going to be very important to us later on as we get deeper into what the Fourier Transform does, but for now I want to use the frequency domain to look at how we might change our basic single frequency Sine wave into something more interesting, something, a little more musical.

So let’s go back to the Sine Wave we had before. Here it is in the Time Domain … and here it is in the Frequency Domain. What happens now if we add another Sine wave? You can see our new Sine wave represented in the frequency domain as a new peak to the right of the first one meaning that it is a Sine wave with a higher frequency than the first. The peak is smaller, telling us that it is not as loud. Now let’s look at the affect it has had on our Sine Wave in the time domain.

Now we’re going to add another Sine Wave at another frequency, and another one, and another. Each Sine wave appears as a new peak in our Frequency domain graph and changes the shape of the waveform in our time domain graph. As we add more and more sine waves to our signal and begin to play a little with their amplitudes, our waveform begins to actually sound like something real, a Violin in fact, playing the note A.

This adding together of Sine waves is what Fourier meant when he said that:

“Any function of a variable…”

…In this case our function is a sound wave and our variable is time,

“…can be expanded in a series of sines of multiples of the variable.”

In other words, when we apply Fourier’s theory to our sound experiment, we see how the sound wave made by this Violin is actually made up of a series of lots of different sine waves at different frequencies and amplitudes all added together.

This series of sine waves is what is known as a Fourier Series.

I said before that there were 3 properties of a Sine wave we could play with; frequency, amplitude and one other. In the next lecture we’re going to deal with the 3rd and final property – Phase.