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# Unit 4 AOS 2: Waves

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## Emma vL

on 22 October 2018

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#### Transcript of Unit 4 AOS 2: Waves

String fixed both ends:
Waves
http://www.acs.psu.edu/drussell/Demos/waves/Lwave-v8.gif
http://www.acs.psu.edu/drussell/Demos/waves/Twave.gif
http://www.walter-fendt.de/ph14e/stwaverefl.htm
Harmonics:
Diffraction
http://www.acoustics.salford.ac.uk/schools/teacher/lesson3/flash/whiteboardcomplete.swf
Waves can be:
Waves transmit energy, not matter
2 types of waves:
Longitudinal
Sound is a Longitudinal wave
Freeze time and draw wave as
pressure variation
Hey.
What is the wavelength of the 365 Hz note?

By gradually increasing the frequency another higher resonance (the second harmonic) is found. What is the frequency of this second harmonic?

How will they know when they've found this second harmonic?

Draw the vertical displacement against distance when the first overtone is being applied.

What is the wavelength of the wave produced?

If the fundamental frequency is 90 Hz, what is the speed of the wave?

Sketch the shape of the next two standing waves, labeling the nodes and anti-nodes and state their frequencies

Example 1
Example 2
What?
Two speakers are set up behind a TV screen, one optimised for high frequencies and one for low.
The sound is of much lower fidelity than when the tv is not there. Explain how and why.

Stage
x
y
Engineers are measuring the sound intensity of 300 Hz and 3000 Hz. At point X they are the same but at point Y they are different.
How and why?

Low f
both ears hear
High f
one ears hears
Long wavelength diffracts
Short wavelength does not diffract
the "bending" of waves around an object
or
the "spreading out" of waves thru a gap

examples:
d
L
Double slits
Transverse
http://www.acs.psu.edu/drussell/Demos/waves/Lwave-v8.gif
longitudinal
transverse
http://www.acs.psu.edu/drussell/Demos/waves/Twave.gif
Direction of wave
motion
Wave properties
Same for longitudinal
Amplitude
: maximum pressure variation
Wavelength (m)
: length of one complete wave

Frequency (Hz)
: number of complete waves passing a point per second

Period (s)
: time for one complete wave to pass a point
Velocity (m/s)
: speed of wave
Can read from graph:
Can't read from graph:
Compression
Rarefaction
Peak
Trough
Pressure variation
Two types of waves:
Amplitude
Wavelength ( )
f =
1
_
T
_
T =
1
f
_
_
v = f
wave equation:
depends only on the
medium
(not f or )
Eg sound waves in air: v=340 m/s,
water: v= 1500 m/s etc.
In air v = 340 m/s
change f -> change pitch
change amplitude -> change intensity
http://www.acs.psu.edu/drussell/demos/superposition/superposition.html
Standing waves on string - free/fixed
Superposition of waves
destructive superposition
constructive superposition
http://www.acs.psu.edu/drussell/Demos/StandingWaves/StandingWaves.html
Pressure variation harmonics gifs/java:
Node
Antinode
superposition of a wave and it's
own reflection
Harmonics
String fixed both ends
1st harmonic (fundamental frequency)
2nd harmonic (1st overtone)
3rd harmonic (2nd overtone)
4th harmonic (3rd overtone)
L
/
2
/
1
= 2L
2
= L
3
=
2L
_
3
_
_
4
=
L
2
_
_
standing sound waves in pipes
know you've found a harmonic because it sounds louder
Pressure variation
pipe open both ends
Harmonics
String fixed one end only
n
=
2L
_
n
_
_
Harmonics
n
= = nf
nv
2L
_
_
_
f
n
=
4L
_
n
_
_
n
=
nv
4L
_
_
_
f
String fixed one end only:
n = 1, 3, 5, ...
(odd only)
n = 1, 2, 3, ...
1st harmonic
(fundamental frequency)
3rd harmonic (1st overtone)
5th harmonic (2nd overtone)
1
= 4L
5
=
4L
_
5
_
_
3
=
4L
_
3
_
_
Resonance:
forced oscillation matching objects natural frequency
NB:
Some variation over naming of these harmonics
(cps/txtbook/internet/within!exams)
Sometimes 3rd harmonic is called the 2nd and so on.
Read question carefully, should be clear which they want.
I use this naming system as that way the n in formulas matches the name.
fixed end = N, free end = AN
NB:
Free end = AN
Sound & Light waves (point sources) obey the
Inverse Square Law

