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BIT1002 Waves and Sound

By the end of this you should understand waves and sound, which means understanding
  1. Simple Waves
  2. Interference
  3. Speed of Waves
  4. Sound
  5. Intensity
  6. The Doppler Effect

Simple Waves

Easiest to visualize are water waves or waves in string One dimensional waves: e.g.

Pulses

If wave travels a distance 
δx in δt = t₂-t₁

then 

 v = δx/δt

Continuous waves

Need not be periodic

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

Standing waves

e.g. vibrating violin string.
Waves can be
  1. Longitudinal
  2. Transverse

Longitudinal Waves

Waves can be

Transverse waves

where the movement is perpendicular to the velocity

Wave Function

The wave function describes the shape of the wave: if the wave is non-dispersive (i.e. doesn't change shape)

y(x-vt) describes the shape at any time t

A travelling wave:

What is correct description of this wave?
  • What is λ?
  • What is the velocity of the wave v?
  • What is f?
  • What is A?
 t = 0.0s
t = 0.1s
t = 0.2s

Speed of Waves

What governs the speed of a wave?

Sound is often produced by strings: speed of wave probably depends on

Interference

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}}
so
\color{red}{ \begin{array}{l} y\left( x \right) = y_2 \left( x \right) + y_1 \left( x \right) \\ = A\left[ {\sin \left( {kx} \right) + \sin \left( {kx + \delta } \right)} \right] \\ \end{array}}
giving
\color{red}{ y\left( x \right) = 2A\cos \left( {\frac{\delta }{2}} \right)\sin \left( {kx + \frac{\delta }{2}} \right)}
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

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

Standing waves

Waves which do not travel (!) Try the animation

Sound

Practically: we are very interested in sound, since it forms such a large part of our communication.
Most sound waves are set up by standing waves 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

e.g. a violin string has an area of 10-6 m2, a density of 3000 kg m-3 and a tension of 200 N.
  1. What would be the speed of a wave in it?
  2. What notes would it produce if it has a length of 40 cm?

Notes and Music

Nano-guitar.

Notes and Music

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.
A beer bottle is about 25 cm high. What would you expect the fundamental note to be at? (speed of sound = 330 ms-1)
  1. 33 Hz
  2. 330 Hz
  3. 3.3 kHz
  4. 33 kHz

In general, each frequency will have a different amplitude, and the sound depends on this.

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}}}
Units of I? watts/m2: check

e.g. maximum overpressure the ear can stand is ~ 30 Pa (atmos pressure ~105 P)
β is loudness in decibels

e.g. my amplifier puts out 100 W. How much louder does it sound at 100 W than it did at 20?
We also use db to express the signal to noise ratio:

Doppler Effect

Waves coming from a moving source have their frequency and wavelength changed. Watch 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 $$ \color{red}{ \lambda ' = \lambda \left( {1 + \frac{v}{c}} \right),f ' = \frac{f}{{\left( {1 + \frac{v}{c}} \right)}}} $$
e.g. a train whistle has a normal frequency of 550 Hz. If it is heard with a frequency of 580 Hz, is it moving towards or away from the observer? e.g. a train whistle has a normal frequency of 550 Hz. If it is heard with a frequency of 580 Hz, what is the speed of the train?

Ultrasound

Very short wavelength sound can be used to image objects.
Pulses
  1. are produced by transducer (f ~ 6Mhz, c ~ 1500 ms-1. What would λ be)?
  2. travel through body
  3. are reflected by interfaces: e.g. weakly where one kind of tissue changes to another, strongly by bone
  4. reflected back to transducer
  5. turned into picture of body by measuring time delay. What would delay time for pulse reflected from backbone be?

Needs either transducer moved or (better) array of transducers to see full image. Non-invasive technique, with much less risk involved than with X-rays. Can also use this with Doppler effect to see moving organs: e.g. to pick up foetal heart-beat. Can step up power (to 107 W m-2) to zap kidney stones Part of physics research at Carleton

http://www.physics.carleton.ca/~bog/index.html

and now we talk about Temperature and Heat