NOTE: MiniTest 5 is part of the course Final Exam, the part covering previously untested material.
| Date |
Lecture topics |
Equations |
| Mon 11/19 |
General Properties of Waves
- The shape of a traveling wave is totally unrestricted.
- Periodic waves repeat both in time and in space. From this we define a periodic wave's period, frequency, and wavelength.
- Waves can be represented by graphs showing the wave in space at one time (like a photograph), or by graphs showing the motion of one point in the medium versus time. These graphs are easy to confuse, but are not the same thing.
- Period is different from wavelength!
- Considering the relation between these graphs reveals the useful equation v = λ f.
- The v = λ f equation can be used to analyze some things which aren't really waves, but resemble them. For example, a passing train.
- Keep in mind: wave speed is totally different from the speed of a particle in the medium.
- We'll focus specifically on sinusoidally-shaped waves.
- With sinusoidal waves, the motion of each piece of the medium is SHM.
- We can borrow the idea of amplitude from SHM.
- As with SHM, the size of the amplitude is independent of all other wave characteristics (e.g. wave speed, wavelength, frequency)
- Looked at "House of Sound" memnonic for remembering which parameters are independant of which other
Longitudinal / Compression Waves
- Graphical representation of longitudinal waves: displacement viewpoint and pressure/density viewpoint.
- The pressure/density graph is shifted up on the vertical axis, because the pressure/density never goes smaller than zero.
- The vertical axis on pressure graphs is usually chosen to be the deviation of pressure from equilibrium, so that the graph is again equally above and below the horizontal axis..
- The position of maximum displacement is NOT the point of maximum density!
- The pressure/density graph is shifted to the right by 1/4 cycle, or 90° of phase.
- Increased density (compression) implies increased pressure. However, density and pressure are not proportional.
- Pressure is used to describe a force that is spread over an area. (symbol p, unit Newton/meter2 = Pascal = Pa)
- The air around us applies a pressure of 100,000 Pa=100kPa to everything (more or less, depending on the weather).
- That is quite a lot: 15 pounds/square inch in more familiar units.
- We don't notice this because we are immersed in the air. Force applied by the air on one side is balanced by a force applied by air on the other side.
- Sound passing through the air varies this pressure only slightly: a variation of Δpmax=1Pa would be a very loud sound.
- The specific equation relating Intensity and pressure amplitude is given to the right.
|
v = λ/T =λ f
p = F/A
I = (Δp2)/(ρs)
ρs = 400 kg/m2s |
| Mon 11/26 |
General Properties of Waves
- Reflection
- First we looked at reflection from the end of a giant spring, then on the wave machine.
- When a wave bounces off a fixed boundary, the shape inverts.
- When a wave bounces off a boundary where the medium is free to move, the shape does not invert.
- Generally, when a wave encounters any boundary where either the wave speed or medium density changes, there will be some reflection and some transmission. The incoming wave (before it reaches the boundary) is called the incident wave. The greater the difference, the more will reflect and the less will transmit.
- As far as inversion of the reflection, heading into a heavier medium is like a fixed boundary (reflected pulse is inverted), and heading into a lighter medium is like a free boundary (reflected pulse is upright).
- Transmitted waves are always upright (never inverted).
- A table describing all this is available on the course web page.
- For air in tubes, we found that an open end of a tube is a place where pressure is fixed. Therefore, pressure pulses invert when they reflect from an open end. At closed tube ends, the pressure is free.
- The above may seem backwards. But we have seen before that places of maximum pressure are different from places of maximum displacement (for longitudinal waves). At an open end, the air is FREE to be displaced, but it is at a FIXED pressure.
- If a medium is linear, then:
- waves can pass through each other without affecting the other
- while overlapping, the medium displacement is the place-by-place sum of the displacement due to each of the waves. This is superposition, which we've seen before along a time axis.
- Kinetic energy can only be possessed by things with mass. Potential energy can only be possessed by things which can be deformed.
- Kinetic energy is usually hidden from view in a photograph.
- Superposition can result in a shape that looks like there is no wave (e.g. when two pulses of opposite direction meet). But really, it's just that the wave is total in the velocity of the medium, instead of in the displacement of the medium.
- A wave doesn't have mass, so waves don't "have" kinetic energy. Any kinetic energy is in the parts of the medium.
- Nevertheless, waves "carry energy." What we really mean by that is that waves cause energy to move from one part of a medium to the next
Standing Waves
- Vibration of strings: the waves don't look like the traveling waves we've considered so far. Call them standing waves.
- Nodes are places on standing waves that don't move (have zero amplitude of vibration). When counting nodes, include any that might happen at the end of a string.
- Antinodes are places on standing waves that move the most (maximum amplitude).
- A loop is the football shaped part between two adjacent nodes.
