Waves and Sound

1
Waves and Sound
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Remember Periodic Motion?

• Motion which repeats in a regular cycle
• Pendulum, vibrating spring, vibrating guitar
  string

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Simple Harmonic Motion

• Motion around a point of equilibrium
• Force proportional to displacement of object from
  equilibrium

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What is a wave?

• Wavedisturbance that carries energy through
  matter or space
• Note that the actual matter does not travel far
  but the energy can- the energy in this wave could
  have traveled from Alaska!

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Classification of Waves

• Waves Are
• Mechanical or Non-Mechanical
• One (or More) Pulses or Periodic
• Longitudinal or Transverse or Combined

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Mechanical Waves

• Require A Medium For Transmission
• Medium Mass / Atoms / Material
• Transmitted Via Vibration Of Particles In The
  Medium Around A Rest Position
• Examples
• Sound
• Water Wave

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Non-Mechanical Waves

• No Medium Is Required For Transmission
• Can Be Transmitted Through Empty Space
• Examples
• Visible Light
• Infrared Or Ultraviolet Light
• Radio/TV Waves Microwaves
• Any Electromagnetic Radiation

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Pulse vs. Periodic

• Pulse
• A Single Vibratory Disturbance

• Periodic Wave
• A Series Of Regular Disturbances
• Regular Identical Evenly Timed

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Transverse waves

• Disturbance is perpendicular to the motion of the
  wave
• http//www.youtube.com/watch?vcPKGa2DsIs0

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Longitudinal Waves

• Disturbance is parallel to motion of wave
• Ex- sound waves
• Fluids usually only transmit longitudinal waves

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Surface Waves/Elliptical Waves

• Underwater, waves are longitudinal but at the
  surface they have elements of both longitudinal
  and transverse
• Motion of a particle on the surface is an ellipse

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Torsional Waves

• Twist around a central axis
• Like Tacoma Narrows Bridge

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Wave properties

• Equilibrium
• Crest
• Trough
• Amplitude

• Phase
• Wavelength

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Amplitude

• Maximum displacement of a particle in a wave from
  the equilibrium
• Examples brightness of a light, loudness of a
  sound

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Wavelength

• Distance between 2 corresponding locations
• Usually measured from crest to crest or trough to
  trough
• Symbol is ?

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Amplitude and Wavelength

• These waves have the same wavelength but
  different amplitudes
• These waves have the same amplitude but different
  wavelengths

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Phase

• Points On A Periodic Wave Are In Phase If They
  Have
• Same Displacement From Rest Position
• AND
• Same Direction Of Motion

• C and F are In Phase

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Phase

• Points that are in phase act the same- they are
  a whole multiple of a wave apart
• Since wavelength is one complete cycle, we
  usually refer to it as 360?
• So in phase n360

• Points that are out of phase are not a whole
  multiple of 360? apart- they can be any of
  degrees apart
• We usually look at 90?, 180?, and 270? apart

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Phase Problems-

• Using A as a reference, which point(s) are
• 360? in phase
• 90?out of phase
• 180? out of phase
• 270?out of phase

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Frequency

• Number of vibrations per second
• Symbol is f
• Unit is Hz (1/s)

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Period

• Time to complete one cycle
• Symbol is T
• Unit is s
• T1/f

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Speed

• Speed of a wave wavelength x frequency
• v ?f
• Examples- we see the baseball hit the bat before
  we hear it b/c light wave travels faster than
  sound wave

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Comparing Wave Speeds

• Light 3.00 x 108 m/s
• Sound 3.31 x 102 m/s
• We See The Lightning Flash Before We Hear The
  Thunder.
• We See The Bat Hit The Ball Before The Crack Is
  Heard

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Speed of a Wave on a String

• For faster waves tighter string (more tension)
  or lighter string (less mass per length)
• Mass/length is known as the linear mass density

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Speed of wave problems

• A ball of string is purchased at a local hardware
  store. According to the manufacturer, the package
  contains 100 yards (91.5 meters) of string and
  has a mass of 12 oz (341 grams)
• What is the string's linear mass density?
• If the string's tensile strength is 90 N, what is
  the maximum speed a pulse could travel along the
  string?

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solutions

• Mass/length 3.73 x 10-3 kg/meter
• Speed155.3 m/sec

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Wave Graphs- same shape but different info

• Vibration graph- shows behavior at one spot
• Waveform graph shows wave behavior in many spots
  at one time

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Problems

• A periodic wave goes through twenty complete
  cycles of its motion in 4.0 seconds
• What is the frequency of the wave?
• What is its period?
• Determine the frequency of a wave whose period is
  5.0 seconds

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Wavefront

• The Locus Of Adjacent Points Which Are In Phase
• Such As The Crest Of A Water Wave

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Spherical Wavefront
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Spherical Wavefront
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Periodic Wave Phenomena

• Superposition/Interference
• Resonance
• Doppler Effect
• Diffraction
• Reflection
• Refraction

