Wave Phenomena - 2D and 3D
How can you remember whether to add or subtract for the doppler moving source formula?
STS - Source Towards? Subtract
Define refraction
the change in direction of a wave due to a change in speed when it crosses a boundary between two different media at an angle
define path difference
the difference in the distances two waves must travel from their sources to a given point
Define path length
the distance a wave must travel from its source to a given point
What are the approximate ratios of successive intensity maxima compared to the central maxima of a single slit diffraction pattern?
5%, 2%, 1%
Draw a basic intensity distribution graph. Now, draw how it changes when using blue (a shorter wavelength) and red (A longer wavelength) light
Blue graph should have closer bumps, red should have further away, with peaks being slightly shorter.
Light passes through a single slit and shines on a flat screen located .4 meters away. The width of the slit is 4x10^-6 meters. Determine the width of the central bright fringe when the wavelength of light in a vacuum is 690 nm
.138 meters Solving for 2s. Y/b = s/D
Two speakers with identical frequencies of 300 Hz are placed in a line. Standing in front of one speaker, you can hear no sound. How far apart are the speakers?
.563 meters Change in L = .5 Y for the minimum path difference. We know that the speakers are half a wave apart, so they create destructive interference
What are some possible n-values associated with the destructive interference of two point sources?
0.5, 1.5, 2.5..
What are some possible n-values associated with the constructive interference of two point sources?
1, 2, 3...
A diffraction pattern forms on a screen 2.5 meters away when light of a wavelength 675 nm passes through a single slit. 1. Determine the angle that locates the first dark fringe if the width of the slit is 18 um 2. How wide is the central maximum?
1. .0375 radians looking for angular half-width. theta = Y/b don't multiply theta by 2 2. .1875 m looking for s x 2 s/D = theta theta x D = s s x 2 = linear width
In a double slit experiment, light whose wavelength is 6x10^-7 meters is shone through two slits that are .1 mm apart, onto a screen that's 2.5 m away. The small angle approximation applies here. 1. At what angle from the central maximum will the first bright fringe appear? 2. At what angle from the central maximum will the second bright fringe appear? 3. What is the distance between the central maximum and the first bright fringe? 4. What is the distance between the central maximum and the second bright fringe? 5. What is the distance between any two adjacent bright fringes?
1. .344º nY/d = sin0 2. .688º. twice the first one, or 2Y/d = sin0 3. .015 m s = nY x D /d or Xn/Dn = sin0 4. .030 m s = nY x D /d or Xn/Dn = sin0 or twice the first one 5. 0.015 m
Monochromatic light from a laser with wavelength 632.8 nm is incident normally on a diffraction grating containing 6000 lines per cm. 1. Find the distance between the slits on the grating 2. Find the angles at which one would observe the first order fringes (light and dark) 3. Determine how far each of these fringes is from the central maximum on a screen 2.5 meters away
1. 1.67x10^-6 m 1 cm/6000 slits 2. light: 22.3º Dark: 11º theta = sin^-1 (y/d) Dark fringe, n = .5 so 22.3º /2 = 11º 3. 1.02 meters SMALL ANGLE APPROX doesn't apply!! Tan0 = x/2.5
in the sodium emission spectrum there are two wavelengths that are close to one another. These are 589 nm and 589.59 nm. for these two lines to be resolved by a diffraction grating, determine the resolvance number of slits that must be illuminated for the first order maxima number of slits that must be illuminated for the second order maxima
1. change in y = .59 nm or 5.9x10^10 m R1= 589/.59 = 998 R2 = 589.59/.59 = 999 R ~ 998.5 2. 1000 mn = y/delta y m = 1, solve for n 3. 500 mn = y/delta y m = 1, solve for n
what equation do you use when finding the minimum resolution angle for a circular aperture
1.22 y / b
how do you account for the refractive index number of a human eye in a resolution equation? What equation (in variables) should you use?
1.22(y/n of eye)/b = s/d
A soap film (n=1.33) is surrounded on both sides by air. Sunlight strikes the film nearly perpendicularly and several colored fringes are seen. One particular shade of red light, whose wavelength is 655 nm is seen. What is the minimum thickness of the film at this point to cause constructive interference for this color?
