HUMS 4100 Waves and Sound

Simple Waves

e.g.

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

Speed of Waves

What governs the speed of a wave?

Standing waves

Waves which do not travel (!)
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 were 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

http://mat140.bham.ac.uk/~richard/talks/bessel/drum_harmonics/table.html

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

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)
β is loudness in decibels

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

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
Formally this is done by adding two sine waves

\color{red}{ \begin{array}{l} y_1 \left( x \right) = A\sin \left( {kx} \right) \\ y_2 \left( x \right) = A\sin \left( {kx + \delta } \right) \\ \end{array}}
What if the frequencies of the waves are not exactly the same?

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

Doppler Effect

Waves coming from a moving source have their frequency and wavelength changed. Try the animation
  • During the time taken to emit one wavelength, the emitter moves away a distance v ΔT, i.e. \color{red}{\lambda ' = \lambda + v\delta t}
  • Time taken to emit one λ is \color{red}{ \delta t = \frac{\lambda }{c} }.
  • Hence $$ \color{red}{ \lambda ' = \lambda \left( {1 + \frac{v}{c}} \right),f ' = \frac{f}{{\left( {1 + \frac{v}{c}} \right)}}} $$

Two-Dimensional Waves

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 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? (more or less 3rd expt)

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. Quadrupole nature means that grav. radiation cannot be produced by monopole or dipole system: e.g. supernova collapse (which has plenty of energy) is probably symmetric, so no radiation

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 need to detect motions of << atomic radius in a one ton sapphire crystal.

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, Testing relativistic gravity with binary and millisecond pulsars, in General Relativity and Gravitation 1992, eds. R.J. Gleiser, C.N. Kozameh, O.M. Moreschi. Institute of Physics Publishing (Bristol).

Hence 1993 Nobel Prize