The mathematics of pure tones —

tuning forks and the basics of sound

We've now seen a few audiograms and sound sure can look messy. As scientists we want to look for the simplest kinds of sounds and then build any sound out of our fundamental building blocks.

Just as chemistry tries to understand the properties of a substance by looking at the molecular structure which in turn is studied by considering the individual atoms, we want to understand sound by finding "atoms of sound," understanding these, and then combining the atoms into molecules of sound, and finally the molecules combine to form general sounds, like a symphony, my voice or the wind rustling through the leaves.

What should these pure sounds be? Let's think a bit about music. Those of you have ever played an instrument (and even some of you who haven't) may have seen a musical score. At its most basic level, the score lays out a sequence of notes to be played, and the duration which the notes need to be held. So, running left to right is time, and up and down is pitch (middle C, A sharp, etc.). So the "pure" tones are the notes, which are realized differently on each instrument, but in fact, they are characterized as particular kinds of sound waves that occur at specific "frequencies". We'll explain this idea now.

Different instruments realize these notes in different ways — an A on a guitar sounds different from an A on a trombone or a flute. That's because the physics of a plucked string is different from the physics of a open-ended tube. As a way of understanding pure tones, or our atoms of sound, the simplest model is the tuning fork.

We ring a tuning fork by whacking it against something (your hand, your knee, your best friend's forehead) and then standing it up and allowing the vibrations of the tines to disturb the surrounding air, sending it to the waiting ears of everyone in the vicinity.

The tines move back and forth, sending the air molecules to your ear where they go crashing against your eardrum. So, what might the audiogram look like?

Well, let's concentrate our attention on the tip of one of the tines. In fact, imagine that we put a drop of glow-in-the-dark paint on the the tip, turn the lights off and follow the glowing dot. We se it move back and forth nice and regularly (quick - think, does a stiffer fork vibrate faster or slower than a "flabby" fork?). The air molecules then get sent away, and our ear drums more or less follow the same pattern. So, one possibility is something like this:

Notice the regular up and down. It is a periodic wave, in the sense that there is a simple pattern that is repeated over and over again. The number of full copies of the pattern that fit in a single second is the frequency of the periodic function, and as we've already remarked, is measured in Herz, or cycles per second. The duration of a single copy of the basic pattern is called the period of the periodic function. The period and frequency are related by

period = 1/frequency

Now one possible drawback of the "triangle" wave above (we call it that because it's a triangle above and below the x-axis) is that it isn't smooth. The graph makes it look like as the tine reaches it's maximal displacement that it immediately whips back in the other direction. Our experience is that the change of direction is a bit more subtle than that — that there is a gradual slowing down as the apex is reached, and then just for a moment, it's as if the tine stops, and then a gradual speeding up, and so on and so on...So, maybe instead, the graph is more like a smoothed out version of the triangle, maybe something like this:


Some of you may recognize this as the graph of the *sine* function! In fact, this is the right graph (well, up to a very good approximation, but we won't get into that!), and before we convince you of that mathematically, it's worth pointing out a few things about this graph.

We've already talked about its frequency, or equivalently its period. Here is a picture of a few different kinds of sine functions of differing frequencies (but the same amplitude) all superimposed over one another. Higher frequency means the peaks and troughs are squeezed more closely together.

The picture below shows a lot of sinewaves, all at different frequencies, but at only one amplitude.

Perceptually, higher frequency (usually) means a higher pitch, although in fact this is sort of complicated and we'll discuss that later.

A few other things about these graphs. There are other things that could have been varied (besides the frequency) in making the pictures. We could have also varied the height (or equivalently the depth) of the wave. This parameter is called the amplitude of the wave. Lastly, there is an arbitrary nature to where in time we started to measure the displacement. If we had started a moment later or earlier we would have obtained a shifted version of this picture while still having a sine wave.

The figure below shows what happens when we have a lot of sinewaves, all at the same frequency, all at the same amplitude, but different phases (and different colors, isn't that cute?).

Strangely enough, when we add up all those amplitude/frequency similar but phase different sinewaves, we get, you guessed it, one pretty ordinary sinewave!

This last parameter is called a phase shift. The moral is that any sinusoidal wave is determined by three parameters: frequency, amplitude and phase shift. We'll be studying how the values of these parameters affect what we hear.

Getting to the math

Newton's second law states that the force felt by an object in motion is in proportion to the acceleration experienced by the object. (It is well known that the first law is to rotate your tires every 30,000 miles.) For example think about driving in your car — you and the contents of your fine automobile are the objects in motion. You are speeding down the interstate, brimming coffee cups in the cup holders and compact discs on the seat next to you. You're cruising at 80 mph, so no acceleration, and life is good. You're slouched in your seat, coffee isn't spilling, CDs stay in place. Suddenly, you pass a state trooper hidden in the median - you hit the brakes (while simultaneously rehearsing your story about being late to your best friend's kidney transplant), causing a deceleration (a negative acceleration) and now the calm of your car is disturbed. CDs go slding to the floor, coffee sloshes agains the side of the cup, you find yourself thrown forward a bit in your seat. Everything is experiencing a force! (and may the force be with you!) The harder you brake, the stronger the force.

In shorthand we write

Force = (mass) x (acceleration)

or
F= m a

So, greater acceleration means greater force.

Newton's second law is a general law, and so it also applies to the force felt by the tip of the tine of our tuning fork (say that three times fast!), or the "end of (the) tine". But we can also formulate another law for the force felt at the end of tine.

Think about how the force at the tine depends on how far from rest the tine is displaced. If you move it a little to the left, then a little restorative force, bringing the tine back to rest, is felt moving the tine to the right. Similarly, if the tine is displaced a lot, then there is a big force felt trying to move it back the other way. It's not too hard to believe that in general, the Force felt is proportional to the displacement, but in the opposite direction. We also need to account for the fact that different tuning forks may have different stiffness. So we write

Force = -(tuning fork constant) x (displacement)

or

F= -kx

where the minus sign indicates that the force is in the direction opposite to the displacement. The symbol k will be a number that depends on the tuning fork. The higher the value, the stiffer the fork, and the faster it vibrates.

Now, when a mathematician receives his or her Ph.D., they sign an oath, which says that if two things are equal to the same thing, then you have to then set them equal to one another and see what you get, so we are forced (and just try to stop us!) to now write the following equation:

m a = - k x

This is in fact a famous equation, a type of differential equation which is the basic harmonic oscillator. What it says, is that for something like the tine on a tuning fork (or a spring), that the acceleration experienced by the tine is related to its displacement. Rember that both the displacement of the tine and its acceleration are functions of time. So, a solution to this differential equation (or "diff e. q." as we say) is a function that describes the motion of the tine and relates the position and acceleration as in the equation. It turns out that if you have an object whose displacement is a sine function (meaning that at time t, the displacement is sin(t) ) then you can solve for its acceleration at time t (we'll see that it is described by -sin(t) ) so that it satisfies the harmonic oscillator for the values k=1 and m=1. The general solution is still a sine function, but with a different frequency. In fact, it is

x(t) = sin( (square-root(m/k)) t)