1.25 x 10-7 m 2dn = (0 + 1/2) y
A thin film of gasoline floats on a puddle of water. Sunlight falls almost perpendicularly on the film and reflects up. Determine the minimum thickness of the film that will result in destructive interference for blue light whose wavelength is 469 nm. nGasoline = 1.40
1.675 x 10^-7 m 2dn = 1 x Y y/2n = d
The security alarm on a parked car goes off and produces a frequency of 960 Hz. An observer drives toward this parked car at 20 m/s How fast is the sound moving relative to the air? The observer? What is the frequency and wavelength the observer percieves?
343 m/s, 363 m/s 1016 Hz, .357 m
A high-speed train is traveling at a speed of 45 m/s when the engineer sounds the 415 hz warning horn. The speed of sound in air is 343 m/s. An observer stands at the railroad crossing that the train is approaching. How fast would the train be moving, and in which direction, if the observer hears a whistle whose frequency is only 90% of what it is at rest?
38.1 m/s away Explanation: To determine direction: F' = .90 F = 373.5 Hz. Since this is lower than the original, the train must be moving away. To determine value: F'/f = .9 .9 = V/(V+Us) Us = V/.9 - V 343/.9 - 343 = 38.1 m/s
A high-speed train is traveling at a speed of 45 m/s when the engineer sounds the 415 hz warning horn. The speed of sound in air is 343 m/s. An observer stands at the railroad crossing that the train is approaching. What are the frequency and wavelength of the sound as perceived by the observer?
477.67 Hz .718 m
A thin film of magnesium fluoride is often applied to the surface of high quality lenses in cameras and telescopes. Find the minimum thickness of a layer of magnesium fluoride (n=1.38) on flint glass (n=1.66) that will cause destructive interference of reflected light of wavelength 550nm
9.98x10^-8 m two hard reflections = 2dn = (m+1/2) Y
A star is moving away from Earth at a speed of 3 x 10^5 m/s. One of the elements in the star emits light with a frequency of 5 x10^14 Hz. By how much is the frequency shifted when it is received by a telescope on Earth? What is the frequency received by the Earth-bound telescope? What frequency would be received on Earth if the star were moving toward the Earth at the same speed?
A. change in F = 3x10^5 / 3x10^8 x 6x10^14 = .006x10^14 Hz B. 5.994 x 10^14 Hz
Describe the relationship between the actual and minimum angles when two sources of light are just resolved
Actual (S/D) equals minimum (y/b)
Describe the relationship between the actual and minimum angles when two sources of light are well resolved
Actual (S/D) is greater than minimum (y/b)
What are 3 features of the single slit intensity pattern? Discuss central maximum, secondary maxima, and interference
Central maximum is wide and bright Secondary maxima are equal and dim different parts of the wave interfere with itself as it spreads
What do the variables mean in the doppler formula for EM radiation?
Change in F = frequency shift (f - f') Change in Y = wavelength shift (y-y') V = relative speed of source and observer (original speed - speed they're going)
In a double slit scenario, will the fringes get closer or further apart if the screen is brought closer to the slits?
D decreases, so s decreases, fringes are closer
Define phase difference
Difference in the phases of two waves when they meet/overlap
For a double slit formula scenario, define Do, Dn, d, X, and S
Do = long leg of the triangle, marks the central maximum Dn = hypotenuse of triangle, marks the edge X/S = linear displacement from the central maximum d = distance between slits
for what kinds of waves should you use the doppler formula/relationships for Em radiation
EM radiation: radio waves, light, x-rays, etc.
What do all the variables in the Doppler Formula (moving observer) equation stand for?
F = original frequency F' = perceived frequency V = speed of sound in medium Vo = speed of observer relative to medium
What do all the variables stand for in the Doppler (moving source) equation?
F = original frequency F' = shifted frequency V = speed of sound in medium Us = speed of source relative to medium
What does s stand for in a single slit scenario? What equations can you use to find it?
In a single slit scenario, s = central half-width, the distance from the central maxima to the first dark spot off to one side. Sin theta = Y/b = S/D
What direction will a wave refract when it goes from air to glass?
FAST - from a fast medium to a slow medium, it will bend toward the normal
What is the refraction rule for remembering which direction a wave will bend?
FAST - into fast, bend away. Into slow, bend towards
For an observer moving away from a stationary source, how does frequency, wavelength, and sound of a wave change?
