Waves and Optics / Name:
Waves & Wave Motion
Terms to describe waves include: speed (v), wavelength (λ), crest, trough, amplitude, frequency (f – measured in Hz or 1/sec) and period (T). Recall that period and frequency are inverses of each other (f = 1/T).
Types of Waves
Longitudinal (sound) – the wave and the medium move parallel to each other
Transverse (water and light) – the wave and the medium move perpendicular to each other
Torsional – the medium travels in a circular motion perpendicular to the wave
Basic Wave of Equation for Speed of a Wave
- For light and other parts of the electromagnetic spectrum (radio waves to gamma rays), the speed in air or a vacuum is c = 3 x 108 m/s. As light enters more optically dense transparent materials, its apparent speed will decrease. When the light reenters air, the speed is again “c.”
- For sound, the speed in air at room temperature is about 340 m/s. As the temperature goes up, the speed will increase (v = 331 + .6T). Sound travels faster in most other substances, going fastest in solids.
- For a string on an instrument, the speed depends on the tension in the string and the mass/unit length of the string.
Wave Reflection
Fixed End Reflection: occurs when a wave goes from a higher speed, less dense medium to a slower speed, denser medium. A crest will reflect back as a trough; a trough will reflect back as a crest. This is a ½ λ shift or change in the wave. An example of this is when sound travels in air down a tube and meets water.
Free End Reflection: occurs when a wave goes from a lower speed, denser medium to a faster speed, less dense medium. A crest reflects back as a crest; a trough reflects back as at trough. There is no shift or change in the wave.
Law of Superposition
When two or more waves meet, the law of superposition predicts the resulting wave. Typically we represent sound and light waves by a repeating sine curve. Superimpose all waves (showing their amplitudes as a function of time) and add all amplitudes at each time to find the resultant wave. Constructive Interference results when two identical waves meet with their crests aligned. The result is a single wave with the same frequency and wavelength, but greater amplitude. Waves that are a whole number of wavelengths apart (1, 2, 3 etc.) when they meet will undergo constructive interference. Destructive Interference results when two identical waves meet with their crests and troughs aligned. The result is that the waves cancel. Waves that are a half of wavelength apart (1/2, 3/2, 5/2 etc.) when they meet will undergo destructive interference.
Doppler Effect
If a wave source is moving, the frequency heard by a detector changes. The heard frequency is higher if the source is approaching the detector; the heard frequency is lower if the source is moving away. In this case, the apparent wavelength changes but the speed remains the same as if the source were not moving.
If the detector is moving, the frequency heard by a detector changes as well. The heard frequency is higher if the detector is approaching the source; the heard frequency is lower if the detector is moving away from the source. In this case, the apparent speed changes, but the wavelength remains the same as if the source was not moving.
The amplitude of a wave determines how strong or bright or loud it is. The increased loudness as a sound source gets closer to the detector has nothing to do with the Doppler Effect! In fact, the intensity of any wave coming from a point source falls off as the inverse of the square of the distance separating the source and the detector. It is the apparent change in frequency that is the Doppler Effect. Close analysis of light being received from distant galaxies has enabled us to measure the speed of the galaxy by the change in frequency. Thus we know galaxies farthest from the earth are going away from us faster than galaxies closer to our Milky Way.
Basic Wave Behavior
Reflection
Laws of Reflection:
- The angle of incidence equals the angle of reflection: θi = θr
Note: the angle of incidence and the angle of reflection are measured from the NORMAL (line perpendicular to the reflecting surface where the ray hits the surface)
- The incident ray, reflected ray and normal are all in the same plane
Refraction
Refraction is the change in direction of a wave at the boundary of two substances due to a change in the wave’s speed. If the incident wave is perpendicular to the boundary, the angle of incidence is zero, and no refraction occurs. The ray continues straight and the angle of refraction is zero. The frequency remains the SAME in both media; the wavelength changes in the same proportion as the speed change.
The Index of Refraction (n) of a transparent substance indicates the speed change of light as compared to air. Thus:
where v is the speed in the transparent media
The Index of Refraction of air is 1.0, for water 1.33, for glass 1.5 – 1.6. It has no units. It describes a physical property of the material. When light speeds up, the ray bends away from the normal. When light slows down, the ray bends towards the normal.
