Example An of Resonance
Pendulum A is set in motion The others begin to vibrate due to the vibrations in the flexible beam Pendulum C oscillates at the greatest amplitude since its length, and therefore frequency, matches that of A
Other Examples of Resonance
Child being pushed on a swing Shattering glasses Tacoma Narrows Bridge collapse due to oscillations by the wind Upper deck of the Nimitz Freeway collapse due to the Loma Prieta earthquake
Using a Tuning Fork
As the tuning fork continues to vibrate, a succession of compressions and rarefactions spread out from the fork A sinusoidal curve can be used to represent the longitudinal wave
Crests correspond to compressions and troughs to rarefactions
Representations of Waves
Wave fronts are the concentric arcs
The distance between successive wave fronts is the wavelength
Rays are the radial lines pointing out from the source and perpendicular to the wave fronts
Plane Wave
Far away from the source, the wave fronts are nearly parallel planes The rays are nearly parallel lines A small segment of the wave front is approximately a plane wave
Superposition ..AKA....Interference
One of the characteristics of a WAVE is the ability to undergo INTERFERENCE. There are TWO types.
We call these waves IN PHASE.
We call these waves OUT OF PHASE.
The Central Maximum
Suppose you had TWO sources each being allowed to emit a wave through a small opening or slit. The distance between the slits denoted by, d. The distance from the slit spacing to the screen is denoted by the letter, L. If two waves go through the slit and then proceed straight ahead to the screen, they both cover the SAME DISTANCE and thus will have constructive interference. Their amplitudes will build and leave a very bright intense spot on the screen. We call this the CENTRAL MAXIMUM.
Diffraction Path difference
Notice that these 2 waves are IN PHASE. When they hit they screen they both hit at the same relative position, at the bottom of a crest. How much farther did the red wave have to travel? Exactly ONE WAVELENGTH
The call this extra distance the PATH DIFFERENCE. The path difference and the ORDER of a fringe help to form a pattern.
Path difference
Notice that these 2 waves are OUT OF PHASE. When they hit they screen they both hit at the different relative positions, one at the bottom of a crest and the other coming out of a trough. Thus their amplitudes SUBTRACT. How much farther did the red wave have to travel? Exactly ONE HALF OF A WAVELENGTH The call this extra distance the PATH DIFFERENCE. The path difference and the ORDER of a fringe help to form another pattern.
0 .5
Path difference
Notice that these 2 waves are IN PHASE. When they hit they screen they both hit at the same relative position, at the bottom of a crest. How much farther did the red wave have to travel? Exactly ONE WAVELENGTH
The call this extra distance the PATH DIFFERENCE. The path difference and the ORDER of a fringe help to form a pattern.
Interference of Sound Waves
Sound waves interfere
Constructive interference occurs when the path difference between two waves' motion is zero or some integer multiple of wavelengths
path difference = n
Destructive interference occurs when the path difference between two waves' motion is an odd half wavelength
path difference = (n + )
Two speakers placed 1.00 m apart are driven by the same oscillator. A listener is originally at point O, which is located 2.00 m from the center of the line connecting the two speakers. The listener then walks to point P, which is a perpendicular distance 1.0 m from O, before reaching the first minimum in sound intensity. What is the frequency of the oscillator? Take the speed of sound in air to be v = 343 m/s.
Diffraction
AP Physics B
Diffraction
When light OR sound is produced by TWO sources a pattern results as a result of interference.
Interference Patterns
Diffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings
The Central Maximum
Figure 2 Figure 1 Here is the pattern you will see. Notice in figure 2 that there are several bright spots and dark areas in between. The spot in the middle is the BRIGHTEST and thus the CENTRAL MAXIMUM. We call these spots FRINGES. Notice we have additional bright spots, yet the intensity is a bit less. We denote these additional bright spots as ORDERS. So the first bright spot on either side of the central maximum is called the FIRST ORDER BRIGHT FRINGE. Figure 1 represents the intensity of the orders as we move farther from the bright central maximum.