Frequency decreases as fewer waves per second are detected. Wavelength, the control variable, is constant. Perceived speed decreases.
For an observer moving towards a stationary source, how does frequency, wavelength, and speed of sound of a wave change?
Frequency increases as more waves per second are detected. Wavelength, the control variable, is constant. Perceived speed increases.
What happens to wave intensity as the distance from the source doubles?
I = 1/4
A person stands 3 meters away from a 100 watt speaker. Determine the intensity of the sound they hear from this location, and from 6 meters away
I = 100 J/s / 4 pi (3m)^2 = .88 w/m^2 and .22 w/m^2
What is the proportional relationship between distance from source r and wave intensity I?
I is proportional to 1/r^2
What is intensity proportional to when using diffraction grating?
I is proportional to the number of slits squared
What is the wave intensity formula? (Not in EQ book. Hint in wave characteristics section: I is proportional to A^2) Hint 2: Think of the surface area of a sphere.
I= P/4 pir^2
Draw five intensity vs angle from straight through position graphs showing diffraction patterns for 1, 2, 3, 4, and 5 slits
Increase number of bumps between each. Overall shape is still like a single slit pattern (normal curve-ish). Quick way to remember: the small bumps in between two maxima, plus those two maxima themselves, should equal number of slits For example, when there are 5 slits, you would count two maxima plus three small bumps in between them. so it would be one bump, 3 small ones, one bump, 3 small ones, etc, overall following the shape of the normal curve (big bumps don't exceed overlay of normal curve).
Rank these in order from longest to shortest wavelength: red light, blue light, UV rays, infared rays
Infared, red, blue, UV
Describe what happens to a wave's frequency, speed, and wavelength when it goes from a fast medium into a slow medium
Into a slow medium, speed v decreases. Frequency must be constant, so wavelength must decrease. v = fY
Describe how the graph of a single slit diffraction pattern changes as the slit is wider and narrower
Narrower slit = wider and lower central maxima, more spread out graph Wider slit = taller and thinner central maxima, all bumps closer together
How can we remember whether to add or subtract when using the moving observer doppler formula?
OTA - observer towards? Add
Describe what an observer hears as a truck moves at constant velocity past them
One constant high pitch when its moving toward them and one constant low pitch as it moves away
A square is 3.5 m on one side, and point A is the midpoint of one of its sides. On the side opposite this spot, two in-phase loudspeakers are located at adjacent corners. Standing at point A, you hear a loud sound. As you walk along the side of the square toward either empty corner, the loudness gradually diminishes but doesn't entirely disappear UNTIL you reach either empty corner, where you hear no sound at all. 1. Find the wavelength of the sound waves 2. The frequency of the sound emitted by both speakers increases to 700 Hz. As you walk along the side of the square from A to an empty corner, you hear the loud sound at A repeatedly diminish to no sound, then increase to maximum again. By the time you reach the corner, you have noticed the sound disappear and reappear 3 times. Estimate a speed for this sound.
Part 1: 2.9 m Path difference to point B: 3.5(square root 2) - 3.5 = 1.449 meters since that spot is the first instance of destructive interference, we know that n= 0, so path difference = 0.5 Y so, 1.449 x 2 = Y = 2.9 meters Part 2: 338.1 m/s we know that there is now constructive interference at the corner, and n=3. Path difference equation: change in l = nY, = 3.5(square root 2) - 3.5 = 1.449 meters. This value = 3Y So, to find the length of 1 wavelength, 1Y, divide 1.449 by 3 Finally, use the equation c = yf to solve for c
What two things determine which wavelengths interfere constructively and destructively in a thin film interference scenario?
Path difference and phase difference
Imagine two point sources (like speakers) that interfere constructively with each other. Describe the phase and path differences (how to calculate phase difference) using variables if necessary
Phase difference = 0 Path difference = 0 Y
Imagine two point sources (like speakers) that interfere destructively with each other. Describe the phase and path differences (how to calculate phase difference) using variables if necessary
Phase difference: 180º or pi Path difference: 1/2Y
Describe what an observer hears as a truck with increasing velocity passes them
Pitch increases until it meets them and then decreases after it passes
Define wave intensity. What's it's defining equation?
