1.1 - Pressure

If you listen to CBC in the mornings, they'll give you the news, and the sports and the traffic and the weather. Part of the weather report is to tell you that the barometric pressure is something around 100 kPa (kilopascals). What does this mean? Well, the air particles around you are all under pressure due to things like gravity and the weight of the air particles above them and other meteorological things that are outside the scope of this course. That pressure determines the amount of physical space between molecules in the air. When there's a higher barometric pressure, there' less space between the molecules than there is on a day with a lower barometric pressure.

These molecules like to stay at the same pressure all over, so if you bunch them up in one place in a room somehow, they'll move around to try and equalize the difference. Kind of like when you pour a glass of water into a bucket, the water level of the entire bucket equalizes and therefore rises, rather than the water from the glass all bunching up in a little mound of water where you poured it in.......

Let's say hypothetically for a moment, that you are sitting alone in a sealed room on a day when the barometric pressure is 100 kPa. Let's also say that you have a clarinet with you and that you play a concert A. What physically happens to convert air coming out of your mouth into a concert A coming in your ears?

To begin with, let's pretend that a clarinet is just a tube with a hole in each end. One of the holes has a springy piece of wood next to it which, if you press on it, will close up the hole.

1 - When you blow into the hole, you bunch up the air particles and create a little area of high pressure inside the mouthpiece.

2 - Blowing into the hole with the reed on it also has the effect of pushing the reed against the hole and sealing it so that no more air can enter the clarinet.

3 - At that point the little high pressure area moves down the clarinet and leaves a low pressure behind it.

4 - Remember that the reed is springy, and it doesn't like being pushed up against the hole in the mouthpiece, so it bounces back and lets more air in.

5 - Now the cycle repeat and goes back to step 1 all over again.

6 - In the meantime, all of those high and low pressure areas move down the clarinet and radiate out the bell into the room like ripples on a lake when you throw in a rock.

7 - From there, they get to your ear and push your eardrum in and out (high pressure pushes in, low pressure pulls out)

Those little fluctuations in the air pressure are small variations in the 100 kPa weather report. They're usually very small, never more than about �1 Pa (though we'll elaborate on that later....)

To see an animation of what this looks like, check out this page.

1.2 - Simple Harmonic Motion

Take a weight (a little one...) and hang it on the end of a Slinky which is attached to the ceiling and wait for it to stop bouncing.

Measure the length of the Slinky. This length is determined by the weight and the strength of the Slinky. If you use a bigger weight, the Slinky will be longer - if the slinky is stronger, it will be better able to support the weight and therefore be shorter.

This is the point where the "system" is at rest or equilibrium. [check the terminology]

Pull down on the weight a little bit and let go. The Slinky will pull the weight up to the point of equilibrium and pass it.

By the time the whole thing slows down, the weight will be too high and will want to come back down to the equilibrium point, which it will do, stopping at the point where we let it go in the first place (or almost anyway.......)

A couple of questions:

Question : At what point is the velocity of the weight the greatest?
Answer : when it's crossing the equilibrium point.

Question : At what point is the weight stopped?
Answer : at the two points where it's furthest from the equilibrium point.

1.2.1 Overtones and Harmonics

If we go back to the clarinet example, it's pretty obvious that the pressure wave that comes out the bell won't be a sine wave. This is because the clarinet reed is doing more than simply opening and closing - it's also wiggling and flapping a bit - on top of all that, the body of the clarinet is resonating (more on this topic later in the year) various frequencies as well, so what comes out is a bunch of different frequencies simultaneously.

We call these other frequencies "overtones" which are mathematically related to the bottom frequency (called the "fundamental") by simple multiplication.... that is to say, the first overtone is twice the frequency of the fundamental, the second overtone is three times the frequency of the fundamental and so on. (this is an oversimplification which we'll straighten out later in the semester....)

Some people call the fundamental and its overtones "harmonics." If you're using this terminology, then the multiplication is simpler. The frequency of the first harmonic (aka the fundamental) is multiplied to find the frequencies of the other harmonics. The second harmonic is twice the first harmonic, the third harmonic is three times the first harmonic and so on and so on.....

1.3 - Longitudinal vs. Transverse Waves

There are basically three types of waves used to transmit energy through a medium or substance.

1 - Transverse

2 - Longitudinal

3 - Torsional

We're only concerned with the first two.

Transverse waves are the kind we see every day in ropes and puddles. They're the kind where the motion of the particles is perpendicular to the direction of the wave propagation. What does this mean? It's easy to see if we go fishing..... A bobber on the surface of a lake will sit there bobbing up and down as the waves roll past it. The waves are travelling towards the shore along the surface of the water, but the water itself only moves up and down, not sideways (we know this because the bobber would move sideways as well if the water was doing so....)
So, as the water molecules move vertically, the wave propagates horizontally.

