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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.

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

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

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

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.

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

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

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

- We have some terms to define and understand

Mass

- 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?

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

Calculus review?

- What does a derivative mean in mathematical terms?
- Example:

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.

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.

Newton’s Second Law

- Relation between force mass and acceleration

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?

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

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.

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)

Dust off those old calculus skills

- First differential
- Second differential

Put it all together

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

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.

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?

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

Example

- 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.

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.

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

- 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

- Picture that makes SHO a little bit clearer.

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

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

ex has a simple derivative

Sin and cos can be described by infinite series

- Sin(x)
- Cos(x)

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

- 3+4i = 5[cos(0.93) + i sin(0.93)]

Can we put sin and cos series together to get ex series? Not if x is real. But with i…

eix series

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.

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

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