I
1
_
r
2
-
-
Distance and
intensity ( I )

-
-
Standing waves
Reflected waves
N
i
r
Law of reflection:
i = r
amount of diffraction
_
_
w
Low pitch sounds diffract more than high pitch sounds
http://www.acoustics.salford.ac.uk/schools/teacher/lesson3/flash/whiteboardcomplete.swf
Diffraction
little diffraction
much diffraction
Destructive superposition
Destructive interference
Constructive interference
A 75 cm string is fixed at one end only
Organ pipe closed at one end produces a note of frequency 365 Hz. The speed of sound in the pipe is 340 m/s.

NB: 'second harmonic' for closed pipe is what in formulas?
( resonates')
Checkpoints
(2014)

872-900, 946-947
where w = size of object/gap
-
-
1
Sketch the fundamental vibration and determine its wavelength.

Sketch the other harmonics up to the forth harmonic and determine their wavelength.

What are the frequencies of the second, third and forth harmonics?

Calculate the speed of the vibration as it travels along the string

A harp string, 70 cm long when at rest, is vibrating with a fundamental frequency of 330 Hz.

Example
1
1
=
v
2L
_
_
f
recall:
v
_
f =
_
_
_
_
_
2
= = 2
v
L
_
f
1
f
3
= = 3
3v
2L
_
_
f
1
f
4
= = 4
L
_
f
1
f
2v
_
3
= = 3
3v
4L
_
_
f
1
f
1
=
v
4L
_
_
f
5
= = 5
5v
4L
_
_
f
1
f
ODD HARMONICS ONLY
http://www.walter-fendt.de/ph14e/stlwaves.htm
= nf
1
Interference between two coherent (in phase) wave sources
(NB: sometimes still called 2nd, 3rd, 4th etc.
read question carefully should be clear)
Other notes
Make transverse and longitudinal waves
Measure:
time for 3 waves to pass a point
amplitude
(transverse only)
wavelength
Calculate:
period
frequency
speed of wave
Extension:
Send a wave from either end towards each other and observe what happens
Try and create a series of 'standing waves'
pic: http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Spectra/travelwave.gif
Wavelength ( )
Period (T) and Frequency (f)
Wavelength(m)
: length of one compete wave
Period (s)
: time for one complete wave to pass a point
Frequency (Hz):
number of complete waves passing a point per second
i
i
i
Properties of waves:
f =
1
_
T
_
T =
1
f
_
_
v = f
wave equation:
Amplitude
http://biologicalexceptions.blogspot.com.au/2014/07/lets-get-loud.html
Better resolution:
l
l
/
Pitch and Intensity
Checkpoints
(2015)

878-902,945-946, 950-951, 959
http://resource.isvr.soton.ac.uk/spcg/tutorial/tutorial/Tutorial_files/Web-basics-pointsources.htm
Acoustic Monopole
constructive superposition
destructive superposition
_
_
_
_
_
_
_
_
_
_
_
_
_
_
'
Doppler Effect
The apparent change in wavelength (or frequency) of a wave as perceived by an observer moving relative to the source
low pitch
high pitch
Doppler Effect
Speed of Light
___________
Electromagnetic wave model of light
Light is a transverse wave
http://www.walter-fendt.de/ph14e/emwave.htm
Electric and magnetic fields oscillating perpendicular to each other
How do we know light is transverse?
IT CAN BE
POLARISED

Longitudinal waves cannot be polarised
Transverse waves can
Polarised light has all the electric field part of the EM wave oscillating in one direction
(& the magnetic field part perpendicular to it - electric field only depicted above)