Power per unit area. I=P/Area
what do all the variables stand for in the resolvance of diffraction grating equation?
R = resolvance m = 1,2,3 = diffraction band # N = total number of slits
What mode should your calculator be in when working with the single slit diffraction formulas?
RADIANS
What are the two conditions for total internal reflection?
Ray of light must be traveling from the more optically dense to the less optically dense medium Ray must strike the boundary at an angle greater than the critical angle
What 3 phenomena cause a rainbow? (atmospheric rainbows, diffraction gratings, thin films)
Refraction/dispersion (atmospheric rainbows), diffraction/interference (diffraction gratings), reflection/interference (thin films)
What is diffraction
The bending or spreading of a wave when it passes through a small opening or around a barrier
What is total internal reflection?
The complete reflection of light at a boundary (no refraction)
What variables impact the ability to resolve two sources of light?
The distance away from the aperture (D), the distance between two sources (S), wavelength (Y) and the size of the aperture (b)
What are the two key features of an ideal intensity distribution graph? for waves.
a series of bumps, with the peak of the center one lining up with x=0. bumps have equal spacing and widths All equal brightness
What is the proportional relationship between total energy and amplitude for a mechanical wave? (hint in EQ book)
Total energy is proportional to I, which is proportional to amplitude squared so total energy is proportional to amplitude squared
Rank these in order from highest to lowest frequency: red light, blue light, UV rays, infared rays
UV, Blue, Red, Infared
Describe how thin film interference works, with reference to the two resulting light rays. Also, explain what determines whether or not you'll see a rainbow.
When an incoming light ray hits the upper boundary (the thin film on top of a substance), it both reflects (Ray A) and refracts (Ray B). Light rays A and B will then interfere either constructively or destructively as seen when viewed from above, depending on whether they're in phase or not when they recombine. If the incoming light ray was white light, destructively interfering wavelengths won't be seen and constructive will. This produces the rainbow effect.
When can the phase of a wave change? (thin film interference)
When it reflects
What is the small angle approximation?
When using double slit formulas: if theta is small, then sin theta = theta in radians, and sin theta = tan theta
What are the destructive fringe separation equations for a double slit scenario? Construct using Xn (aka S), Dn, Y, d, and theta. It can be helpful to draw a triangle showing a double slit scenario
Xn/Dn = (n+1/2)Y /d = sin0n In a double slit scenario we're working with triangles- looking at the picture, we know from soh cah toa that sin theta = S/D REMEMBER: IMAGINARY N+1/2 is multiplying the Y In the equation book we have s=YD/d. plug YD/d into S/D equation and add in the invisible n multiplier and you get (n+1/2)Y/d
What are the constructive fringe separation equations for a double slit scenario? Construct a series of equivalencies using Xn (aka S), Dn, Y, d, and theta. It can be helpful to draw a triangle showing a double slit scenario
Xn/Dn = nY/d = sin0n How to get these equations: In a double slit scenario we're working with triangles- looking at the picture, we know from soh cah toa that sin theta = S/D REMEMBER: IMAGINARY N is multiplying the Y In the equation book we have s=YD/d. plug YD/d into S/D equation and add in the invisible n multiplier and you get nY/d
What is a wavefront
a line or arc joining neighboring points that have the same phase or displacement
What is diffraction grating
a piece of glass or plastic with many equally spaced parallel slits on it
Describe the relationship between the actual and minimum angles when two sources of light are not resolved
actual (S/D) is less than minimum (y/b)
Draw a graph showing the double slit interference pattern modulated by the single-slit diffraction effect. On this graph, label the two different s distances associated with each pattern AND their associated equations
amplitude of double-slit pattern, modulated by single slit: bunch of bumps like a double-slit pattern in the shape of a single-slit diffraction pattern (looks sorta like a normal curve) double slit s: distance between center bump and the first bump off to one side. Equation: sin theta = ny/d = s/D single slit s: distance from central maxima to first dark spot- the single slit half width. Equation sin theta = Y/b = S/D
What is a critical angle 0c?
angle of incidence for which the angle of refraction is 90º. Light is infinitely faint here.