Snell’s Law – the law of refraction
where the angles are measured from the NORMAL
Total Internal Reflection
When a light ray travels from a high index of refraction material to a low index refraction material (i.e. the light ray is speeding up and bending away from the normal), a critical angle (θc) is reached when the light is totally reflected. It happens when the refracted angle equals or tries to exceed 90º. Thus:
where n2 < n1
Diamond has a very high index of refraction, about 2.2. It internally reflects a lot of light. This gives diamond it high luster. Fiber optic materials make use of internal reflection to send light long distances and around corners.
Ray Diagrams for Image Location
Three rays drawn from any one point on an object may be used to locate the image formed by all rays coming from that point. These rays are:
- An incoming ray parallel to the principal axis will be reflected (or refracted) to the focal point.
- An incoming ray seeming to come from the focal point will be reflected (or refracted) parallel to the principal axis.
- For a mirror: any incoming ray coming form the center of curvature will be reflected straight back. For a lens: any incoming ray going through the center of a lens can be assumed to go straight through.
If the three reflected/refracted rays converge and meet, you have a real image. A real image can be projected onto a screen. A real image formed by just one mirror or one lens is always inverted. If the three reflected/refracted rays diverge, you have a virtual image located at the common origin of the diverging rays. A virtual image can NOT be projected onto a screen. A virtual image formed by just one mirror or one lens is always upright.
The Mirror/Lens Equation
An image may also be located algebraically by using the mirror/lens equation. This equation is derived using ray diagrams.
- so is the distance between the object and the mirror or lens. It is ALWAYS positive
- si is the distance between the image and the mirror or lens. It is positive for REAL images and negative for VIRTUAL images.
- f is the focal length of the mirror or lens. It is where incident parallel light rays would meet after being reflected or refracted. For converging mirrors (concave) and lenses (convex), f is ALWAYS positive. For diverging mirrors (convex) and lenses (concave), f is ALWAYS negative. For spherical mirrors ONLY, the radius of curvature = twice the focal length (R = 2f). For a lens, there is no correlation as the material AND the radius of the lens determines the focal length.
Magnification compares the size of the image to the size of the object.
- ho is the height of the object. It is ALWAYS positive.
- hI is the height of the image. It is positive for virtual images, as they are usually upright. It is negative for real images, as they are usually inverted.
CAUTION: as you use these equations remember when each variable is positive and when it is negative.
Dispersion
Glass has a different index of refraction depending upon the color of the light. The index of refraction is higher for violet light than it is for red light. This means that the speed change for violet light will be greater than for red light. As a result, violet light refracts more as it enters or leaves a glass prism. This spreads the white light into the spectral colors and is called dispersion.
Polarization
Certain materials take light, thought of as a transverse wave oscillating in all directions, and convert it to a transverse wave oscillating in one plane. If glass plates made of this polarizing material are placed on top of each other and the top one rotated 90º with respect to the bottom plate, the resulting transmitted light is blocked, and no light passes through.
Diffraction
Diffraction is when parallel waves become circular waves when going through an opening/slit that is the same size or smaller than the wavelength of the wave. The smaller the opening, the more complete the conversion to a circular wave. If the opening is very small compared to the wavelength, the resulting wave from diffraction is just like a circular wave originating at the opening. When sound (wavelength of about 1 m) passes through a door or window, it diffracts. Light whose wavelength is much smaller than the opening does not diffract at all. Thus, you can hear someone talking before you see the person through an open door or window.
Double and Multi-Slit Interference
Young sent parallel light through two small slits separated by a small distance, d. Since the slits were small compared to the wavelength of the light used, each slit diffracted the light and two circular waves of equal frequency and wavelength were produced. Where the two circular waves overlapped, by the law of superposition, you get interference. This was a famous experiment that established beyond dispute that light has wave properties.
Where the diffracted waves of light underwent constructive interference, bright lines or maxima (i.e. antinodes) appeared. They are numbered starting from the center (n = 0) and increase by 1 left or right. The number of the maximum corresponds to the difference in wavelength of the two interfering waves. For example, at the 3rd maximum, the two interfering waves must be 3 wavelengths out of phase with each other. As this is a whole number of wavelengths, constructive interference occurs!