Light can be polarised by reflection of surfaces
"glare"
Boundary conditions
Fixed end
180 phase change" occurs
Free end
No phase change
o
"
Only certain wavelengths will form standing waves...
1st harmonic
2nd harmonic
3rd
4th
5th
6th
Harmonics
String fixed one end only
1st harmonic
(fundamental frequency)
2nd harmonic (1st overtone)
3rd harmonic (2nd overtone)
1
= 2L
2
= L
3
=
2L
_
3
_
_
1st harmonic (fundamental freq)
1
= 4L
5
=
4L
_
5
_
_
3
=
4L
_
3
_
_
Free end = AN
Open end = N
1
=
v
2L
_
_
f
2
= = 2
v
L
_
f
1
f
3
= = 3
3v
2L
_
_
f
1
f
3
= = 3
3v
4L
_
_
f
1
f
1
=
v
4L
_
_
f
5
= = 5
5v
4L
_
_
f
1
f
ODD HARMONICS ONLY
3rd harmonic (1st overtone)
5th harmonic (2nd overtone)
recall:
v
_
f =
_
_
_
_
_
boundary is a node
boundary is an anti node
(java)
Fixed end = N
Other examples of standing waves...
Light travels at c in a vacuum
Wave properties
Refraction
Wavelength decreases, velocity decreases but frequency (& hence energy) stays the same
v =
f
f =
1
T
_
T =
1
f
_
Young's double slit experiment
http://science.sbcc.edu/physics/flash/2%20slit%20interference.swf
P.D
P.D
Nodes:
PD = (n- )
2
_
1
Anti-nodes:
PD = n
n=1,2,3...
n=0,1,2,...
Light as a wave
Light is produced by accelerating charges:
An accelerated charge produces a changing electric field
A changing electric field produces a changing magnetic field
This is the definition of light!
Identify wavelength, frequency and uses in society
Applications: optical fibers
Snells Law
speed of light in vacuum
speed of light in material
=
refractive index
of a material (n)
How fast does light travel in water?
What percentage of c does light travel in diamond?
When the angle between the normal and a ray in fluid (with n=1.45) is increased beyond a certain critical angle, the light ray does not emerge from the fluid.

