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

The Simple Harmonic Oscillator

… good vibrations…The Beach Boys

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

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:

)sin(tAy

)cos(tAdxdy

Sin and Cos curves

y

t

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)

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)

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

maF

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

• 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

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.

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

Dust off those old calculus skills

• First differential

• Second differential

)cos( tAdtdx

)sin(22

2

tAdt

xd

Put it all together

• Substitute parts into the equation

• Conclusion (after cancellations)

)sin()sin(2 tAmKtA

mK

2

General form of SHO

xdt

xd 22

2

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

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?

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

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.

What is the phase here?

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)

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

...!4!3!2

1432

xxxxex

ex has a simple derivative

axax

xx

aeedxd

eedxd

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

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.

Argand diagram

-4

-3

-2

-1

0

1

2

3

4

5

6

-3 -2 -1 0 1 2 3 4

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…

...!4!3!2

1432

xxxxex

...!7!5!3

)sin(753

xxxxx

...!6!4!2

1)cos(642

xxxx

eix series

)sin()cos(

...)!5!3

(...)!4!2

1(

...!5)(

!4)(

!3)(

!2)(1

5342

5432

xixe

xxxixxe

ixixixixixe

ix

ix

ix

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

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.