St. Louis Public Schools



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Wave Phenomena

Completely out of phase: (A,C) (B,D) (A,G)

The phase difference between

any points completely out of phase is

Phase: The phase of any particle is its position in its cycle of oscillation.

In phase: (A,E,I) (B,F) (D,H) (C,G)

The phase difference between any points in phase is

Principle of Linear Superposition: When two or more waves meet, the displacement of the resultant wave is the vector sum of the displacements of the component waves.

Superposition and Interference

Phase Difference:

Reason: Rope applies upward force on support (incident pulse). By Newtons 3rd law, support applies downward force on rope (reflected pulse).

Reflection at a Boundary between Two Media

Phase Difference:

II. Fixed End Reflection (Hard Reflection)

I. Free End Reflection (Soft Reflection)

Component waves

Pulses with unequal amplitudes

Pulses with equal amplitudes

Destructive Interference: superposition of two or more pulses or waves out of phase

Pulses with unequal amplitudes

Pulses with equal amplitudes

How is a standing wave on a string formed?

A traveling wave moving in one direction in the string is reflected off the end of the string. This sends a reflected wave traveling in the opposite direction in the string which is nearly identical with the first traveling wave. It has the same frequency, the same wavelength, and almost the same amplitude. These two traveling waves moving in opposite directions in the string are the component waves. These component waves interfere with each other creating the standing wave whose amplitude at any point is the superposition of the components’ amplitudes. This standing wave is the resultant wave of the two component traveling waves.

Constructive Interference: superposition of two or more pulses or waves in phase

Below is a time-elapsed picture of one possible standing wave that could be made on a string.

Anti-Node:

Standing (stationary) wave - resultant wave formed when two waves of equal amplitude and frequency traveling in opposite directions in the same medium interfere

Standing Waves (Stationary Waves)

Node:

| |Standing (Stationary) Wave |Traveling Wave |

|Energy |Energy is not transferred by the wave, but it does have energy stored in |Energy is transferred by the wave. |

| |it. | |

|Amplitude |All points on the wave have different amplitude so we say it has variable |All points on the wave have the same amplitude so we say it has |

| |amplitude. The maximum amplitude is 2A at the antinodes and 0 at the |fixed amplitude (provided energy is not dissipated.) |

| |nodes. | |

|Wavelength |Same as the wavelength of the component waves. |Equal to the shortest distance along the wave between any two |

| |Equal to twice the distance between any two consecutive nodes (or |points that are in phase. |

| |antinodes). | |

|Frequency |All particles oscillate in SHM with the same frequency as the component |All particles oscillate in SHM with the same frequency. |

| |waves. | |

|Phase and phase |All points in one section between two consecutive nodes (or antinodes) are|All points within one wavelength have a different phase. |

|difference |moving in phase. All points in the next section are 1800 out of phase |Thus, all phase differences are possible. |

| |with all points in this section. | |

| |Thus, the only possible phase differences are 00 and 1800. | |

3rd Harmonic

2nd Harmonic

Transverse Standing Wave: String fixed at both ends

3rd Harmonic

Longitudinal Standing Waves: Pipe open at both ends

v =

L =

λ2 =

f2 =

v =

L =

λ1 =

f1 =

1st Harmonic (fundamental)

Boundary conditions:

Fundamental wavelength

and frequency

(1st harmonic):

Other resonant frequencies

of vibration (harmonics)

(resonant modes):

v =

L =

λ3 =

f3 =

v =

L =

λ1 =

f1 =

1st Harmonic (fundamental)

2nd Harmonic

v =

L =

λ3 =

f3 =

v =

L =

λ2 =

f2 =

Boundary conditions:

Fundamental wavelength

and frequency (1st harmonic):

Other resonant frequencies

of vibration (harmonics)

(Resonant modes):

How is this standing wave formed?

Vibrations of air at one end produce a traveling (longitudinal) wave that reflects off the closed end of the pipe which causes a second traveling wave in the opposite direction. These two component waves interfere to produce a standing wave if they have a frequency which is one of the resonant frequencies of the pipe.

3rd Harmonic

1st Harmonic (fundamental)

5th Harmonic

v =

L =

λ5 =

f5 =

v =

L =

λ1 =

f1 =

v =

L =

λ3 =

f3 =

Boundary conditions:

Fundamental wavelength

and frequency (1st harmonic):

Other resonant frequencies

of vibration (harmonics)

(Resonant modes):

Other substances:

Speed: all EM waves travel at 3.00 x 108 m/s in a vacuum

Formula:

Sunlight is unpolarized. When sunlight is incident on an object, part of the light will be absorbed by the object (or refracted inside the object) while other parts of the light will be reflected by the object. This reflected light is partially to completely polarized, depending on the angle the light reaches the object. The polarization direction will be in the same direction as the surface of the object, that is, the plane of polarization for the reflected light is parallel to the surface of the object. Sunlight striking the surface of water or a road are examples where reflected light will be polarized. This is often called glare.

