physical manifestations of periodic functions matthew koss college of the holy cross july 12, 2012...
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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
Data Studio 500
Simple Harmonic Motion
Simple Harmonic Oscillations
A Amplitude
w 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
or
y t A t
sin ( )
sin ( )
x t A t
or
y t A t
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
ete
rs)
( ) cos ( )y t A t
1
2
fT
f
k
m
( ) cosk
y t A tm
Another Representation
2( ) cos
2( ) cos
x t A tT
or
y t A tT
Amplitude
2 Total Angle ( )
Initial Angle
Period
A
tT
T
or
( ) cos 2
( ) cos 2
x t A ft
or
y t A ft
Amplitude
2 Total Angle ( )
Initial Angle
Frequency
A
ft
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 t
dt dt T
da t v t
dt
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 t
dt dt T
AA t t
T T T T
d y dv da A t
dt dt dt T T
A t A tT T T T T
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
2
1
2
2
fT
f
k
m
k
T m
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
or
x t A t x
Atom Can Execute Simple Periodic Motions
SHM is the Projection of Circular Motion
Illustration
y(t)
y2(t)
y1(t)
y2 y1
A A
y(t)
Simple Pendulum
( ) cos( ),g
t A tL
mg
TF 2L
Tg
Same as a simple pendulum, but…
Distance from pivot to cm or cg.L
2
mgL
I
IT
mgL
Physical Pendulum
axis
cm
L
Oscillations on a String
( ) cos 2
( , ) ( ) cos 2
y t A ft
y x t A x ft
( , ) sin cos 2n
y x t A x ftL
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 t
T
Mathematical Relationships
A Amplitude
kx-wt+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
or , /
1 2
2
v wave speed vT
v f v k
T period
f fT
wavelength k
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
, 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
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
1
2
, 1,2,3,
Tn
L
T
L
n
Fnf n
L
Ff
L
f nf n
1f f
1 22f f f
1 33f f f
( ) ( ) cos 2y t A x ft
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
2
Tt
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)
amp
litu
de
(m
)
-2
-1
0
1
2
0 50 100 150 200 250
Time (sec)
am
pli
tud
e (
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
pli
tud
e (
m)
-2
-1
0
1
2
0 50 100 150 200 250
Time (sec)
am
pli
tud
e (
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 cos
2 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:
iPads & I Phones
More Complex Sounds
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/