syllabus overview
DESCRIPTION
Syllabus overview. No text. Because no one has written one for the spread of topics that we will cover. - PowerPoint PPT PresentationTRANSCRIPT
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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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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.
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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 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:
)sin(tAy
)cos(tAdxdy
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Sin and Cos curves
y
t
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Position versus time graph-what does the slope mean?
-8
-6
-4
-2
0
2
4
6
8
10
0 5 10 15
Time (seconds)
Posi
tion
alon
g x-
axis
(met
ers)
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Velocity versus time graph—what does slope mean?
-2
-1
0
1
2
3
4
5
6
0 5 10 15 20
time (seconds)
velo
city
(met
ers
per s
econ
d)
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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: 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
maF
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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?
xF Ksp
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• Newton's second law
• Substitute spring force relation
• Write acceleration as second derivative of position versus time
aF m
ax mK
2
2
dtxdmK x
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Final result
xmK
dtxd
2
2
•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 are constants; x, t are variables. is determined by the physical properties of the oscillator (e.g. k and m for a spring)
)sin( tAx
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Dust off those old calculus skills
• First differential
• Second differential
)cos( tAdtdx
)sin(22
2
tAdt
xd
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Put it all together
• Substitute parts into the equation
• Conclusion (after cancellations)
)sin()sin(2 tAmKtA
mK
2
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General form of SHO
xdt
xd 22
2
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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.
)sin( tAx
)cos(v tAdtdx
)sin(22
2
tAdt
xda
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Example
• What is the amplitude, A?• How can we find the angular frequency, ?• 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(t) but that means that the oscillator is at x=A at t=0. The general solution has another component –PHASE ANGLE
)sin( tAx
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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 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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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.
VlAvf s
2
(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
...!4!3!2
1432
xxxxex
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ex has a simple derivative
axax
xx
aeedxd
eedxd
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Sin and cos can be described by infinite series
• Sin(x)
• Cos(x)
...!7!5!3
)sin(753
xxxxx
...!6!4!2
1)cos(642
xxxx
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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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Argand diagram
-4
-3
-2
-1
0
1
2
3
4
5
6
-3 -2 -1 0 1 2 3 4
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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…
...!4!3!2
1432
xxxxex
...!7!5!3
)sin(753
xxxxx
...!6!4!2
1)cos(642
xxxx
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eix series
)sin()cos(
...)!5!3
(...)!4!2
1(
...!5)(
!4)(
!3)(
!2)(1
5342
5432
xixe
xxxixxe
ixixixixixe
ix
ix
ix
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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.
)]sin()[cos()( titAAey ti
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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.