PhysicalManifestations

ofPeriodic

FunctionsMatthew Koss

College of the Holy CrossJuly 12, 2012

IQR Workshop: Foundational Mathematics Concepts for the High School to College Transition

Simple Block and Spring

http://phet.colorado.edu/en/simulation/mass-spring-lab

Data Studio 500

Simple Harmonic Motion

Simple Harmonic Oscillations

A Amplitudew t + f Phase (radians)/Angle

(radians)f Phase Constant (radians)w Angular Frequency (rad/s)T Period (s)f Frequency (Hz)

cos ( )

cos ( )

x t A t

ory t A t

w f

w f

sin ( )

sin ( )

x t A t

ory t A t

w f

w f

Simple Harmonic Motion

for Block and Spring

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3 3.5

X Postition (meters)

Y (m

eter

s)

( ) cos ( )y t A tw f

1

2

fT

f

km

w

w

( ) cos ky t A tm

f

Another Representation2( ) cos

2( ) cos

x t A tT

or

y t A tT

f

f

Amplitude2 Total Angle ( )

Initial Angle Period

A

tT

T

f

f

or

( ) cos 2

( ) cos 2

x t A ft

ory t A ft

f

f

Amplitude2 Total Angle ( )

Initial Angle Frequency

Aft

f

f f

Review

maxx

minx t T

2

A Periodic Function (sine or cosine) is the Recorded History ofthe Oscillations of an object attached to a spring.

Position, velocity, and acceleration

2( ) cos

2( ) ( ) cos

( ) ( )

y t A tT

d dv t x t A tdt dt T

da t v tdt

f

f

If you know calculus

Calculus Approach

2

2

2

2cos

2cos

2 2 2 2sin sin

2 2sin

2 2 2 2 2cos cos

y A tT

dy dv A tdt dt T

AA t tT T T T

d y dv da A tdt dt dt T T

A t A tT T T T T

f

f

f f

f

f f

If Not, then …

2

2( ) cos

2 2( ) sin

2 2( ) cos

x t A tT

v t A tT T

a t A tT T

f

f

f

2

1

2

2

fT

f

km

kT m

w

w

Zero Offset

• Oscillations do not always occur about the zero point.• To account for this, there is one additional term called the

zero offset which is middle value in the oscillations.• So, more completely:

( ) cos ( )

( ) cos ( )

offset

offset

y t A t y

orx t A t x

w f

w f

iPads and

Video Physics

Physics Toolkit

Atom Can Execute Simple Periodic Motions

States of Matter Simulation

SHM is the Projection of Circular Motion

Illustration

y(t)

y2(t)

y1(t)

y2 y1

A A

y(t)

PhET Rotation Simulation

Simple Pendulum

( ) cos( ), gt A tL

w f w

mg

TF 2 LTg

PhET Pendulum Simulation

Same as a simple pendulum, but…

Distance from pivot to cm or cg.L

2

mgLI

ITmgL

w

Physical Pendulum

axis

cm

L

Oscillations on a String

( ) cos 2

( , ) ( ) cos 2

y t A ft

y x t A x ft

f

f

( , ) sin cos 2ny x t A x ftL f

Tangent on Traveling WavesA wave is a disturbance in position propagating in time.

v A

Many traveling waves are periodic in both position and time, e.g.

2 2siny A x tT

f

Mathematical Relationships

A Amplitudekxwt+f Phase (radians)w Angular Frequency

(rad/s)T Period (s)f Frequency (Hz)k (Angular) Wave number Wavelength

2 2sin

sin( )

y A x tT

y A kx t

f

w f

or , /

1 2

2

v wave speed vT

v f v k

T period

f fT

wavelength k

w

w

In general: ( , ) and ( )y f x t y f x vt

Specifically:Periodic

Sine Waves

Waves and Oscillations Compared

An oscillation in time is a “history” of a wave at a particular place.

An oscillation in space is a “snapshot” of a wave at a particular time,

, sin( )

sin ( )

y x t A kx t

y t A t

w f

w f

, sin( )

sin( )

sin( ),

sin( )

sin( ),

specific

specific

specific

specific

y x t A kx t

y t A kx t

A t kx

y x A kx t

A kx t

w f

w f

w f

w f

w f

Sum of Two Traveling Waves Makes Standing Waves

Last Slide of

Digression

Standing Waves on a String, or

Oscillations on a String

1

1

, 1, 2,3,2

12

, 1,2,3,

Tn

L

T

L

n

Fnf nL

FfL

f nf n

1f f

1 22f f f

1 33f f f

( ) ( ) cos 2y t A x ft f

String Vibrates the Air

Guitar Strings

The strings on a guitar can be effectively shortened by fingering, raising the fundamental pitch.

The pitch of a string of a given length can also be altered by using a string of different density.

Sound is a Periodic Oscillation of the Air

0t

2Tt

v

v

Bv

2

Tuning Forks

Data Studio 500 Redux

BeatsIf the two interfering oscillations have different frequencies they will superimpose, but the resulting oscillation is more complex. This is still a superposition effect. Under these conditions, the resultant oscillation is referred to as a beat.

-2

-1

0

1

2

0 50 100 150 200 250

Time (sec)

ampl

itude

(m)

-2

-1

0

1

2

0 50 100 150 200 250

Time (sec)

am

plitu

de (m

)

-2

-1

0

1

2

0 50 100 150 200 250

Time (sec)

ampl

itude

(m)

-2

-1

0

1

2

0 50 100 150 200 250

Time (sec)

am

plitu

de (m

)

-2

-1

0

1

2

0 50 100 150 200 250

Time (sec)

ampl

itude

(m)

Beat Frequency Mathematics

-2

-1

0

1

2

0 50 100 150 200 250

Time (sec)

ampl

itude

(m)

fBeat = f1 -f2

1 1 2 2

1 2

1 2 1 2

2 11 2

( ) sin(2 ) & ( ) sin(2 )

sin(2 ) sin(2 )

2 2 2 22sin cos

2 2

22 ( )( ) 2 sin cos2 2beat

I t I f t I t I f t

I f t I f t

f t f t f t f t

f ff fI t I t t

Amplitude (I) of Sound Oscillations

I0 is taken to be the threshold of hearing:

The loudness of a sound is much more closely related to the logarithm of the intensity.

Sound level is measured in decibels (dB) and is defined as:

MacScope II

Audacity

iPads & I Phones

More Complex Sounds

Fundamental/Normal Modes

http://phet.colorado.edu/en/simulation/fourier

Time and Frequency Domains

Sample Musical

Instrument Sounds in the

Frequency Domain

Web References/ResourcesPhET Simulationshttp://phet.colorado.edu/en/simulations/category/new

Springshttp://phet.colorado.edu/en/simulation/mass-spring-labRotationhttp://phet.colorado.edu/en/simulation/rotationAtomic Oscillationhttp://phet.colorado.edu/en/simulation/states-of-matterPendulumhttp://phet.colorado.edu/en/simulation/pendulum-labNormal Modeshttp://phet.colorado.edu/en/simulation/normal-modesMaking Waveshttp://phet.colorado.edu/en/simulation/fourierVideo Physicshttp://itunes.apple.com/us/app/vernier-video-physics/id389784247?mt=8Physics Toolkithttp://physicstoolkit.com/MacScope & Physics2000http://www.physics2000.com/Pages/Downloads.htmlAudacityhttp://audacity.sourceforge.net/download/