Modules 8

1
Modules 8 9 Waves

• Serway Faughn Chapters 13 14

2
Module Study Objectives

• Vibrations
• Simple harmonic motion
• Wave motion
• Interference

• Sound
• Doppler effect
• Standing waves

3
Vibrations Waves

• Vibrations are common in nature
• They produce waves
• Sound
• Ocean waves
• Bridges buildings

4
Simple Harmonic Motion

• Simple harmonic motion (SHM) occurs when the
  force on an object is proportional to the
  displacement from the equilibrium position.
• Motion is periodic.

5
Hookes Law SHM

• When Hookes law applies we get SHM

6
Acceleration Under SHM

• Newtons 2nd law tells us acceleration under SHM
  is proportional to displacement and opposite to
  it.

7
Elastic Potential Energy

• Energy stored in a stretched/compressed elastic
  material

8
Velocity As a Function of Position

• Conservation of energy for an oscillating system
  provides an expression for velocity.

9
SHM Uniform Circular Motion

• The projection of uniform circular motion on a
  plane is SHM.
• Locomotives use piston motion (SHM) to make
  wheels turn.

10
Period and Frequency

• Period is time for rotation/oscillation
• Frequency inverse of period
• Period for SHM depends on mass and spring
  constant

11
Position As a Function of Time

• The equivalence of SHM and circular motion
  provides displacement at a given time.

A
??t
?
x
12
The Pendulum

• Exhibits periodic motion
• Approximates SHM for small angles (lt150)

13
Damped Oscillations

• Oscillations are damped by friction, which
  reduces the mechanical energy of the system over
  time.

14
Wave Motion

• The world is full of waves sound, ocean waves,
  earth tremors, radio, TV etc
• Wave motion of a disturbance (not the medium it
  travels in)

15
Types of Waves

• Transverse (eg pond ripple)
• Longitudinal(eg sound)

16
Amplitude

• Amplitude A of a wave is the maximum distance
  from equilibrium

A
17
Wavelength

• Wavelength lambda is the distance between two
  successive points that behave identically.

?
18
Frequency, Amplitude Wavelength

• The definition of amplitude wavelength enable
  the derivation of a relationship between wave
  velocity, frequency wavelength

19
Example

• The middle C note on a piano has a frequency of
  262Hz and a wavelength of 1.31m, so using vf? we
  get the speed of sound in air343 m/s.

20
The Speed of Waves in Strings

• Depends on the tension in the string and the mass
  per unit length.

21
Superposition of Waves

• If two or more travelling waves are moving
  through a medium, the resultant wave is found by
  adding together the displacements of the
  individual waves point by point.

22
Interference

• Constructive waves in phase that produce a
  bigger amplitude
• Destructive waves out of phase that can cancel

23
Reflection of Waves

• When a wave pulse reflects from a rigid boundary,
  the pulse is inverted.
• When the boundary is free, the reflected pulse is
  not inverted.

24
Sound

• Longitudinal wave
• Compression and rarefaction of air (or water)
• Requires medium
• Slower than light

25
Sound Waves

• Audible waves (20Hz-20kHz)
• Infrasonic waves (lt20Hz) eg earthquakes
• Ultrasonic waves (gt20kHz) eg dog whistles, blood
  flow meters

26
Speed of Sound

• Depends on compressibility and inertia.
• Affected by temperature
• 331 m/s _at_ 00C
• 343 m/s _at_ 200C

27
Energy and Intensity

• Intensity I is the rate at which energy flows
  through a unit area A perpendicular to direction
  of travel.
• Threshold 10-12 Wm-2 Pain 1 Wm-2

28
Intensity Levels in Decibels

• Relative intensity level aims to mimic human
  ears approximately logarithmic response
• Reference intensity I0 1.0x10-12 Wm-2.

29
Example

• Find the decibel levels for 1 and 2 machines,
  each of sound intensity 1.0x10-5 Wm-2.
• ?110log(10-5/10-12)70dB ?2 (2
  machines)10log(2x10-5/10-12)73dB.

30
Spherical and Plane Waves

• Sound radiates out from a point, falling in
  intensity with distance according to an inverse
  square law.

31
Doppler Effect

• Relative motion between wave source and observer.
• Approach observer detects more waves (higher
  frequency)
• Recession observer detects fewer waves (lower
  frequency)

Higher pitch
Lower pitch
32
Doppler Effect

• Can calculate Doppler shift in frequency for
  moving observer o and source s.
• Signs depend on direction of motion.

33
Example

• Determine Doppler-shifted frequency for a train
  with a 500Hz whistle approaching at 40m/s. (Use
  sound speed 345m/s)
• Answer
• 500?345/(345-40)566Hz

34
Shock Waves

• Motion faster than sound speed produces a shock
  wave.
• When aircraft exceed the speed of sound, we get
  sonic booms.

35
Interference of Sound Waves

• Can sum or cancel
• Depends on relative phases
• Applications stereo speakers, reducing aircraft
  cabin noise

36
Standing Waves

• Two oppositely travelling waves in a string can
  combine to produce a standing (non-moving) wave.
• Example musical instruments

37
Standing Wave Formula

• For a string fixed at both ends, the possible
  standing wave patterns follow a sequence of
  frequencies.

38
Forced Vibrations and Resonance

• Resonance when a string is forced to vibrate at
  its natural resonant frequency and so vibrates
  with a large amplitude.
• Example Tacoma narrows bridge disaster wind
  produced a resonant oscillation.

39
Standing Wave in Air Columns

• If a pipe is opened at one end, the column of air
  is free to move at one end.
• A different pattern of standing waves results
  compared to the case of two ends fixed.

40
Beats

• Sound alternates between louder and softer.
• Two slightly different frequencies alternate
  between constructive and destructive
  interference.
• Can be used to tune a musical instrument.

41
Quality of Sound

• Quality or timbre of a sound is the mixture of
  harmonics
• Human speech and music involve complicated
  harmonics

42
The Ear

• Complex structure, related to distant sea-going
  creature ancestors
• Has some natural protection against loud sounds
  but this is not enough for modern civilisation

43
End of Module

• Any questions?