Waves

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Waves
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Wave math I

• f(x-vt) represents a positive moving wave at wave
  speed v.

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Pure sine wave one particular wave type

• y A sin(kx-wt)
• What is k? Wave number, k2p/l.
• Does this formula have the yf(x-vt) form? Yes!
• To appreciate the physical significance of the
  wave formula with two variables (x and t) freeze
  one and look at the function. Freezing t is like
  taking a snap shot of the wave. Freezing x is
  like looking at one point on the wave as time
  passes.

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Why are sine waves so important?

• Sine waves are the fundamental building blocks
  out of which any wave shape can be constructed.
• Mathematically they form a complete orthogonal
  set of functions.
• Sine functions define the idea of frequency
  only an infinite sine wave has a single pure
  frequency. All other waves have some combination
  of frequencies.

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Fourier transforms and spectrum

• We typically think of sounds and signals in the
  time domain however, representing those signals
  in the frequency domain can give much more
  information than a time representation alone.
• The process of converting from time to frequency
  is known as spectral analysis. As you learned in
  PHYS1600the ear is a mechanical spectrum
  analyzer.

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Fourier series and spectrum

• Any periodic function can be broken down into the
  sum of harmonics of sine and cosine waves.
• Building a signal up by adding sine (and cos)
  waves is called synthesis. Breaking a signal up
  to find the coefficients is called spectral
  analysis or Fourier analysis.

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How can the coefficients be found?

• How can we determine the an and bn values for a
  given function f(t)?
• Orthogonality of the sine function
• m n gt 1

m ? n
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Coefficient formula

• This shows that it is possible to find the
  coefficients
• In practice, we will let the computer do this
  operation numerically (the Fast Fourier
  Transform).

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Odd and Even functions

• Odd and even refer to symmetry about zero.
• Is the sine function odd or even?
• What about cosine?
• Consequenceodd functions can be made up of the
  addition of sine waves alone, even cosine
• What about non-symmetric functions?

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Quick question

• Odd or even?

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Next?
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Finally
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Odd, even or some combo

• The combination of odd and even in most functions
  means that the Fourier transform consists of
  sines and cosines. But with complex exponential
  notation we already have both. The amplitude A
  can be complexthis mixes sines and cosines

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Wave stuff you should remember!
Note k is called wave number, w is called
angular frequency
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Complex wave representation

• Remember complex exponentials are just sine or
  cosine functions in disguise!

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The Wave Equation

• Just as for simple harmonic motion, all wave
  motion is described by a mathematical relation
  (technically a partial differential equation)
• Every wave function yf(x-vt) satisfies this
  equation.

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Wave speed

• The wave speed given by nfl or by the wave
  equation is the wave speed for a pure sine wave
  of a given single frequency. This is called the
  phase velocity.
• In audio range acoustics the speed of sound is
  essentially constant for all frequencies.
• If the velocity changes with frequency then a
  pulse (many superposed sine waves) travels with a
  different velocitythe group velocity.

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Wave properties

• Superposition of waves
• Interference
• Diffraction
• Reflection and refraction
• Acoustic impedance concept

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Superposition

• Waves can occupy the same part of a medium at the
  same time without interacting. Waves dont
  collide like particles.
• At the point of overlap the net amplitude is the
  sum of all the separate wave amplitudes. Summing
  of wave amplitudes leads to interference.
• Constructive versus destructive interference.

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Superposition II

• We use the additive property of superposition
  when we synthesize waveforms. We create a bunch
  of separate sine waves of different frequencies,
  amplitudes, and relative phases and just add
  them.
• Note that when we make sound waves numerically in
  the computer we do not need to include kx term.
  Why not?

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Diffraction

• Bending of waves around objects and through
  openings.
• Huygens principleevery point of a wave front
  becomes a point source for new wave fronts.
• Transmission line matrix method demos.

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Reflection and Refraction

• http//webphysics.ph.msstate.edu/jc/library/24-2/s
  imulation.html
• http//www.sciencejoywagon.com/physicszone/lesson/
  otherpub/wfendt/huygens.htm
• Refraction does not come up too much in acoustics.