- Standing waves also repeat their motion in time, so that they have a period and a frequency
- If you look closely, neighboring loops are vibrating out of phase. Therefore, the standing wave repeats every two loops. Like with traveling waves, we call this repetition length the wavelength.
- For sinusoidal standing wave: the wave amplitude is the amplitude of vibration at an antinode. That is, it is the maximum displacement that occurs anywhere in the wave.
|
|
| Fri 11/30 |
- Showed that superposition of two counter-propagating waves (with same wavelength) create a sinusoidal standing wave.
- Thus, a standing wave can be considered to be made of two traveling waves. All three waves have the wave wavelength, period, and frequency.
- Most important consequence: v = λf still works for standing waves..
- Standing Wave Shapes
on strings
- For a uniform string, wave speed will be the same everywhere. Frequency of motion also is the same everywhere. Thus, from v = λf, the wavelength must be the same everywhere.
- On a rope with two fixed ends:
- Nodes at the ends limits the possible wavelengths. The specific sinusoidal possibilities are called modes of vibration, and they are numbered according to the number of loops they have.
- The frequencies of the modes are harmonically related. The mode with lowest frequency is called the fundamental mode, and others are called harmonic
- If string is fixed at one end and free at the other, then a different set of shapes is allowed.
- The frequencies of the modes are still harmonically related. However, only odd harmonics are possible (that is, their frequencies are odd multiples of the fundamental frequency).
- The formula for the fundamental is also different, with a 4 in the denominator.
- Regardless of whether either end is fixed or free, the allowed frequencies are harmonics (mutliples) of the lowest allowed frequency.
- When working with these modes, remember that usually L≠λ
- It is also possible for an end to be between the fixed and free cases, so that neither a node or anitnode are at the end. But for many situations, fixed or free are good models.
- Similar standing waves of air can occur in a tube. As before, can focus on either displacement or pressure of air. We will focus on pressure.
- Recalling how pulses inverted or remained upright upon reflection, we already knew that an open end is "fixed," while a closed end is "free." This may seem backwards, but you have to focus on pressure, rather than displacement.
- An open tube end gives a fixed pressure (pressure fixed at atmospheric; can't change pressure because there is nothing to push against). This implies a pressure node.
- A closed tube end gives a free pressure end (can change pressure by pushing air against end). This implies a pressure antinode.
- "Open tube" means open at both ends, "closed tube" means closed at one end and open at the other.
- It is possible to have a situation where both ends are free. Then the frequencies are the same as when both ends are fixed.
- Instead of worrying about fixed/free and open/closed, you can just check whether the ends are the same (use the first frequency equation) or the ends are different (use the second equation)
Musical Intervals
- Pitches sound nice together when their frequency ratio can be expressed with small whole numbers. Some names of intervals are
- octave (freq ratio 2/1)
- fifth (freq ratio 3/2 or 1.5/1) [has nothing to do with the fraction 1/5]
- fourth (freq ratio 4/3 or 1.333/1) [has nothing to do with the fraction 1/4]
- To change a pitch by some interval always multiply or divide.
- Because 12 fifths is almost equal to 7 octaves, we tend to be driven towards 12 notes per octave. But this doesn't quite work, because 12 fifths aren't exactly 7 octaves.
- In modern western music, this is handled by dividing the octave into 12 "equal" smaller intervals, semitones. That means that no note combinations are perfect fifths or fourths, but many combinations come very close. This is called equal temperament.
- "Equal" semitones means they have equal frequency ratios. Going up 12 semitones makes an octave. Thus, to go up a semitone, one needs to multiply the frequency by the 12th root of 2.
|
fn = n v/(2L)
fm = m v/(4L)
fn = nf1
fm = mf1
fourth: f1/f2=4/3
fifth: f1/f2=3/2
octave: f1/f2=2/1
semitone: f1/f2=12Ö2/1
12Ö2 = 1.05946 |
| Mon 12/3 |
Standing Waves
- Fourier's theory with standing waves: natural modes can be combined (via superposition) to give any other shape that is possible. Shapes will tend to change with time.
- When multiple modes are vibrating at the same time, the period (& frequency) of the total motion will be the same as the period (& frequency) of the fundamental. In an instrument, each mode will create a partial in the sound spectrum.
Getting the right pitches from tubes and strings
- From the frequency equation, we basically have three ways we can get different pitches: vary the mode, the wavespeed, or the length.
- Changing pitch by changing mode:
- Harmonics are evenly spaced on a linear frequency scale, but octaves are evenly spaced on a logarithmic frequency scale. This means that for a given fundamental, higher octaves contain more harmonics of that fundamental.
- This is one way brass intruments obtain different notes. It is the only way bugles do (taps, revele).
- Changing pitch by changing wave speed
- String instruments can vary the wave speed by varying the string tension.
- A "washtub bass" uses this to play notes.
- More commonly, this is used to "tune" the instrument, but not to play melodies.
