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Syllabus overview. No text. Because no one has written one for the spread of topics that we will cover.

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syllabus overview
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).
  • 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, don’t miss quizzes, hand in all homework on time, etc. you should end up with an A or a B.
web page
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

  • 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.
areas of emphasis
Areas of emphasis
  • Room and auditorium acoustics
  • Modeling and simulation of acoustics effects
  • Digital signal analysis
    • Filtering
    • Correlation and convolution
    • Forensic acoustics examples
the simple harmonic oscillator

The Simple Harmonic Oscillator

I’m pickin’ up good vibrations…

The Beach Boys

simple harmonic oscillator sho
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…
ingredients for sho
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
  • Boy, this sounds like the easy one to start with; but you’ll be amazed at how confusing it can get!
  • Gravitational mass and inertial mass. Say what!
  • What is the difference between mass and weight?
  • 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 calculus…aargh
calculus review
Calculus review?
  • What does a derivative mean?
  • Example:
  • 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.
newton s second law
Newton’s Second Law
  • Relation between force mass and acceleration
apply newton s second law to mass on a spring
Apply Newton’s second law to mass on a spring
  • Linear restoring force—one 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.
Newton's second law
  • Substitute spring force relation
  • Write acceleration as second derivative of position versus time
final result
Final result
  • Every example of simple harmonic oscillation can be written in this same basic form.
  • The solution to the SHO equation is always of the form
  • To show that this is a solution differentiate and substitute into formula.
  • Note: A and w are constants; x,t are variables
dust off those old calculus skills
Dust off those old calculus skills
  • First differential
  • Second differential
put it all together
Put it all together
  • Substitute parts into the equation
  • Conclusion (after cancellations)
why is this solution useful
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.
  • 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?
one other item phase
One other item: phase
  • The solution as written is not complete. The oscillator always is at x=0 at t=0. We could use the solution x=Acos(wt) but that means that the oscillator is at x=A at t=0. The general solution has another component –PHASE
  • To find the phase angle look at where the mass starts out at the beginning of the oscillation, i.e. at t=0.
  • Spring stretched to –A and released.
  • Spring stretched to +A and released
  • Mass moving fast through x=0 at t=0.
helmholtz resonator
Helmholtz Resonator
  • Trapped air acts as a spring
  • Air in the neck acts as the mass.

(vs is the speed of sound)

helmholtz resonator ii
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.
sho relation to circular motion
SHO : relation to circular motion
  • Picture that makes SHO a little bit clearer.
complex exponential notation
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
exponential function
Exponential function
  • Very common relation in nature
  • Number used for natural logarithms
  • Define (for our purposes) by the infinite series
imaginary numbers
Imaginary numbers
  • Concept of √-1 = i
  • i2 = -1, i3 = -i, i4 = ?
  • Not a “real” number—called an imaginary number.
  • Cannot add real and imaginary numbers—must keep separate. Example 3+4i
  • Argand diagram—plot real numbers on the x-axis and imaginary numbers on the y-axis.
two ways of writing complex numbers
Two ways of writing complex numbers
  • 3+4i = 5[cos(0.93) + i sin(0.93)]
complex exponential solution for simple harmonic oscillator
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.
relation to circular motion
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.