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Path length difference and phase

• In many cases you can determine the existence of
  constructive or destructive interference by
  examining the path length difference between
  interfering waves.
• Math to convert path length difference to phase
  difference
• For pure constructive or destructive Df mp
• Constructive for m0,2,4,6..
• Destructive for m1,3,5,7

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Diffraction interference example
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Simple caseLloyds mirror

• What wavelengths will interfere destructively?
  (Assume no inversion on reflection)

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Speaker enclosures and baffles

• What is the purpose of a baffle?
• Prevent destructive interference between front
  and back emitted waves from a speaker.
• Why are circular baffles bad?
• Baffle Step
• 6 db difference between low frequencies and high
  frequencies cross-overs.

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Baffle Step

• Low frequencies are diffracted in all directions
• High frequencies are more directional in the
  forward direction

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Circular baffle example

• The dip is at about 460 Hz. Does this agree with
  a simple interference calculation?
• Plot is relative to infinite baffle

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What is the consequence of a circular baffle?
Spectral hole.
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Speaker placed off center in a rectangular baffle
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Edge Diffraction
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Pressure variation from a sphere

• Normal incidence (q0), high frequency, why the 6
  dB rise? y axis is in relative db

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Edge diffraction interference
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Diffraction of sound around the head

• Diffraction as a function of angle around head
  for three different frequencies.
• Why the big variation with frequency?

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Sound, pressure, and thermodynamics

• Sound in air is the result of air molecule
  movement (displacement).
• More air molecules in a given volume of space
  equals an increase in air pressure
• Kinetic model of a gaslittle molecules whizzing
  around banging into each other and the walls of
  the container
• Ideal gas equation PVnRT

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Pressure and displacement
Animation courtesy of Dr. Dan Russell,
Kettering University
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Physical model of gases

• Air consists of mainly nitrogen (78) molecules,
  along with oxygen (21).
• At room temp the average molecule is moving at
  about 400 m/s.
• The average mass of a molecule is 5.4x10-26 kg
• The average size of a molecule is 2x10-10 m
• The average spacing between molecules is 30 x
  10-10 m

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What causes air pressure?

• Pressure is caused by the reaction force of the
  collisions of gas molecules with any surface
  exposed to the gas.
• Pressure increases with the number of gas
  molecules because there are more collisions.
• Pressure increases with temperature (for same
  density of molecules) because the molecules are
  moving faster.

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Ideal Gas Equation

• PVnRT
• P pressure (Nm-2)
• V volume (m3)
• n number of moles of gas
• R gas constant 8.31 Jmol-1K-1
• T temperature in degrees Kelvin (K)
• Isothermal versus adiabatic processes

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Isothermal example

• T is a constant. If n is a constant (R is always
  constant) then Right Hand side of equation is a
  constant
• P1V1P2V2
• If we reduce the volume the pressure rises
• Big change in V use formula
• Small DV we can show that

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Adiabatic example

• Adiabatic processno heat flows so the
  temperature of the gas can vary.
• Sound wavesthe pressure variations happen so
  fast so that heat cannot be redistributed. Thus,
  sound pressure variations are adiabatic.
• In a fixed volume of space through which a sound
  wave passes what factors in the ideal gas law are
  constant?

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Adiabatic processes and sound

• PVgconstant
• g depends on the gas involved usually 1.333
• We can show that for small changes
• Look back at Helmholtz resonator derivation

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Sound is an adiabatic process

• At the high and low pressure regions of a sound
  wave the temperature is slightly high and low
  respectively.
• If very large amplitude sound waves can be formed
  the temperature difference can be used to make
  acoustic coolers (refrigerators).
• Adiabatic nature sets speed of sound.

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Relation between Displacement and Pressure
Amplitude

• Back in PHYS1600 we learned that displacement and
  pressure amplitude are p/2 (a quarter wavelength)
  out of phase.
• Redo that old argument quickly.
• Now we can also relate the relative amplitudes of
  pressure amplitude and displacement

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Definition of the variables

• p0 is the pressure amplitude of the wave.
• r0 is the density of air (1.29 kg m-3)
• w is the angular frequency
• vs is the speed of sound in air
• eo is the displacement amplitude

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Review of Sound Pressure Level

• You should be able to convert SPL to pressure
  amplitude.
• You should be able to convert a pressure
  amplitude to a decibel value in SPL.
• Example What is the displacement amplitude of a
  10 dB SPL pure tone at 1000 Hz?
• Convert SPL to p0
• Use p0 and e0 formula

