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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

  • 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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  • 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.

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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

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  • 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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  • 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?

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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 calculus…aargh

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Calculus review?

  • What does a derivative mean in mathematical terms?

  • Example:

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  • 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: y=cos(t)

  • Decisions: Sampling rate and number of bits of digitization.

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Newton’s Second Law

  • Relation between force mass and acceleration

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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.

  • x and F are vectors for position and force—the 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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  • 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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Why is this solution useful?

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

  • We can calculate the velocityat any time.

  • We can calculate the accelerationat any time.

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  • 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 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 ANGLE f

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  • 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.

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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 (t=0) 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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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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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.

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Two ways of writing complex numbers

  • 3+4i = 5[cos(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.