Orders, m
We use the letter, m, to represent the ORDER of the fringe from the bright central.
Second Order Bright Fringe First Order m=2 Bright Fringe m=1
First Order Central Bright Fringe Maximum m=1
Second Order Bright Fringe m=2
It is important to understand that we see these bright fringes as a result of CONSTRUCTIVE INTERFERENCE.
Bright Fringes
The reason you see additional bright fringes is because the waves CONSTRUCTIVELY build. There is a difference however in the intensity as you saw in the previous slide. As you can see in the picture, the BLUE WAVE has to travel farther than the RED WAVE to reach the screen at the position shown. For the BLUE WAVE and the RED WAVE to build constructively they MUST be IN PHASE. Here is the question: HOW MUCH FARTHER DID THE BLUE WAVE HAVE TO TRAVEL SO THAT THEY BOTH HIT THE SCREEN IN PHASE?
Path Difference
2 1 2 1 The bright fringes you see on either side of the central maximum are multiple wavelengths from the bright central. And it just so happens that the multiple is the ORDER.
Therefore, the PATH DIFFERENCE is equal to the ORDER times the WAVELENGTH
P.D. = m ( Constructive)
Diffraction Dark Fringes
We see a definite DECREASE in intensity between the bright fringes. In the pattern we visible the notice DARK REGION. These are areas where DESTRUCTIVE INTERFERENCE has occurred. We call these areas DARK FRINGES or MINIMUMS.
Dark Fringes
First Order Dark Fringe m=1
ZERO Order Dark Fringe m=0
ZERO Order Central Dark Fringe Maximum m=0
First Order Dark Fringe m=1
It is important to understand that we see these dark fringes as a result of DESTRUCTIVE INTERFERENCE.
Path difference
Notice that these 2 waves are OUT OF PHASE. When they hit they screen they both hit at the different relative positions, one at the bottom of a crest and the other coming out of a trough. Thus their amplitudes SUBTRACT. How much farther did the red wave have to travel? Exactly ONE HALF OF A WAVELENGTH The call this extra distance the PATH DIFFERENCE. The path difference and the ORDER of a fringe help to form another pattern.
0 .5
Dark Fringes
On either side of the bright central maximum we see areas that are dark or minimum intensity. Once again we notice that the BLUE WAVE had to travel farther than the RED WAVE to reach the screen. At this point , however, they are said to destructively build or that they are OUT OF PHASE.
Here is the question: HOW MUCH FARTHER DID THE BLUE WAVE HAVE TO TRAVEL SO THAT THEY BOTH HIT THE SCREEN OUT OF PHASE?
Diffraction Path Difference
1.5 1.5 The dark fringes you see on either side of the central maximum are multiple wavelengths from the bright central. And it just so happens that the multiple is the ORDER. Therefore, the PATH DIFFERENCE is equal to the ORDER plus a HALF, times A WAVELENGTH 0.5 0.5
P.D. = (m+1/2) ( Destructive)
Path Difference Summary
For CONSTRUCTIVE INTERFERENCE or MAXIMUMS use:
P.D. = m
For DESTRUCTIVE INTERFERENCE or MINIMUMS use:
P.D. = (m+1/2)
Where "m", is the ORDER. Refer to slides 6 and 11.
Young's Experiment
In 1801, Thomas Young successfully showed that light does produce an interference pattern. This famous experiment PROVES that light has WAVE PROPERTIES. Suppose we have 2 slits separated by a distance, d and a distance L from a screen. Let point P be a bright fringe.
P
y d B.C.
We see in the figure that we can make a right triangle using, L, and y, which is the distance a fringe is from the bright central. We will use an angle, , from the point in the middle of the two slits. We can find this angle using tangent!
L
Diffraction Another way to look at This right triangle is "path difference" SIMILAR to the one
P
made by y & L.
d
B.C.
d P.D.