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Waves at An Interface

• Interface
• A Boundary With A Different Medium
• Part Of The Wave Is Reflected
• Part Is Transmitted Through The Second Medium
• Part Is Absorbed (Turns Into Heat)
• Speed can change

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Reflection

• At a rigid boundary, when wave hits with an
  upward force, the boundary medium will react with
  a downward force so reflected wave is
• INVERTED

• If boundary is nonrigid (it can move) wave will
  reflect in same orientation

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Refraction

• When a wave enters a new medium velocity can
  change causing wave to bend

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Doppler Effect

• A Variation In Observed Frequency When There Is
  Relative Motion Between A Source And An Observer

• Approaching
• Higher Frequency Observed
• Receding
• Lower Frequency Observed

• Sound
• Pitch Changes
• Light
• Color Changes

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Doppler Effect Examples

• Siren Passing

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Doppler Effect
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Calculations involving Doppler Effect

• Let fs be the source frequency and fd be the
  detected frequency
• If source moving towards you, frequency will
  increase so choose or - accordingly
• fd(vvd)/(vvs) fs
• Thus, if moving away, frequency will be lower
• If moving towards, frequency will be higher

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Example Doppler

• A car is traveling 20 m/s away from a stationary
  observer. If the cars horn emits a frequency of
  600Hz, what frequency will the observer hear?
• Use v340m/s for the speed of sound

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Solution

• Since car is traveling away from observer,
  frequency will be lower
• fd(3400)/(34020) 600Hz 567Hz

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Breaking the sound barrier

• Speed of sound varies in different mediums
• When something travels faster than the local
  speed of sound it breaks the sound barrier

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Breaking the sound barrier
Regions of constructive interferenceSHOCK WAVES
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Superposition of waves

• When 2 waves meet, the displacement in the medium
  is the sum of the individual displacements
• They then continue, unchanged by their meeting

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Constructive Interference

• Maximum Constructive Interference Occurs When The
  Phase Difference Is 0 In Phase

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Destructive Interference

• Maximum Destructive Interference Occurs When The
  Phase Difference Is 180 Out of Phase

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Interference Patterns

• Symmetrical Patterns Produced By Sources In Phase
  In The Same Medium

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Interference Can Produce Colors
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Interference Patterns
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Interference Patterns

• Maximum destructive interference produces nodes
• Maximum constructive interference produces
  anti-nodes
• http//www.physicsclassroom.com/mmedia/waves/ipd.c
  fm

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Nodes and Antinodes

• Nodes-
• net displacement0
• Antinodes-
• Net displacement max

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Nodes Anti-Nodes

• Nodes
• Points of NO DISPLACEMENT
• Anti-Nodes
• Points of MAXIMUM DISPLACEMENT
• Line of Nodes
• Smooth Area

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Standing Waves

• Each object has a natural frequency at which it
  is willing to vibrate
• If you force a vibration at this frequency,
  object will resonate or vibrate at increasing
  amplitude
• Next shower, try to test this- sing different
  notes until you reach one that is significantly
  louder (increased amplitude)
• We can make this using a wave reflecting off a
  boundary at the same frequency, amplitude, and
  wavelength

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Resonance

• If small, regular forces applied at just the
  right time it can increase the amplitude of
  vibration
• Ex- trampoline, maybe Tacoma Narrows Bridge?
• In a string, this depends on its length- always
  draw!
• http//www.ngsir.netfirms.com/englishhtm/StatWave.
  htm

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Nodes, Antinodes in Standing Waves

• Nodes and antinodes alternate
• Each node is 1/2? from the last
• We use this to determine how standing waves form

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Standing Waves
Notice there are distinct wavelengths that can
produce these standing waves. Note 1st
overtone2nd harmonic
2nd harmonic 3rd harmonic 4th harmonic
fundamental
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Harmonics vs overtones

• Lets take a guitar string the fundamental
  (also known as 1st harmonic) is when there is ?/2
  wavelengths

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2nd harmonic (1st overtone)

• The second harmonic occurs when you have twice
  the fundamental so 2(?/2)1 wavelength

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3rd harmonic

• Now we have 3x the fundamental or 3(?/2)3?/2 or
  1.5 wavelengths
• Are we seeing the pattern here?
• NOTEuse the term harmonic since it matches the
  math

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Determining harmonic frequencies

• Consider an 80-cm long guitar string that has a
  fundamental frequency (1st harmonic) of 400 Hz.
• What is the wavelength of the 2nd harmonic?

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• wavelength is 160 cm or 1.60 m.
• Now what is the speed of the wave?

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• speed frequency wavelength
• speed 400 Hz 1.6 mspeed 640 m/s
• Now, the speed of the other harmonics is the
  same- you can use their wavelengths to determine
  the frequency of each harmonic
• What is the frequency of the 2nd harmonic?