Plane of Polarization:

2. Stress analysis: Some materials are optically active under stress and allow different colors to pass through at different angles. Engineers can build models out of plastic and subject them to stress. Then when they are placed between a polarizer and an analyzer and viewed, points of probable mechanical failure due to high stress can be determined.

1. Determining the concentration of solutions: Sugar solutions, such as glucose, are optically active. The angle by which polarized light is rotated when passing through the solution is related to the concentration of the solution. Therefore, if a container with a sugar solution is placed between a polarizer and an analyzer and the analyzer is rotated until the intensity of the light passing through it is maximum then from measuring the angle of rotation the concentration of the solution can be calculated.

Insert a third polarizer between the original polarizer and the analyzer.

Some component of light from the first will make it through the second and some component of the second will make it through the third.

The intermediate polarizer “rotates the plane of polarization” at the cost of lost intensity.

How can light be transmitted through “crossed polarizers?”

Optically Active Substance –

1)

2)

Natural, unpolarized light of intensity 6.0 W m-2 is incident on two polaroids oriented at 600 to each other. Find the intensity of the light transmitted through each of them.

The diagrams below represent the wavefronts produced by two point sources. The solid circles represent wave crests and the dashed circles represent wave troughs.

Formula:

Malus’ Law – The intensity of transmitted polarized light is equal to the product of the incident intensity and the square of the cosine of the angle between the transmission axes of the polarizer and the analyzer.

What happens when the analyzer is neither parallel nor perpendicular to the polarizer?

When the transmission axis of the analyzer is perpendicular to that of the polarizer . . .

When the transmission axis of the analyzer is parallel to that of the polarizer . . .

Analyzer –

How do polarized sunglasses reduce glare?

A polarizer allows the parallel component of any wave to pass through and blocks the perpendicular component of any wave.

Why?

NOTE:

A simple model of a polarizer using a wave on a rope

Transmission axis of polarizer is perpendicular to the plane of polarization of the wave.

Transmission axis of polarizer is parallel to the plane of polarization of the wave.

Transmission axis –

Polarizer –

Example: What is the Brewster angle for sunlight reflected off a lake?

What is glare?

If unpolarized light is incident on a polarizer with intensity I0, what is the intensity of the transmitted polarized light?

Brewster’s Law – When light is incident on a surface at such an angle that the reflected and transmitted rays are perpendicular, the reflected ray is completely plane polarized. Then the index of refraction of the substance is equal to the tangent of the angle of incidence.

I0 =

Polarization by reflection

A more sophisticated model of a polarizer using light

Unpolarized Light –

Polarized Light –

Production of EM waves: oscillating electric charge produces varying electric and magnetic fields

Transverse: vibration is perpendicular to direction of motion of energy

Polarization of Light

Through an Opening

Around a Barrier

Diffraction: the bending or spreading of a wave when it passes through a small opening (aperture) or around a barrier

Diffraction

Condition for noticeable diffraction:

3. The two speakers now emit a frequency of 300 Hz and are placed in a line. Location A is the first place along the line joining them where you can stand and hear no sound. How far apart are the speakers?

The larger the aperture size (slit width) compared to the wavelength of the wave, the less the wave diffracts.

Path Difference (Δℓ) – difference in the distances two waves must travel from their sources to a given point

Circular Aperture

Interference of Waves in Two Dimensions

Conditions for Nodal Line

Type of interference:

Phase difference:

Path difference:

Conditions for Anti-nodal Line

Type of interference:

Phase difference:

Path difference:

1. A diffraction pattern forms when light of wavelength 675 nm passes through a single slit. Determine the angle that locates the first dark fringe if the width of the slit is 18 μm.

2. The same set-up as above is used but now the frequency of sound emitted by both speakers is increased to 700 Hz. This time, as you walk along the side of the square from A toward an empty corner, you hear the loud sound at A alternately diminish to no sound and then increase to a maximum again. By the time you arrive at the corner, you have noticed the sound disappear three times. Use this information to estimate a speed for sound.

1. A square is 3.5 m on a 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 and as you walk along the side of the square toward either empty corner, the loudness diminishes gradually but does not entirely disappear until you reach either empty corner, where you hear no sound at all. Find the wavelength of the sound waves.

Small Angle Approximation:

The larger the barrier size compared to the wavelength of the wave, the less the wave diffracts.

Single Slit Diffraction and Interference

Why is there an interference pattern when light travels through a single slit?

Position of first minimum

4. Calculate:

Wide slit

Narrow slit

The intensity pattern for single slit diffraction

Features of the Single Slit Intensity Pattern

a)

b)

c)

3. Draw small right triangle perpendicular to rays. Wavelet pairs interfere destructively if the path difference is ½ λ.

2. Match points in pairs that are ½ slit width apart:

1 and 3, 2 and 4 (ignore 5)

1. Break wavefront into separate point sources. Assume screen is very far from the slit so that the rays are parallel when they interfere destructively at the first dark fringe (first minimum).