Calculate this critical angle in degrees.
Draw what happens to the light incident at angles greater than the critical angle.
Example
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
n
=
c
v
_
v = c/n = 2.26E8 m/s
v/c x 100% = 1/n x 100%
= 1/2.42x100%
= 41% of c
Refractive index
Critical angle & Total internal reflection
sin (c) =
n
n
2
1
_
_
43.6
o
sin(x) = n2/n1
= 1/1.45
x =
Travels at c in a vacuum
Slower in other mediums
The bending of light as it passes from one medium to another
Bends away from normal if speeding up
Bends towards normal if slowing down
N
N
all light reflected
Other examples
Dispersion
Refractive index is different for different colours!
bends least
bends most
Excellent animation:
this is tricky concept which you can just take at face value but here is a more in depth explanation if you want it...
1st minimum
1st maximum
2nd minimum
2nd maximum
Central maximum
1st minimum
2nd minimum
2nd maximum
1st maximum
Calculate the path difference at each dark/light band shown
Example
Effect of wavelength, d and L
The
path difference
to the second maximum is 723 nm longer than the
path difference
to the first minimum.
What is the wavelength of the light used?
x
_
_
L
d
_
Dark and light bands observed
longer wavelengths have more spaced out bands
So:
use lasers (already coherent & monochromatic)
put incoherent polychromatic light through a single slit (-> coherent) and wavelength filter (-> monochromatic)
Sunlight:
many colours (
polychromatic
) and waves not in phase (
incoherent
)
LED
: one colour (
monochromatic
) and waves not in phase (
incoherent
)
Laser
: one colour (
monochromatic
) and waves in phase (
coherent
)
Coherence of light
no filter
green filter
What would happen to the pattern spacing if a shorter wavelength of light was used?
Node (dark band):
PD = (n- )
2
_
1
n=1,2,3...
Anti-node (light band):
PD = n
n=0,1,2,...
Youngs double slit experiment
Interference between two
coherent (in phase)
light sources is observed
"Young's double slit experiment"
Can occur when n1 > n2 (i.e. light speeds up & bends
away
)
Snell's Law becomes:
Particle model prediction:
Wave model prediction:
Hint: add at each peak and intersection with zero of the higher frequency wave
Draw the wave resulting from this superposition:
max displacement
no displacement
L
Sound:
Constructive interference = loud
How are these maxima and minima formed?
AN
N
but what is this & how do we get it?
At 568μm in length Hydrothermal worms are deep sea creatures, almost as small as bacterium
Imaging limits
Pollen from a variety of common plants: sunflower (Helianthus annuus), morning glory Ipomoea purpurea, hollyhock (Sildalcea malviflora), lily (Lilium auratum), primrose (Oenothera fruticosa) and castor bean (Ricinus communis). The image is magnified some x500, so the bean shaped grain in the bottom left corner is about 50 μm long.
Effect of wavelength & gap size
know (qualitatively) what effect changing these variables has on the pattern spacing
x
_
_
w
http://www.walter-fendt.de/ph14e/singleslit.htm
L
w
x
Diffraction of Light: Single Slit
Blue Light
Red Light
(or w) = less spacing
(or w) = more spacing
red spreads the most, blue the least
Can use to separate the colours in white light !!
if object size (w) too small for light... electrons can be used!
ELECTRON MICROSCOPE!
w
Single slit diffraction/interference!
< w
https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle
More:
https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle
More:
=
~
wavefront kind of like many point wave sources
a path difference!
Where these dark bands come from??
Pollen!! the bean shaped grain in the bottom left corner is about 50 μm long!
Too much
diffraction makes images hard to resolve
>> W
W
~
<
clear pattern
a big blur
<< W
light opposite hole only
but light and dark bands are observed! why?
Some waves arrive out of phase and cancel out = dark band!
little diffraction
much diffraction
for Sound this means:
Double vs single slit patterns
Light Production
Sunlight:
many colours (
polychromatic
) and waves not in phase (
incoherent
)
LED
: one colour (
monochromatic
) and waves not in phase (
incoherent
)
Laser
: one colour (
monochromatic
) and waves in phase (
coherent
)
Coherence of light
Sunlight/incandescent light bulb
random thermal motion of valence electrons in collisions
LEDs
Laser
( within 1/100th of w)
(little diffraction)
Synchotron
https://i.imgur.com/tvFKaZy.gifv
Levitation with standing sound waves!
it's easier to tell where high pitch sounds are coming from
Waves are made in a ripple tank from two sources X and Y.
2 cm
The wavelength is 2 cm.
x
Y
A
B
C
Calculate the distances:
a) (YA - XA)
b) The path difference to the first minima (B)
c) (YC - XC)
Example
Central maxima
1st minima
1st minima
1st maxima
1st maxima
2nd min
2nd min
2
2
2nd max
2nd max
0
/2
/2
3 /2
3 /2
Path difference
...
...
AN
N
PD = (n- )
2
_
1
PD = n
...don't worry if you don't get it though
Kind of like a car would do....
=
PD
The pattern shape matches the gap shape!
We are looking at this one
We want THIS when
studying the
light/gap itself
We want THIS when
trying to view an
object with light (imaging.....)
( > 100 x w)
( < w/100)
Predicting diffraction
x
_
_
w
(much diffraction)
(some diffraction)
light:
Maxima
Minima
n=0,1,2,...
n=1,2,3...
x
_
_
L
d
_
=
PD
(Path difference)
Min if:
Max if:
Distance between consecutive max
Central maxima
1st minima
1st minima
1st maxima
1st maxima
2nd min
2nd min
2nd max
2nd max
...
...
2
2
0
/2
/2
3 /2
3 /2
Path Difference
PD = (n- )
2
_
1
PD = n
n=0,1,2,...
n=1,2,3...
Min if:
Max if:
Path difference
PD = (n- )
2
_
1
PD = n
n=0,1,2,...
n=1,2,3...
PD
(Path difference)
Min if:
Max if:
Double Slit Diffraction
The double slit experiment done with light by Young provided evidence for the wave nature of light
EMR can be produced by accelerating charges:
An accelerated charge produces a changing electric field
A changing electric field produces a changing magnetic field
This is the definition of light!
this is tricky concept which you can just take at face value but here is a more in depth explanation if you want it...
...don't worry if you don't get it though
http://www.walter-fendt.de/ph14e/singleslit.htm
R

O

Y

G
B
I

V
for light this means:
Diffracts more
Diffracts less
This is called an
Interference Pattern
:
seen here in a ripple pond
Constructive interference = loud
Destructive interference = slient
Two speakers produce a sound of 680 Hz in air (v=340 m/s)
What is the wavelength of the sound used?
Students 2, 4 and 6 report hearing a loud sound while students 1, 3, 5 and 7 report hearing a quiet sound.

How far apart the students standing?
4 m
24 m
Example 3
Explain why the central band is bright not dark.
The path difference to X in found to be 750 nm
Calculate the wavelength of light used
What is the path difference to bright band P?
A laser is shone onto double slits:
Example 2
http://www.walter-fendt.de/html5/phen/doubleslit_en.htm
2
/
/
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