Oscillations, Sound and Music




A drumhead (Talman, MCSD Denver)

 In music we perceive
    • Loudness
    • Pitch
    • Timbre
    What do they mean?
  • Why does "fingering" change a note?
  • Why does tightening a string change the note?
  • Can we hear any note?
  • Why do we hear in octaves?

Simple Harmonic Motion

We are interested in waves, but it is usually easier to start with oscillations (one-D waves) There are many forms of oscillatory motion is nature: e.g. First some nomenclature

The Spring

Easiest mathematically is the spring. One more special force:
  • x is extension (not the original length)
  • - sign to get the direction right
  • k is the spring constant
  • \color{red}{ P = 2\pi \sqrt {\frac{m}{k}} }
  • stiff springs (large k) vibrate faster
  • large masses (large m) vibrate slower

Simple Waves

Pulses

Easy to see what we mean by velocity/speed of wave
If wave travels a distance 
δx in δt = t₂-t₁

then 

 v = δx/δt

Periodic Waves

Continuous waves which repeat are periodic waves
Because it is mathematically simple, we will often use harmonic waves: i.e. sine waves: As before v is speed at which one crest (or trough) moves

Longitudinal Waves

Waves can be

Transverse waves

where the movement is perpendicular to the velocity

Damped S.H.M.

Reality rearing its ugly head again. If there is friction in the system, the frequency is almost the same, but the amplitude decreases exponentially with time. The extreme cases are

Forced oscillations and resonance

Up to now, we have assumed that the system is started off at some time and allowed to oscillate. However, can arrange for the system to be driven, so that driving force varies in time. Watch the demo:

If driving frequency ω0 matches natural frequency of system ω, then size of oscillations grows

Usually some damping, so does not grow without limit. e.g.

Speed of Waves

What governs the speed of a wave?

Interference

Back to waves in general
e.g pulses: Waves will pass through each other with no (permanent) effect on each other.

Total displacement will consist of sum of individual displacements e.g. collision of two pulses

Harmonic Waves

We are usually more concerned about interference between harmonic waves

See how two sin waves add
What if the frequencies of the waves are not exactly the same?
If this is a sound wave, we hear it as "beats":
  1. 400 + 401 Hz
  2. 400 + 410 Hz
  3. 400 + 420 Hz
Note closer frequencies are the wider the separation

from Joe Wolfe http://www.phys.unsw.edu.au/music/index.html

Standing waves

Waves which do not travel (!)
Now we can start on sound and music:

Simplest is a guitar string
  • Ends don't move: can only fit a fixed number of half-waves into the string
  • L = (n/2) λ, n = 1,2,3,4,
  • These are known as "harmonics": n=1 is fundamental, n = 2 is "first harmonic" and so on.
  • e.g. guitar has all wavelengths that satisfy (n/2)λ = L so frequencies are 
  • f0 = 1/2v/L   fundamental 
f1 = 2/2v/L   1st harmonic 
fM = M/2 v/L  (M-1)th harmonic
  • In principle, all of these can be excited, but in practice the amount of energy required for higher harmonics is larger, so that they are less easily excited.Hence actual note heard is superposition of many frequencies

Nodes are points where string does not move: points half way between with maximum motion are antinodes.

Wind Instruments

Most wind instruments have one open end e.g. flute, oboe, beer-bottle, organ...
Watch the animation

Note what is really happening is that the molecules move a lot at the antinodes but are stationary at the nodes.In particular,there is no motion at the closed end. The fundamental in this case has 1/4 wave 
 
λ = 4 L

λ = 4L/3
This is 3/4 wave.

Percussion

Drum harmonics

Notes and Music

Various aspects become obvious: In terms of this physics,
In general, each frequency will have a different amplitude, and the sound depends on this. Why does same fundamental on different instruments sound different?:
Can analyse this:
Not obvious that this gives us the original string shape, but we can add in successive harmonics to produce the original shape
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Means we can look at different shapes and find the harmonic content
Note that once we understand Fourier analysis, we can produce "impossible" sounds (see http://asa.aip.org/sound.html )
e.g. Shephard scale
e.g. Rissett scale

Nano-guitar.