Longitudinal waves are a little tougher to see, but they involve the compression and refraction of the particles in the medium such that the motion of the particles is parallel with the direction of propagation of the wave. The easiest way to see a longitudinal wave is to stretch out a Slinky between two people, squeeze together a small section of it an let go. The compressed part will appear to move back and forth bouncing between the two ends of the spring. This is essentially the way sound travels through air particles.

Torsional waves don't apply to anything we're doing in this course, but they're wave in which the particles rotate around the axis along which the wave propagates (like a twisting rod....). This type of wave can be seen on a Shive wave machine at physics demonstrations and science and technology museums......

The amplitude of a wave is simply an measurement of the height of the wave if it's a transverse, or the amount of compression and refraction if it's longitudinal. In terms of sound, it's measured in Pascals, since sound waves are variation in atmospheric pressure. If we were measuring waves on the ocean, the unit of measurement would be metres......

There are a number of methods of defining the amplitude measurement - we'll be using three, and you have to be careful not to confuse them.

- Peak - This is a measurement of the difference between the maximum value of the wave and the point of equilibrium.
- Peak-Peak - This is a measurement of the difference between the minimum and maximum values of the wave.
- RMS - This is a measurement based on the amount of power in the wave. It's equivalent to 0.707 of the Peak value if the signal is a sine wave. In other cases, the relationship between the RMS and the Peak value is different.

Go back to the clarinet example. If we play a concert A, then it just so happens that the reed is opening and closing at a rate of 440 time per second. This therefore means that there are 440 cycles between a high and a low pressure coming out of the bell of the clarinet each second.

We normally use the term Hertz (indicated Hz) to indicate the number of cycles per second in sound waves. Therefore 440 cycles per second is more commonly known as 440 Hz.

In order to find the frequency of a note one octave above this pitch, multiply by 2 (1 octave = twice the frequency). One octave below is 1.2 the frequency.

Always remember that a complete cycle consists of a high and a low pressure. One cycle is measured from a point on the wave to the next identical point on the wave (i.e. the positive-going zero crossing to the next positive- going zero crossing or maximum to maximum.....)

If we know the frequency of a sound wave (i.e. 440 Hz), then we can calculate how long it takes a single cycle to exit the bell of the clarinet. If there are 440 cycles each second, then it takes 1/440th of a second to produce 1 cycle.

The usual equation for calculating this amount of time (known as the period) is :

T=1/f
where T is the period
f is the frequency

there is a small deviation of c with frequency, though this is small and therefore generally ignored

Frequency	Deviation (ppm)
100	-30
200	-10
400	-3
1.25 k	0
4 k	+5
10 k	+10

Changes in humidity change the g factor for air (they are proportional) therefore humidity is proportional to c

Humidity (%)	Deviation (ppm)
0	0
20	415
40	1136
60	1860
80	2590
100	3320

This last difference (of 0.33 %) is bordering on our ability to detect a pitch shift.

Also - in case you were wondering, "ppm" stands for "parts per million." It's just like "percent" really, except that you divide by 1000000 instead of 100 so it's useful for really samll numbers. Therefore 1000 ppm is 1000/1000000 = 0.001 = 0.1 %.

The equation we normally use for c is

c = 332 m/s + (0.6 * T)
where c is the speed of sound in air
T is the temperature in �C

The wavelength (abbreviated l) is the distance from a point on a periodic (repeating) waveform to the next identical point. (i.e. crest to crest, or positive zero-crossing to positive zero crossing)

it is determined by a combination of the velocity of propagation of the wave and the frequency

the equation is

wavelength=c/f
where c is the speed of sound
if is the frequency of the periodic waveform

2.1 - Basic Human Limits

According to textbooks, our auditory perception is confined to the following basic limits:

Frequency Response
20 Hz - 20,000 Hz

Threshold of hearing (softest audible sound)
20 �Pa @ 1 kHz

Threshold of pain (loudest tolerable sound)
200 Pa

2.2 - Decibels

Click here for info.

Go throw a rock in the water on a really calm lake. The result will be a bunch of high and low water levels that expand out from the point where the rock landed. The highs are slightly above the water level that existed before the rock hit, the lows are lower. This is analogous to the high and low pressures that are coming out of a clarinet, being respectively higher and lower than the equilibrium pressure that existed before the clarinet was brought into the room.

Now go and do the same thing out on the ocean as the waves are rolling past. The ripples that you create will cause the bigger wave to rise and fall on a small scale. This is essentially the same as what was happening on the calm lake, but now, the level of equilibrium is changing.

How do we find the final water level? We simply add the two levels together, making sure to pay attention to whether we should be adding a positive value (higher water level) or negative value (lower water level.)