Where the diffracted waves of light underwent destructive interference, dark regions or minima (i.e. nodes) appeared. They are numbered in relation to the central maxima. For a dark region to occur, the two interfering waves must be a multiple of ½ wavelengths out of phase with each other. At the first minima (n = 1), the waves are 1 – ½ or ½ wavelength out of phase. At the second minima (n = 2), the waves are 2 – ½ or 1 ½ wavelengths out of phase, and so on . . .
This led to the following important double slit equations:
where:
x is the distance between the observed maximum or minimum and the central maximum
L is the distance form the center of the slits to the central maximum (i.e. screen)
n is the number of the maximum or minimum being observed
d is the distance between the slits
NOTE: the AP exam only includes the first equation. It is really all you need if you think of n as the path difference (in terms of wavelength) that must exist between the two waves interfering at that point. For example, the first maxima would have a path difference of 1λ, the second maxima would be 2λ etc. For minima, the firs would have a path difference of ½ λ, the second 3/2 λ and so on.
The same equations work if there are more than two slits (ie. a multiple slit diffraction grating). For a multiple slit diffraction grating, the effects overlap each other. The resulting maxima and minima are more distinct (narrower) and brighter than the pattern formed by a double slit diffraction grating. As a result, measuring actual wavelengths of light is best done with a multiple slit diffraction grating.
From experiments using diffraction gratings, the wavelength of visible light has been measured. Visible light extends from violet (wavelength of about 400 nm) to red light (wavelength of about 750 nm). Each color is a range of wavelengths. It was not until lasers were invented (about 50 years ago) that monochromatic (one wavelength) light sources became easily available.
Sound waves from two identical sources can also interfere to form maxima and minima. Instead of seeing bright and dark areas of light, you would hear loud and soft/no areas of sound. The two/multi slit equations work for sound as well as light!
Single Slit Interference
The resulting interference pattern from a single slit came as a surprise and led to the theory that upon going through a small slit, many circular waves were produced. To predict where the maxima (antinodes) are located, use the equation that worked for minima (nodes) with double or multiple slits. To predict where minima (nodes) are located, use the equation that worked for maxima (antinodes) for double or multiple slits. That’s right, the equations are reversed!
A single slit interference pattern has lines – both dark (minima) and light (maxima). The bright lines are not as sharp or well defined as they are with double or multiple slit interference patterns. The lower intensity can be explained since light is only allowed to come through one slit!
Thin Film Interference
The colored bands formed when light reflects from a soap bubble or oil slick on another surface is caused by the interference of two light waves. For the soap bubble, light strikes the outside surface. Some of the light is transmitted into the soap bubble and some is reflected off the surface of the soap bubble. As the light that was reflected was trying to enter a more optically dense medium (i.e. the index of refraction increases), fixed end reflection occurred. This means that the reflected light was shifted ½ wavelength. Some of the light that was transmitted into the soap bubble gets reflected as it tries to leave the soap bubble. As the light that was trying to enter a less optically dense medium (i.e. the index of refraction decreases), free end reflection occurred and there was no shift in wavelength.
At just the right film thickness, constructive interference will occur. For example, if the soap thickness is equal to ¼ of a wavelength, the transmitted light travels an extra ¼ wavelength down and then ¼ wavelength back up (after it is reflected) for a total of ½ wavelength. It now meets the original reflected wave that has shifted ½ wavelength due to fixed end reflection. The two waves are ½ + ½ = 1 wavelength out of phase. Remember – whole number of wavelengths out of phase add up to constructive interference! A line of light matching the wavelength of the incident light (for example the wavelength of green light) would appear at that point. Other colors of white light have different wavelengths. These would require different thicknesses of soap bubble in order to have constructive interference. When you see more than one color on a soap bubble, it is because the soap bubble is NOT of uniform thickness and different wavelengths thus have the ability to constructively interfere.
You can also have destructive interference due to a thin film. A uniform thickness of transparent material can be added to glass to eliminate any reflection. Designing the thickness to produce destructive interference between light reflected at the front and back of the added coat accomplishes this task.
Practice Problems
2011b
2010b
2010
2009
2008b
2008
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