Reflection of waves

Initially in 1-D: can see reflection and refraction.

Easiest to see with a pulse.

Wave exerts upward force on support

=> downward force on rope

=> inversion of original wave

  • If end is free to move wave is not inverted
  • No force on free end
  • So wave is just reflected
Reflection from an "interface"

Wave is partially reflected and partially transmitted Try the animation
Note: "refracted" wave propagates at a different speed.

What happens when a light wave hits glass?

Some is always reflected

Energy in waves:

Note how the string moves: Not only does the wave move forward, but each individual particle is displaced. This means a wave carries energy. Each particle would have

P.E. = 1/2 k x ²

where x is the maximum displacement. Also

K.E. = 1/2 m v ²

so more rapidly oscillating string means larger velocity so...

Energy in sound waves:

We are sensitive to power/unit area: this is intensity I

\color{red}{ I = \frac{{\delta P^2 }}{{2\rho c}}}
e.g. maximum overpressure the ear can stand is ~ 30 Pa (atmos pressure ~105 P): this is only about .03% At maximum allowable level, ear only absorbs about 1 μW (!)
β is loudness in decibels

You can try this out for yourself at the University of New South Wales site
We also use db to express the signal to noise ratio:

Electromagnetic Waves

Light: Also a wave motion but
Note: we will need 
c = λf

repeatedly. The "energy' in the above diagram gets explained later. We "see" only one octave.
Why do we see so little of the spectrum? Answer lies in the transparency of the atmosphere

Two-Dimensional Waves

Back to waves in general

Although it may sound silly, we are going to need a way to distinguish waves from particles.

A point source of waves produces spherical waves. If we see them a long distance from the source, they look like plane waves.

Diffraction and Interference

e.g. a ripple tank
When we have two sources, we can get interference between them
At the green points, we will get constructive interference (crests add)

At the yellow points, we get destructive interference (crest and trough)

This means we get nodal lines: (this shows the lines of maxima)
"Young's slits" is this interference experiment done for light
For water waves, this can be done by two sources.

e.g. a light produces Na (sodium) light (λ = 589 nm) and it passes through two slits separated by 5μ (5x10-6 m). How will it appear on a screen at a distance of 1 m?

Gravitational Waves

A final consequence of General relativity
  • Vibrating charge radiates E.M. waves (i.e. light)
  • Vibrating mass radiates grav. waves
Differences:
  1. Gravitational force between 2 electrons ~ 10-42 electric force Radiation is quadrupole, not dipole, which also means it is still weaker
  2. Can only be produced by assymetric system

Hence (well, more or less hence!) it requires a large amount of mass to produce a grav. wave, and a large amount to see one: e.g. coalescing binary stars

Note what is happening here is that space-time is stretching (!) History:

Hulse and Taylor: Binary Pulsar

PSR1913+16 discovered 1974. Like all pulsars, emits very regular radio pulse every 59 ms. (Frequency is 16.940 539 184 253 Hz: i.e. is better known than atomic clocks)

This consists of two neutron stars, in orbit 106 km in radius, with period of hours. Change in frequency allows orbit to be calculated exactly, and can measure..

Rate of precession = 4.22662 0/yr (i.e. 30,000x that of Mercury)

and that pulsar is losing energy, by gravitational radiation (mass~1.4 M0, and accns are large)

Decrease of the orbital period P (about 7h 45 min) of the binary pulsar PSR B1913+16, measured by the successive shifts T(t) of the crossing times at periastron; the continuous curve corresponds to $$ \color{red}{ T(t) = \frac{{t^2 }}{{2P}}\frac{{dP}}{{dt}}} $$ given by the general relativity (reaction to the gravitational waves emission).

Reference: Taylor J.H. 1993, General Relativity and Gravitation 1992, eds. R.J. Gleiser, C.N. Kozameh, O.M. Moreschi. Institute of Physics Publishing (Bristol).

Hence 1993 Nobel Prize

Final topic for this part of the course is quantum mechanics