Let's put two small omnidirectional (that is, they radiate sound equally in all directions ) loudspeakers, 34 cm apart in the front of a room. Let's also take a sine wave generator set to produce a 500 Hz sine wave and send it to both speakers simultaneously. What happens in the room?

2.3.1 - Constructive Interference

If you're equidistant from the two speakers, then you'll be receiving the same part of the pressure wave at the same time. So, if you're getting the high point in the wave from one speaker, you're getting a high pressure from the second speaker as well.

Likewise, if you're getting a low pressure from one speaker, you're also receiving a low pressure from the other.

The end result of this overlap is that you get twice the pressure difference between the high and low points in your wave. This is because the two waves are interfering with each other constructively. This happens because the two have a phase relationship of 0 degrees at your position.

Essentially all we're doing is adding two simultaneous points from the first two graphs and winding up with the bottom graph.

2.3.2 - Destructive Interference

What happens if you're standing on a line with the two loudspeakers, so that the more distant speaker is 34 cm farther away than the closer one.

Now, we have to consider the wavelength of the sound being produced. A 500 Hz sine tone has a wavelength of roughly 68 cm. Therefore, half of a wavelength is 34 cm, or the distance between the two loudspeakers.

This means that the sound from the farther loudspeaker is arriving at your position 1/2 of a cycle late. In other words, you;re getting a high pressure from the closer speaker as you get a low pressure from the farther speaker.

The end result of this effect is that you hear nothing (this is not really true for reasons that we'll talk about later in the semester....) because the two pressure levels are always opposite each other.

2.3.3 - Beating, Sum and Difference Tones

Sections 2.3.1 and 2.3.2 assumed that the tones coming out of the two loudspeakers have exactly matching frequencies. What happens if this is not the case?

If the two frequencies (let's call them f1 and f2 where f1 > f2) are different then the resulting pressure looks like a periodic wave whose amplitude is being modulated periodically.

The big question is : what does this sound like? The answer to this question is "it depends on how far apart the frequencies are...."

If the frequencies are close together :

First and foremost, you're going to hear the two sine waves of two frequencies.

Interestingly, you'll also hear beats at a rate equal to the lower frequency subtracted from the higher frequency. For example, if the two tones are at 440 and 444 Hz, you'll hear the two notes beating 4 times per second (or f1-f2).

This is the way we tune instruments with each other. If we have two flutes play two A 440's at the same time (with no vibrato), then we should hear no beating. If there's beating, the flutes are out of tune.

If the frequencies are far apart :

First and foremost, you're going to hear the two sine waves of two frequencies.

Secondly, you'll hear a note whose frequency is equal to the difference between the two frequencies being played.

f1-f2

Thirdly, you'll hear other tones whose frequencies have the following mathematical relationships with the two tones being played. Although there is some argument between different people, the list below is in order of most to least predominant apparent "resultant" or "combination" tones.

2f2-f1, 3f2-f1, 2f1-f2, 2f1-2f2, 3f1-f2, f1+f2, 2f1+f2, and so on....

This is a result of a number of effects.

If you're doing an experiment using two tone generators and a loudspeaker, then the effect is likely a product of the speaker called "intermodulation distortion." In this case, the combination tones are actually being generated by the driver.

If you're using two loudspeakers (or two instruments) then there is some argument as to where the extra tones actually exist. Some arguments say that the tones are in the air, some say that the tones are generated at the eardrum (look up "nonlinearity" as it applies to the hearing mechanism). The most intersting arguments say that the tones are generated in the brain. The proof for this lies in an experiment where different tones are applied to each ear seperately (using headphones). In this case, some listeners still hear the combination tones (look up "binaural beating")

3.1 - Fourier Transforms

Last week we saw how to add sine waves with relatively similar frequencies and amplitudes. Let's take that a step further and add sine waves whose frequencies and amplitudes are mathematically related.

If we add a tone with a frequency "f" and a peak amplitude "a" to a tone with a frequency 3f and a peak amplitude of (1/3)a, we wind up with a third wave form shown below. (If you're wondering why the X axis goes up to about 6.3, it's because it's in radians.....)

If we then add a tone with the frequency 5f and a peak amplitude of (1/5)a, to that waveform, we get the following.

If we keep going with the same pattern up to 17f, we get the following.

If we kept going up to infinity, we'd get a square wave.

About now, you should be saying something like "so what?" Well, the importance of this addition lies in the fact that we only used a bunch of simple sine waves all added together to create the wave. A guy named Jean Baptiste Joseph Fourier (1768-1830) recognized this way back. He decided that any periodic waveform can be simplified into its constituent harmonic content built of a combination of sine waves (which cannot be simplified, remember) with specific frequencies, amplitudes and relative phases.