P.D. = dsin
P.D. Notice the blue wave travels farther. The difference in distance is the path difference.
Similar Triangles lead to a path difference
y
These angles Are EQUAL.
L
d
dsin
Diffraction Putting it all together
Path difference is equal to the following: m (m+1/2) dsin
Therefore, we can say:
Will be used to find the angle!
A viewing screen is separated from a double slit source by 1.2 m. The distance between the two slits is 0.030 mm. The second -order bright fringe ( m=2) is 4.5 cm from the central maximum. Determine the wavelength of light.
Example
L = 1.2m d = 3.0 x10 -5 m = tan -1 ( y ) = tan -1 ( 0.045 ) = L 1. 2 m = 2 y = 0.045m " bright" = Constructive
2.15 degrees
=?
d sin = m (3 x10 -5 ) sin( 2.15) = 2
=
5.62x10-7 m
Example
A light with wavelength, 450 nm, falls on a diffraction grating (multiple slits). On a screen 1.80 m away the distance between dark fringes on either side of the bright central is 4.20 mm. a) What is the separation between a set of slits? b) How many lines per meter are on the grating?
= 450 x10 -9 m L = 1.80m y = 0.0021m = ? d =?
d sin = (m + 1 ) 2 d sin(0.067) = (0 + 1 )450 x10 -9 2 d = 0.0001924 m
0.0021 = tan ( )= 1.8
-1
0.067 degrees
meters d= line lines 1 = d -1 or = 5197.2 lines/m meter d
Diffraction: single slit
How can we explain the pattern from light going through a single slit?
screen
w
x L
Diffraction: single slit
If we break up the single slit into a top half and a bottom half, then we can consider the interference between the two halves.
screen
w
x L
Diffraction: single slit
The path difference between the top half and the bottom half must be /2 to get a minimum.
screen
w
x
L
Diffraction: single slit
This is just like the double slit case, except the distance between the "slits" is w/2, and this is the case for minimum: (w/2) sin() = /2
screen
w
x
L
Diffraction: single slit
In fact, we can break the beam up into 2n pieces since pieces will cancel in pairs. This leads to: (w/2n) sin(n) = /2 , or w sin( n) = n for MINIMUM.
screen
w
x
L
Diffraction: single slit
REVIEW: For double (and multiple) slits: n = d sin( n) for MAXIMUM (for ALL n)
-2 -1 0 1 2
For single slit: n = w sin( n) for MINIMUM (for all n EXCEPT 0)
-2
-1
0
1
2
Diffraction: single slit
NOTES: For double slit, bright spots are equally separated. For single slit, central bright spot is larger because n=0 is NOT a dark spot. To have an appreciable , d and w must be about the same size as & a little larger than . Recall that for small angles, sin() tan() = x/L
Diffraction: circular opening
If instead of a single SLIT, we have a CIRCULAR opening, the change in geometry makes: the single slit pattern into a series of rings; and the formula to be: 1.22 n = D sin( n) .
Interference: Diffraction Grating
With multiple slits, get MORE LIGHT and get sharper bright spots.
lens s1 s2 d s3 s4 s5 4 s2 = s1 + s3 = s2 + = s1 + 2 s4 = s3 + = s1 + 3 s5 = s4 + = s1 + 4 bright bright
Interference: Diffraction Grating
With 5 slits, get cancellation when s = 0.8; with two slits, only get complete cancellation when s = 0.5 .
lens s1 s2 d s3 s4 s5 3.2 s2 = s1 + .8 s3 = s2 + .8 = s1 + 1.6 s4 = s3 + .8 = s1 + 2.4 s5 = s4 + .8 = s1 + 3.2 bright bright dark
Interference: Diffraction Grating
The same Young's formula works for multiple slits as it did for 2 slits.
lens s1 s2 d s3 s4 s5 4 s2 = s1 + s3 = s2 + = s1 + 2 s4 = s3 + = s1 + 3 s5 = s4 + = s1 + 4 bright bright