When can the wavelength of a wave change? (thin film interference)
as it enters a new medium
Describe the general rule for what happens to maxima and minima using diffraction grating, as the number of slits increases
as the number of slits increases, maxima maintain location but become narrower and more intense As the number of slits increases, minima aren't totally dark
What is the formula for a wave's critical angle? Derive it using n1sin01 = s2sin02
at 90º, ni sin0i = nr sin0r. Sin 0r = 90 or more sin 90 = 1 0c = sin^-1 (nr/ni)
What changes in light during the doppler effect
change in frequency
What changes in sound during the doppler effect
change in pitch
Draw a diagram of waves coming off of two point sources in the shape of a semicircle (only draw half of the circle, with the two point sources on the left and the waves going out to the right). On that diagram, label the n-value and path difference for 3 nodal and 3 anti-nodal lines, starting from the center
check page 9 of wave phenomena packet (it's kind of hard to describe what it should look like but I'll try my best) Down the center should be n=0 for constructive interference. Path difference = 0Y To either side should be n=0 for destructive interference. Path difference = 0.5Y To either side of that should be n=1 for constructive interference. Path difference = 1Y Next is n=1 for destructive, path difference = 1.5Y and so on
Draw a basic intensity distribution graph for double slit interference. Now, draw how it changes when the slits are closer, and when the screen is closer
closer slits = energy spread of smaller area. S increases, so bumps are more spread out Closer screen = peaks don't spread as much. A bunch of thin, close-together bumps.
How can you tell if a wave medium is "Fast" or "Slow"
compare n values. A higher index of refraction = a slower medium and a lower index of refraction = a faster medium
Draw a diagram representing the wavefronts produced by two point sources, with solid circles being wave crests and dashed circles representing wave troughs, showing denstructive interference.
concentric rings around two points. The rings overlap at some points. Alternate dashed and solid lines to represent peaks and troughs. Draw a series of dots anywhere on the diagram (or two going near the center) where a peak from one source and a trough from the other meet.
Draw a diagram representing the wavefronts produced by two point sources, with solid circles being wave crests and dashed circles representing wave troughs. Now, draw dots showing constructive interference.
concentric rings around two points. The rings overlap at some points. Alternate dashed and solid lines to represent peaks and troughs. Draw a series of dots down the center (or anywhere on the diagram) where two peaks and two troughs meet
rewrite the constructive and destructive thin film interference equations (in EQ book) when N1 < N2 < N3 (aka when it undergoes two hard reflections instead of a hard reflection and then a soft one)
constructive: 2dn = m (y/n) destructive: 2dn = (m+1/2) y/n
When working with diffraction grating, what is the relationship between d and n, where n is the number of slits per meter and d is the distance between slits?
d = 1/n
In a double slit scenario, will the fringes get closer or further apart if the slits are brought closer together?
d is smaller, so s increases, fringes are further
Define power and its units. What is it's defining formula?
energy per unit time, J/S. P = E/T
What is the Rayleigh criterion?
for two sources to be just resolved, the central maximum of one diffraction pattern is located on top of the first minimum of the other diffraction pattern
why are smaller wavelength lasers used? Why are electrons used instead of light for electron microscopes?
greater resolving power, resolution, and clarity of sound and picture. tracks can be placed closer together. smaller wavelengths = greater resolving power, greater resolution
what does s stand for in a double slit scenario? What equations can you use to find it?
in a double slit scenario, s = distance between maxima. sin theta = nY/d = s/D
Define the doppler effect
the apparent change of frequency of a wave due to the movement of the source or the observer relative to the medium of wave transmission
briefly explain the significance of Young's double slit experiment
it demonstrated that light acts like a wave, and was the first measurement of light's wavelength
Sketch an intensity pattern for two sources of light that are just resolved, well resolved, and unresolved
just resolved has midpoint of one on the minimum of another well resolved has two maxima further separated towards either edge not resolved has two maxima more overlapped, not fitting the Rayleigh criterion
What is a Ray
line indicating direction of wave motion (energy transfer). Perpendicular to wavefronts
for the doppler shift of a moving observer, would a wave with a bigger or smaller wavelength shift more?
longer wavelength has a greater shift. shift in wavelength = (v/c) x y where (v/c) is a constant.
