Constructing Physics

Elementary Physics Course Notes

Adam Johnston
Department of Physics
Weber State University

 

[Note: These notes are based on a collection of different classes and lectures, not all of which you may have been in.]

Adam explained the motivation behind the next couple of sections like this:  

As a physicist or a budding physicist, one has a difficult time meeting people, not to mention attracting them for the purpose of dating, friendships, lifelong partnerships, etc.  While it is not just nor right nor fair, one of the biggest social blunders is to say, "I do physics."  (Don't believe me?  Try it for yourself.  It's amazing how dead a conversation can become after explaining to someone what I do for a living.  "Ohhhhh," they say, "I hate physics.")

To try to combat the very unfair stereotype, a physicist or physics student must have another edge.  It is quite true that the majority of physicists in our department are also musicians, and many credit this skill as the reason it was possible for them to ever meet a future spouse.  I, for example, never would have been given the time of day from Karyn except for the fact that she mistook me for a music major, assuming that playing piano in a dorm lounge was a sign that maybe I was interesting.  (It also helped that the dorm lounge was very dark at night, hiding me and increasing the intrigue.)  It was only later that my secret was revealed.  "Karyn, I have something important to tell you," I admitted after we were engaged and I figured I was safe.  "I'm a physicist."

Anyway, we want to help you at becoming socially acceptable.  Thus, we will study music in physics.  But, to understand music, you must understand sound.  To understand sound, you must understand waves.  And to understand waves, you must understand simple harmonic motion.  Working in this progression, we will start with the simple harmonic motion stuff and work our way up and out of social misfittedness.

Simple harmonic motion may have a complex set of changing forces acting on the moving object (such as a bobbing mass hanging from a spring), but it is beautifully simple in how predictable and regular its continual motion is.  Not only is energy conserved, but it is conserved in exactly the same manner with each cycle of the harmonic motion.  Since harmonic motion displays the same bobbing up and down (or swinging to and fro) over and over again, there are a few features of the motion that we can measure:

bullet Amplitude: the amount of displacement (up and down or back and forth) of the harmonically moving object
bullet Period: the time it takes the object to complete one cycle (up and down or back and forth)
bullet Frequency: how often the object completes a cycle.  (Note that the frequency and period describe the same properties of the harmonic motion, just in different ways.  An object could have a period of 0.2 seconds, which is also a frequency of five oscillations per second.  Either of these pieces of information are equivalent in what they tell us.)

Now imagine some simple harmonic motion that is somehow able to move from one side of the room to another.  It would trace out (on the chalkboard, for example) a wave that looks just like waves in the bathtub or in the ocean. A wave is a way of getting the vibrations communicated from one place to another; a way of transporting vibrational energy.  A wave has the properties of amplitude, period, and frequency, and it also has the properties of:

bullet Velocity: how fast the wave vibrations are moving from one place to another.  (Note that matter/stuff is not moving, just the vibration energy.)
bullet Wavelength: the measure of the length of one complete wave cycle (in the direction of its velocity).

As it turns out (and as it was demonstrated), frequency and wavelength are intimately related.  In fact,

velocity = frequency * wavelength

So, for a wave of a fixed velocity (most waves are), if you increase its frequency, the wavelength will decrease.  And, if you decrease the frequency, the wavelength will increase.  We see this as Adam wiggles a slinky or cord and produces waves.  The faster the wiggles (frequency), the shorter the waves become (the more they get scrunched together).

But what does this have to do with music?  First, we can set up a frequency generator to produce certain frequencies.  This is hooked up to a speaker.  The higher the frequency, the higher the pitch of the sound that we hear.  These frequencies are communicated to the air around the speaker, this sets up a wave of vibrating air molecules which eventually vibrate your ear drums and the desired frequency.

However, you might argue that there are lots of different sounds, but not all of them are "music."  What's more is that a violin played at the same pitch as a trumpet sounds much different, despite being the same frequency of wave.  What else is involved?  Adam introduced the idea that waves can interfere with one another.  In some cases, they interfere constructively (building/adding up), and in other cases they interfere destructively (canceling out).  This was demonstrated by shooting waves in opposite directions and watching how they interacted on the wave propagating cord.

If interference happens, then we could set up patterns of waves which interfere in just the right manner to produce areas of lots of constructive interference (known as "antinodes") and points of lots of destructive interference (known as "nodes").  This is known as a standing wave pattern.  Various standing waves were drawn, produced on a slinky, produced on a saw blade (this was hard to see, so we'll try it again next time), and also produced in a column of air using sound.  For the standing sound wave in the tube of air, we noted that only one particular length of tube produced a standing wave of the correct frequency.  At that point everyone had to leave, but this phenomenon foreshadows our next topic: musical instruments.

bullet Examples of traveling waves, sound waves, and interference can all be found under the sound and waves animations.

* * *

Music is an example of sound.  But what is it that makes some sounds more pleasant and "musical" than others?  This is the topic of the day.

Since we understand waves, we will continue to describe the nature of sound as a wave.  Sound waves are audible between 20 and 20,000 cycles per second (or Hertz), and travel through air at around 330 m/s.  Also, while we continue to draw sounds waves to look like water waves ("transverse waves"), they are actually compression waves ("longitudinal waves").  This last point wasn't brought up in the lecture, but it is described more in the text.

To demonstrate the wave nature of sound, Adam pulled out a device which could show the waveforms produced by sound.  In other words, it allowed us to see music.  (This device is called an oscilloscope, in case anyone ever asks.)  As the pitch of some tones went up, it was clear that the wavelength of these waves got smaller, and that the frequency got higher.  Thus, what we call "pitch" in music is just another way of saying "frequency".

We know that waves can interfere.  So, as a wave, sound should be able to produce interference.  Adam made two speakers destructively interfere with one another, producing a more subdued sound.  Interference is the root of standing waves; and standing waves are what are produced in musical instruments.  Open tubes have antinodes at both ends.  A closed tube (closed on one end, open on the other) has an antinode at the open end and a node at the closed end.  A string (fixed at both ends) would have nodes on both ends.  In all of these cases, various standing waves can be produced.  The kind of sound that is heard is simply a combination of the "fundamental" standing wave (the most simple one, or the one with the longest wavelength) and the higher, more complicated, overtones or harmonics.  These were drawn and described.  (You could refer to your text for some more examples and descriptions.)

In all cases where a sound sounds "musical," we notice that the patterns of waves produced are very regular and symmetrical looking, rather than scattered and random.  This is because the harmonics of musical instruments are created at very regular intervals.  Only notes which are "in tune" with one another are produced in a single sized standing wave container.  Likewise, notes which sound good together tend to be notes which could all fit within the same standing wave container.  Adam played a few examples of some good versus bad sounding music.

We wondered what would happen if Adam decided to quit physics and take his music on the road.  Imagine him playing piano on the back of a truck.  It turns out that the sound waves being produced will be greatly affected by the motion of the truck.  You probably have heard ice cream trucks or sirens or horns and noted (no pun intended) that they sound different when they are moving towards you as opposed to when they are moving away from you.  As described in class (and to the children on Adam's street who witness this phenomenon with the ice cream truck), motion of a sound source compresses wavelengths in the direction which the source is moving, which the wavelengths get longer in the opposite direction.  Where the wavelength is smaller, the pitch is higher; and where the wavelengths are longer, the pitch is lower.  This wave phenomenon is known as the Doppler Effect.

bullet Examples of sound waves, interference, and the Doppler effect can all be found under the sound and waves animations.