This knowledge allows us to represent the square wave above with a different type of graph. Up until now, we have been plotting the pressure amplitude as a function of time. Our new method plots the pressure amplitude as a function of frequency. Therefore, we can plot the square wave (shown above as an amplitude-time graph) in a "harmonic analysis" or "Fourier analysis" shown below.

Note that this method of plotting does not give us any information about the relative phases of the harmonics, in spite of the fact that this is a vital piece of information.

3.2 - Time vs. Frequency

We said in section 3.1 that the upper harmonics of a periodic waveform are multiples of the first harmonic. Therefore, if I have a complex, but periodic waveform, with a fundamental of 100 Hz, the actual harmonic content is 100 Hz, 200 Hz, 300 Hz, 400 Hz and so on up to infinity.

Let's assume that the fundamental is lowered to 1 Hz - we're now dealing with an object that is being struck 1 time each second. The fundamental is 1 Hz, so the upper harmonics are 2 Hz, 3 Hz, 4 Hz, 5 Hz and so on up to infinity.

If we keep slowing down the fundamental to 1 single strike, then the harmonic content is all frequencies up to infinity. Therefore it takes all frequencies sounding simultaneously with the correct phase relationships to create a single click.

If we were to graph this relationship, it would be the following, where the two graphs essentially show the same information.

3.3 - Noise Spectra

The theory explained in section 3.2 that the combination of all frequencies results in a single click relies on an important point that we didn't talk about - relative phase. The click can only happen if all of the phases of the harmonics are aligned properly - if not, then things tend to go awry.... If we have all frequencies with random relative phase, the result is noise in its various incarnations, the two most common of which are white and pink noise.

3.3.1 - White Noise

White noise is defined as a noise that has equal amount of energy per frequency. This means that if you could measure the amount of energy between 100 Hz and 200 Hz it would equal the amount of energy between 1000 Hz and 1100 Hz.

This sounds "bright" (hence "white") to us because we hear pitch in octaves. 1 octave is a doubling of frequency, therefore 100 Hz - 200 Hz is an octave, but 1000 Hz - 2000 Hz (not 1000 Hz - 1100 Hz) is also an octave. Since white noise contains equal energy per Hz, there's ten times a much energy in the 1 kHz octave than in the 100 Hz octave.

3.3.2 - Pink Noise

Pink noise is noise that has an equal amount of energy per octave. This means that there is less energy per Hz as you go up in frequency (in fact, there is a loss of 50% (or a drop of 6.02 dB) of the energy each time you go up an octave)

This is used because it sounds realtively "equal" to us.

3.3.3 - Blue Noise

Blue noise is noise that is the opposite of pink noise in that it doubles the amount of energy each time you go up 1 octave. You'll virtally never see it (or hear it for that matter......)

3.4 - Amplitude vs. Distance

There is an obvious relationship between amplitude and distance - they are inversly proportional. That is to say, the farther away you get, the lower the amplitude. Why?

Let's go back to throwing rocks into a lake. You throw in the rock and it causes a wave in the water. This wave can be considered as a manifestation of an energy transfer from the rock to the water. All of the energy is given to the wave from the rock at the moment of impact - after that, the wave maintains (theoretically) that energy.

The important thing to notice, though, is that the wave expands as it travels out into the lake. Its circumference gets bigger as it travels horizontally (as its radius gets bigger....) Therefore the wave is "longer" (if you're measuring around the circumference). The total amount of energy in the wave, however, has not changed (actually it has gotten a little smaller due to friction, but we're ignoring that effect....) therefore the energy has to be shared across a longer wave. This causes the height (and depth) of the wave to shrink as it expands.

What's the mathematical relationship between the increasing circumference and the increasing radius? Well, the radius is travelling at the constant speed, determined by the density of the water and gravity and other things like the colour of your left shoe.... We know from high school that the circumference is equal to the radius multiplied by about 6.28 (also known as 2Pi). The following graph shows the relationship between the radius and the circumference. You can see that the latter (the steeper line) grows much more quickly than the former. What this means is that as the radius slowly expands out from the point of impact, the energy is getting shared between a "length" of the wave that is growing far faster. (note that, if we double the radius, we double the circumference)

The same holds true with pressure waves expanding from a loudspeaker into a room. The only real difference is that the energy is expanding into 3 dimensions rather than 2, so the surface area of the spherical wavefront (the 3-D version of the circumference of the circular wave on the lake...) increases much more rapidly than the 2-dimensional counterpart. The equation used to find the surface are of a sphere is 4PiR2 where R is the radius. As you can see in the following graph, the surface area of the sphere is already at 1200 units squared when the radius has only expanded to 10 units. The result of this in real life is that the energy appears to be dissipating at a rate of 6.02 dB per doubling of distance. (when we double the radius, we increase the surface area of the sphere fourfold.) Of course, all of this assumes that the wavefront doesn't hit anything like a wall or the floor or you......