What is the formula/relationship associated with a wave changing wavelengths as it enters a new medium? (in EQ book)
n1/n2 = v2/v1 = y2/y1
describe how the perceived frequency, amplitude, and wavelength change as a source moves away from a stationary observer
observer detects fewer waves per second so the frequency is lower (pitch is lower). As frequency decreases, wavelength gets longer. Amplitude decreases (volume)
describe how the perceived frequency, amplitude, and wavelength change as a sources moves towards a stationary observer
observer detects more waves per second, so frequency (pitch) is higher As frequency increases, wavelength decreases Amplitude increases (volume)
List some applications of the doppler effect
police radar devices, medical physics (doppler flow meter to measure blood flow speed) Weather forecasting and astrophysics (radar used to observe the motion of air masses)
General rule for bright fringe distances from central maximum in a double slit scenario
second fringe's distance from maximum = 2x the first one
What are the conditions for noticeable diffraction?
size of opening on width of barrier should be approximately equal to wavelength
What is the red shift for the doppler effect of electromagnetic waves? describe frequency, wavelength, what the movement is
source and observer moving away from each other. lower frequency, longer wavelength
What is the blue shift for the doppler effect for electromagnetic waves? Describe frequency, wavelength, what the movement is
source and observer moving towards each other. Higher frequency, shorter wavelength.
What is the control variable for the doppler effect of a moving source
speed Wave speed is the same relative to the observer, wavelength changes so frequency changes oppositely
In a double slit scenario, will the fringes get closer or further apart if a higher frequency of light is used?
speed of light is constant. Frequency increases, so wavelength decreases. S decreases, fringes are closer
Define resolution (aka resolving power)
the ability to distinguish between two sources of light
What is the law of reflection
the angle of incidence is equal to the angle of reflection when both angles are measured with respect to the normal
The minimum angle between two sources for them to be just resolved is equal to what element of one diffraction pattern?
the angular half-width, theta
How does barrier size compared to the wavelength of a wave impact diffraction?
the larger the barrier size compared to the wavelength of a wave, the less the wave diffracts
How does slit width compared to the wavelength of a wave impact diffraction?
the larger the slit width compared to the wavelength, the less the wave diffracts
For single slit diffraction: What is the main ratio for angular and linear half-widths? (partly in data book)
theta = Y/b = s/D
For single slit diffraction: what is theta, s, and b?
theta = angular half-width. Multiply by 2 to get angular width. s = linear half width. Multiply by 2 to get linear width b = distance between top and bottom of slit
a spy satellite is orbiting 180km above the surface of the Earth. if the diameter of the lens is 45 cm, what is the smallest distance on the ground that can be resolved by the camera on the satellite?
theta min = 1.22y/b = 1.22x5x10^7/.45 m = 1.355x10^-6 rad theta = s/d = x/1.8x10^5 = 1.355x10^-6 rad x = .24 meters
the brightest star in the winter sky in the Norther Hemisphere is sirius. in reality, sirius is a system of two stars that orbit each other with an average separation of 3 x10^12 m. The hubble space telescope (diameter 2.4 m) is pointed ast the Sirius system with is 8x10^16 meters from earth. Can this telescope resolve the two stars, or does it see them as one point of light? assumption: average light has wavelength 500 nm
theta min = 1.22y/b = 5x10^-7 m/2.4 m = 2.5x10^-7 rad theta actual = s/d = 3x10^12m/8x10^16m = 3.75x10^-5 rad Since the angular separation between the two stars is greater than the minimum angle, they can be resolved
When doing single slit diffraction calculations, what should you do if the small angle approximation doesn't work?
use tan theta instead of theta (calculator is in radians!)
what is the control variable for the doppler effect of a moving observer
wavelength wavelength is the same relative to the medium, wave speed increases or decreases relative to observer so frequency changes the same way. EXCEPTION: light, no medium of propagation so relative velocity of source and detector is relevant.
what does it look like when two beams of light are well resolved, just resolved, and not resolved?
well resolved = two very distinct beams of light just resolved = you can tell there are two beams of light, but just barely not resolved = light looks like its coming from one source
When does a wave have a hard reflection vs a soft reflection?
when it goes into a faster medium, it has a soft reflection. When it goes into a slower medium, it has a hard reflection
When is the doppler formula for Em radiation formula valid?
when relative speed is much less than the speed of light (V << C)
When does constructive interference occur?
when two waves/rays are in phase