Syllabus overview

1
Syllabus overview

• No text. Because no one has written one for the
  spread of topics that we will cover.
• MATLAB. There will be a hands-on component where
  we use MATLAB programming language to create,
  analyze, manipulate sounds and signals. Probably
  1 class per week (in computer lab at end of hall
  WPS211) typically Fridays.

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Some good books

• Fundamentals of Acoustics by Kinsler, Frey,
  Coppens, and Sanders (3rd ed.),
• Science of Musical Sounds by Sundberg
• Science of Musical Sounds by Pierce
• Sound System Engineering by Davis Davis
• Mathematics A musical Offering by David Benson.
  (online version available)
• The Science of Sound by Rossing, Moore, Wheeler

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Grading

• Participation is key!
• Attempt all the work that is assigned.
• Ask for help if you have trouble with the
  homework.
• If you make a good faith effort, dont miss
  quizzes, hand in all homework on time, etc. you
  should end up with an A or a B.

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Web page

• Lecture Powerpoints are on the web, as are
  homeworks, and (after the due date) the
  solutions.
• MATLAB exercises are also on the web page
• http//physics.mtsu.edu/wroberts/Phys3000home.htm
  

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Objectives

• Physical understanding of acoustics effects and
  how that can translate to quantitative
  measurements and predictions.
• Understanding of digital signals and spectral
  analysis allows you to manipulate signals without
  understanding the detailed underlying
  mathematics. I want you to become comfortable
  with a quantitative approach to acoustics.

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Areas of emphasis

• The basics of vibrations and waves
• Room and auditorium acoustics
• Modeling and simulation of acoustics effects
• Digital signal analysis
• Filtering
• Correlation and convolution
• Forensic acoustics examples

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The Simple Harmonic Oscillator

• good vibrations
• The Beach Boys

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Simple Harmonic Oscillator (SHO)

• SHO is the most simple, and hence the most
  fundamental, form of vibrating system.
• SHO is also a great starting point to understand
  more complex vibrations and waves because the
  math is easy. (Honest!)
• As part of our study of SHOs we will have to
  explore a bunch of physics concepts such as
  Force, acceleration, velocity, speed, amplitude,
  phase

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Ingredients for SHO

• A mass (that is subject to)
• A linear restoring force
• We have some terms to define and understand
• Mass
• Force
• Linear
• Restoring

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Mass

• Boy, this sounds like the easy one to start with
  but youll be amazed at how confusing it can get!
• Gravitational mass and inertial mass. Say what!
• What is the difference between mass and weight?

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Force and vectors

• What does a force do to an object?
• Why is the idea of vectors important?
• What is a vector?
• What is the difference between acceleration,
  velocity, and speed?
• Acceleration, velocity, and calculusaargh

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Calculus review?

• What does a derivative mean in mathematical
  terms?
• Example

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Sin and Cos curves
y
t
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Position versus time graph-what does the slope
mean?
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Velocity versus time graphwhat does slope mean?
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Summarize

• Position (a vector quantity)
• Velocity (slope of position versus time graph)
• Acceleration (slope of velocity versus time
  graph). Same as the second derivative of position
  versus time.
• Key If I know the math function that relates
  position to time I can find the functions for
  velocity and acceleration.

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Digital representation of functions

• The math you learn in calculus refers to
  continuous variables. When we model, synthesize,
  and analyze signals we will be using a digital
  representation.
• Example ycos(t)
• Decisions Sampling rate and number of bits of
  digitization.

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Newtons Second Law

• Relation between force mass and acceleration

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Apply Newtons second law to mass on a spring

• Linear restoring forceone that gets larger as
  the displacement from equilibrium is increased
• For a spring the force is
• K is the spring constant measured in Newtons per
  meter.
• x and F are vectors for position and forcethe
  minus sign is important! Which direction does the
  force point?

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• Newton's second law
• Substitute spring force relation
• Write acceleration as second derivative of
  position versus time

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Final result

• Every example of simple harmonic oscillation can
  be written in this same basic form.
• This version is for a mass on a spring with K and
  m being spring constant and mass.

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Solution

• The solution to the SHO equation is always of the
  form
• To show that this function is really a solution
  differentiate and substitute into formula.
• Note A and w are constants x, t are variables.
  w is determined by the physical properties of the
  oscillator (e.g. k and m for a spring)

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Dust off those old calculus skills

• First differential
• Second differential

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Put it all together

• Substitute parts into the equation
• Conclusion (after cancellations)

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General form of SHO
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Why is this solution useful?

• We can predict the location of the mass at any
  time.
• We can calculate the velocity at any time.
• We can calculate the acceleration at any time.

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Example

• What is the amplitude, A?
• How can we find the angular frequency, w?
• At which point in the oscillation is the velocity
  a maximum? What is the value of this maximum
  velocity?
• At which point in the oscillation is the
  acceleration a maximum? Value of amax?

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One other item phase

• The solution as written is not complete. The
  simple sine solution implies that the oscillator
  always is at x0 at t0. We could use the
  solution xAcos(wt) but that means that the
  oscillator is at xA at t0. The general solution
  has another component PHASE ANGLE f

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Example

• To find the phase angle look at where the mass
  starts out at the beginning of the oscillation,
  i.e. at t0.
• Spring stretched to A and released.
• Spring stretched to A and released
• Mass moving fast through x0 at t0.

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Worked example

• A mass on a spring oscillates 50 times per
  second. The amplitude of the oscillation is 1 mm.
  At the beginning of the motion (t0) the mass is
  at the maximum amplitude position (1 mm) (a)
  What is the angular frequency of the oscillator?
  (b) What is the period of the oscillator? (c)
  Write the equation of motion of the oscillator
  including the phase.

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What is the phase here?
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Helmholtz Resonator

• Trapped air acts as a spring
• Air in the neck acts as the mass.

(vs is the speed of sound)
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Helmholtz resonator II

• Where is the air oscillation the largest?
• Why does the sound die away? Damping
• Real length l versus effective length l.
• End correction 0.85 x radius of opening.
• Example guitar 1.7 x r.

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SHO relation to circular motion

• Picture that makes SHO a little bit clearer.

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Complex exponential notation

• Complex exponential notation is the more common
  way of writing the solution of simple harmonic
  motion or of wave phenomena.
• Two necessary concepts
• Series representation of ex, sin(x) and cos(x)
• Square root of -1 i

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Exponential function

• Very common relation in nature
• Number used for natural logarithms
• Defined (for our purposes) by the infinite series

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ex has a simple derivative
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Sin and cos can be described by infinite series

• Sin(x)
• Cos(x)

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Imaginary numbers

• Concept of v-1 i
• i2 -1, i3 -i, i4 ?
• Not a real numbercalled an imaginary number.
• Cannot add real and imaginary numbersmust keep
  separate. Example 34i
• Argand diagramplot real numbers on the x-axis
  and imaginary numbers on the y-axis.

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Argand diagram
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Two ways of writing complex numbers

• 34i 5cos(0.93) i sin(0.93)

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Can we put sin and cos series together to get ex
series? Not if x is real. But with i
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eix series
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Complex exponential solution for simple harmonic
oscillator

• Note We only take the real part of the solution
  (or the imaginary part).
• Complex exponential is just a sine or cosine
  function in disguise!
• Why use this? Math with exponential functions is
  much easier than combining sines and cosines.

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Relation to circular motion.

• Simple harmonic motion is equivalent to circular
  motion in the Argand plane. Reality is the
  projection of this circular motion onto the real
  axis.