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Acoustic impedance

• Analogous quantity to electrical impedance.
• Electrical impedance from Ohms law
• ZV/I
• What is V? It is related to the force that
  pushes on the charges.
• What is I? It is related to the velocity of the
  charges in the circuit.
• Acoustic impedance ZacForce/Velocity

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Strings

• The two important physical parameters for a
  string are
• m mass per unit length (kg/m)
• T tension in the string (N)
• Speed of wave, v, on a stretched string is given
  by

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Review of standing wave resonances

• Fundamental and harmonics
• n is the harmonic number 1,2,3
• L is the string length
• v is the wave velocity on the string (
  )

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String impedance

• Impedance for a string
• Different forms of same equation depending on
  what parameters you know.
• Why do the string as an example? Easiest to
  visualize in a reflection configuration.

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Reflection at the junction between two
stringsreflection formula

• What values change as the wave travels from one
  medium to the next and which are the same?
• Tension, mass per unit length, wave velocity,
  frequency, wavelength
• What conditions must be met at the junction
  between two strings with different m?

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Boundary conditions

• At the junction the wave amplitudes must agree,
  otherwise the string comes apart! (frequency must
  be the same in each medium!)
• Harder to see but the slopes at the junction must
  agree. The string must not kink.
• How to calculate the expression for the
  reflection coefficient. Start by imagining the
  situationincident wave goes along to junction
  where it is partially reflected and partially
  transmitted.

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Reflection formula

• Three waves amplitudes, incident A1, reflected
  B1, and transmitted A2
• Continuity of amplitude means that at the
  junction IncidentReflectedTransmitted
• We can choose to set the junction at x0.
• Time term is the same in all three and cancels.

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Reflection formula II

• Condition 1 A1B1A2
• Now continuity of slope is a bit more
  complicated, but leads to
• Condition 2 Z1(A1-B1)Z2A2
• What is the reflection coefficient, r?
• r B1/A1
• What is the transmission coefficient, t?
• t A2/A1

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Solving our two boundary condition equation for r
and t gives

• Reflection
• Transmission

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What can we learn from r and t?

• The transmitted pulse is never inverted.
• The reflected pulse is inverted if Z2 gt Z1.
• Example tied down end of a string has infinite
  Z. (Velocity is always zero independent of force)
  Thus for a wave from a string hitting a tied down
  end r-1. The wave is inverted on reflection.

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Animation

• Light string to heavy string (low m to large m)

Animation courtesy of Dr. Dan Russell,
Kettering University
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Impedance in pipes and ducts

• Sound traveling in pipes and ducts where the
  wavelength is smaller than the dimensions of the
  duct form a one-dimensional wave system much like
  waves on a string.
• The acoustic impedance of a pipe of
  cross-sectional area S is given by
• Where r is the density of air (1.29 kg m-3) and
  vs is the speed of sound.

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Reflection and transmission formulae are identical

• Junction between two pipes leads to reflection
  given by

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Acoustic pipe filters

• A muffler shaped system of pipes acts to filter
  sound of particular frequencies. We can figure
  out which frequencies are reflected by combining
  inversion on reflection, path length differences,
  and interference.

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Calculate the phase

• We will look at the reflected signal.
• Remember Z2gtZ1 means inversion on reflection
  which is the same as a p phase shift.
• For path length phase change Df is given by

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Add phase

• First reflection0 phase shift.
• Second reflectionp phase shift on reflection
  plus (2p/l)2L. If Ll/4 then path length phase
  shift is p. Total is 2p and therefore the 2
  reflections add constructively.

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Conclusion about ¼-wave filters

• If Ll/4 or 3l/4 or 5l/4 then interference will
  be constructive.
• Constructive interference in reflection means a
  big reflected wave and therefore a small
  transmitted wave.
• Example of application is in duct design to
  suppress the transmission of a particular
  frequency (and its odd-numbered higher harmonics)

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Ganged together filters

• We can gang these filters together and get an
  enhanced filtering effect.
• Lets do the math

L
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Transmission through a multi-element quarter wave
filter Experiment
What is the length of one filter segment?
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Figure from paper

• Repeat distance 17 cms