- air instruments can vary pitch by changing the gas in the tube (e.g., helium), but this isn't very practical!
- Changing pitch by changing length
- For most instruments, this is the primary method.
- When comparing lengths and frequencies on the SAME instrument, it makes no difference whether the ends are open or closed. In either case, f µ 1/L.
- Higher frequencies are obtained by going to shorter tubes.
- Caution: Sentences about changing length are sometimes phrased so that they sound like adding, instead of multiplying. For instance, Rossing might write "new length is 5.946% longer than old length." Thus just means the same thing as (new length)=1.05946*(old length).
|
f µ 1/L
|
| Fri 12/7 |
Wind Instruments: Getting the pitches
- Four methods of changing tube length are...
- Multiple pipes (organs, pan pipes)
- Slide (trombone, slide whistle)
- Valves (trumpet, french horn)
- Tone holes in the side of the tube (penny whistle, recorder)
- Somtimes fancy key mechanisms allow opening holes that would be hard to reach with hands (clarinet, flute, saxophone)
- or instruments that vary tube length by putting holes in the side of the tube: A very large hole establishes a new end of the tube. A smaller hole will only be partially effective, so that the new effective tube end is between the original end and the hole.
- Dispell a Myth: Blowing harder is NOT a useful way to change pitch on a wind instrument. In some cases, higher pitches do require blowing harder, but they are not caused by blowing harder.
Wind Instruments: Creating the Sound
- Tubes have natural frequencies, but how do we cause them to actually vibrate? How do we drive the resonator?
- classification of (most common) wind instruments:
| NAME |
VIBRATION SOURCE |
WAYS TO GET NOTES ON RESONATOR |
EXAMPLES |
| brass |
lips |
harmonics |
bugle, natural horn |
| harmonics + slide |
slide trombone |
| harmonics + valves |
most others |
| woodwinds |
flutes |
air stream |
many pipes |
organ, pan pipes |
| holes on tube |
flutes, recorders |
| reeds |
reed |
holes on tube |
clarinet, saxophone, oboe, bassoon |
- For the most part, the relationship between the second and third collumns is just a matter of tradition. There is no reason you couldn't make a slide reed instrument
- The only relationship based in physics is the lips-harmonics relationship. In order to access harmonics higher than the fundamental, you need some control of the vibration source, which is not available with air stream or reeds.
- The "air stream" vibration occurs as air is blown at the edge of a hole into the tube. The air oscillates between going through the hole into the instrument, or across the hole and not into the instrument.
- To really get well defined pitches, we need feedback from the resonator to the vibration source.
- Feedback is whenever the driven element has an effect on the driving element of a system.
- The word is used most commonly for the situation where a microphone and speaker are too close together: the microphone is supposed to drive the speaker, but if the speaker can send its sound back to the microphone then we have feedback.
- In that case, one round trip has to take one period, do that distance between microphone and speaker sets frequency.
- In air instruments, oscillations of tube feed back to control frequency of driving vibrator. The mechanism for this can most easily be understood by thinking of pulses of air:
- flutes: blowing makes pressure pulse which travels the tube length twice per cycle. Pulse inverts at both ends (at driving end because it sucks more air into the tube). Tube is effectively open at both ends.
- reeds and brass: blowing makes pressure pulse which travels the tube length four times per cycle. Pulse inverts only at bottom. At driving end, rarefaction sucks reed or lips closed. Tube is effectively closed at driving end.
- Note connection betwen number of tube traverses per cycle and the number in the bottom of the frequency equation.
- The picture of reflecting pulses is slightly misleading: the actual pulses are much wider.
- fine tuning the behavior of woodwind instruments:
- flutes, etc. are open tubes, and thus have all harmonics
- clarinets are closed cylinders, and thus have only odd harmonics
- all other (common Western) woodwinds are closed cones, and thus have all harmonic
- common brass instruments are also not cones, although they are not cylinders either.
- Wood winds accessing higher modes: A register hole on woodwind instruments (different from the tone holes), when opened, tends to fix the pressure in the middle of the tube. That is, it tends to force a node in the middle.
- Thus the fundamental can't happen, and the instrument plays the next natural mode.
- Note that, depending on which tone holes are open, the desired position of the node for the second partial moves. Thus one would like to have the placement of the register hole move.
- On most instruments, that's not possible. But it turns out that having one register hole in generally the right place works well enough.
- Saxophone does have 2 register holes, and elaborate key mechanisms so that which one is used depends on which tone holes are closed.
- On a guitar, an equivalent to the register hole is used for playing harmonics.
- Instead of just shortening the string (by holding part of it), the string is lightly touched at the appropriate point. A node is forced at that point, meaning that certain modes can't happen. Most used point: If you touch the exact middle, then odd harmonics are eliminated, leaving only the even harmonics.
|
|
HINT: It is more effective to memorize these as relations between concepts. It is less effective to memorize these as strings of letters.