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solution

• Wavelength of 2nd harmonic would be 0.8m
• fv/?
• f640/0.8800Hz
• Now calculate frequency of 3rd harmonic

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Standing waves- strings

• Strings are fixed on both ends
• How does the fundamental frequency compare to the
  length of the string?
• Draw the fundamental and 2 harmonics

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Resonance in Strings- Probs

• Two identical vibrating strings are 2.0 meters
  long and uniform in density. One is fixed on both
  ends, the other is free on one end.
• What is the wavelength of the fundamental
  frequency along the fixed-end string?
• What is the wavelength of the fundamental
  frequency along the free-end string?
• What is the wavelength of the first overtone
  along the free-end string?
• What is the wavelength of the second overtone
  along the fixed-end string?

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Resonance and Wave Speed

• Now put it together with wave speed in a spring
• Suppose a string of length 100 meters has a mass
  of 300 grams determine the fundamental frequency
  in this string when a 5-kg mass is suspended from
  a 1-meter vibrating section.

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Sketch the problem
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solution

• T mg T (5)(9.81) 49.05 N
• mass/length 0.3 / 100 3 x 10-3 kg/m
• for the fundamental, L ½? 1 meter ? 2
  meters
• vw v49.05/(3 x 10-3)
• vw 127.8 m/sec
• vw f?
• 127.8 f(2)
• f 63.9 hz

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Free and Fixed End Reflectors

• Free end reflector one end is fixed and one is
  free
• Reflection is in phase (crest reflects as a
  crest)
• http//www.physicsclassroom.com/mmedia/waves/fix.c
  fm
• Fixed end reflector both ends fixed
• Reflection is out of phase (crest reflects as a
  trough)
• Try it )
• http//www.physicsclassroom.com/mmedia/waves/free.
  cfm

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Standing waves in sound- Closed Tube Sound waves
are longitudinal but we can draw them as
transverse to see them easily

• In a closed tube, the far end is a node
  (particles cant compress)
• The possible length of a tube follows a distinct
  pattern based on the fundamental being 1/4?
• Note that only odd harmonics occur

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Open-tube

• Standing waves can still be established if the
  end is open- the waves reflect off the open air
• In this case, the end is air which can vibrate so
  it is an antinode
• Open tubes can support all harmonics

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Harmonics and music

• String instruments often have multiple harmonics
  vibrating simultaneously- this produces the
  particular timbre of the instrument
• http//dev.physicslab.org/asp/applets/string/help.
  asp
• The pentatonic scale- these overtones and
  fundamental
• http//www.youtube.com/watch?vne6tB2KiZuk

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Sound Waves

• Mechanical
• Need a medium
• No sound in space

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Sound Waves

• Longitudinal
• Vibrations create pressure variations in medium
• Compressions
• Rarefactions
• Feel your throat as you hum
• http//www.physicsclassroom.com/mmedia/waves/tfl.c
  fm

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Visualizing a longitudinal wave

• If you graph the difference in density/pressure
  in the medium, you can graph this as the
  amplitude and visualize it as a sinusoidal graph

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Properties of sound waves

• Share properties of waves
• Speed v?f
• Speed depends on medium and temperature
• Reflection
• Echo, sonar
• Reflections of multiple or rough surfaces called
  reverberations
• Interference
• Dead spots(nodes)
• amplification

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Interference BEATS

• Sound waves interfere just as other waves
• If pitches of 2 waves are close, we hear
  resulting interference as beats- pulsating
  changes in amplitude (loudness)- the beats we
  hear are the areas of constructive interference

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Beats Problems

• Suppose you sound two tuning forks
  simultaneously one fork has a frequency of 256
  hz and the other has a frequency of 260 hz.
• How many beats would be heard each second?
• What is the pitch of these beats?

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Beats

• The frequency of the beats is the difference
  between the 2 original frequencies
• beat frequency f2 - f1
• The pitch of the beats is the average of the 2
  frequencies
• beat pitch ½(f2f1)

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Perception of Sound

• Loudness amplitude
• Measured in decibel (dB)
• Exposure to loud sounds can cause your ear to
  lose sensitivity

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Perception of Sound

• Our perception of the loudness of a sound is not
  directly proportional to the pressure- intensity
  is actually logarithmic so for each 10 decibel
  increase, the intensity goes up 10X
• Also depends on pitch, pure tone vs. combined
  tones

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Speed of Sound

• Depends on medium
• Greater elasticity faster
• Metals conduct sound quickly
• Depends on temp
• Hotfast
• In same medium, all sound waves travel at same
  speed
• When band plays- all sounds reach you at same
  speed regardless of pitch, amplitude, instrument

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Speed of Sound in Air

• In dry air, speed of sound is function of temp
• vw 331 0.6 T
• As air becomes more humid, speed increases
• If air temp not constant, can cause refraction
• When this happens with light, we see a mirage

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Dampening

• Sound waves cause vibrations in the medium so
  energy is lost to heat- thus the wave is damped
• Lower frequency cause less motion so can travel
  farther- thus use low frequency for fog horns

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Physics of Music

• Sound produced by vibrating object which causes
  pressure oscillations in air
• Vibrating string
• Vibrating reed(s)
• Vibrating column of air