Derivation of Single Slit Formula

Half-width of central maximum:

2. Light passes through a single slit and shines on a flat screen that is located 0.40 m away. The width of the slit is 4.0×10-6 m. Determine the width of the central bright fringe when the wavelength of the light in a vacuum is λ = 690 nm (red).

Resolution

The ability to resolve two sources of light depends on . . .

Resolution:

Intensity Patterns

Resolution of two sources through a single slit

Resolution of two sources through a circular aperture

Examples:

Rayleigh Criterion:

The minimum angle between sources for them to be “just resolved” =

Well resolved

Not resolved

Just resolved

Single Slit

Circular Aperture

Conclusion:

Assumption:

1. The brightest star in the winter sky in the Northern Hemisphere is Sirius. In reality, Sirius is a system of two stars that orbit each other with an average separation of 3 x 1012 m. The Hubble Space Telescope (diameter 2.4 m) is pointed at the Sirius system which is 8 x 1016 meters from Earth. Can the Hubble Space Telescope resolve the two stars or does it see them as a single point of light?

Significance of Resolution

Due to diffraction effects, all devices have a limit on their ability to perceive and to resolve between sources of light. For instance, our eyes can never see atoms since atoms are smaller than the wavelength of visible light so light waves will just diffract around them. Here are some cases where diffraction and resolution are important.

1. CDs and DVDs

CDs and DVDs store digital information as “bumps” and “pits” etched into a plastic surface. Music CDs have data tracks approximately 5 x 10-7 m wide with the bumps and pits just over 1 x 10-7 m high. The bumps and pits on a DVD are much smaller so that more data can be stored. The data is read by reflecting a laser beam off the surface. The wavelength of laser light used to read the data and the size of the aperture of the lens used to receive the laser light places a limit on how close together the bumps and pits can be placed, that is, places a limit on the resolution of the data.

3. Radio Telescopes

Astronomers often wish to detect the radio waves emitted by very distant objects like quasars and galaxies. However, since the wavelength of radio waves is much larger than visible light, the ability of a radio telescope to resolve sources is more limited than that of light telescopes. To get around this limitation, astronomers use two or more radio telescopes separated by a large distance, called a Very Large Array (VLA). For instance, in New Mexico, there is a VLA consisting of 27 parabolic dishes each of diameter 25 m arranged in a Y-shape that covers an area of 570 km2.

2. Electron Microscopes

In order to resolve objects beyond the limits imposed by the wavelength of visible light, the wave properties of electrons are used in electron microscopes. The de Broglie wavelength of an electron is much smaller than the wavelength of a photon of visible light so a microscope using an electron beam can resolve objects that are much smaller than those of a light microscope.

EXAMPLE: The Galaxy Cygnus A can be resolved optically as an elliptically shaped galaxy. However, it is also a strong emitter of radio waves of wavelength 0.15 m. The Galaxy is estimated to be 5.0 x 1024 m from Earth. Use of a radio telescope shows that the radio emission is from two sources separated by a distance of 3.0 x 1021 m. Estimate the diameter of the dish required to just resolve the sources.

Angular width:

Linear width:

Angular half-width:

Linear half-width:

Central maximum:

What is the path difference for the spots you drew?

2. Mark all spots on the diagram where the waves overlap and the path lengths differ by a full wavelength.

Resultant Interference Patterns

What is the path difference for the spots you drew?

Single Slit

What is the path difference for the spots you drew?

1. Mark all spots on the diagram that are equidistant from A and B where the waves overlap.

Where:

I =

I0 =

θ =

What is the direction of polarization of glare?

[pic]

3. Mark all spots on the diagram where the waves overlap and the path lengths differ by one-half wavelength.

Where:

Note: Displacement nodes =

Longitudinal Standing Waves: Pipe closed at one end

Note: Displacement nodes =

How is this standing wave formed?

Vibrations of air at one end produce a traveling (longitudinal) wave that reflects off the open end of the pipe which causes a second traveling wave in the opposite direction. These two component waves interfere to produce a standing wave if they have a frequency which is one of the resonant frequencies of the pipe.

Notice that a stable standing wave will only occur at certain discrete frequencies or certain discrete lengths of the string. It is only at these frequencies or lengths that the wave resonates so they are called resonant frequencies or harmonics or resonant modes of vibration of the string. We say that these resonant frequencies are quantized.

This graphic shows what the standing wave looks like after each fraction of a period. Notice that you never see the two component traveling waves but only see the resultant standing wave.

Resultant wave

Component waves

Waves with equal amplitudes and wavelengths

Resultant wave

Waves with equal amplitudes and wavelengths

3. Liquid crystal displays (LCD): Calculators, watches, computer screens and televisions have displays that are made up of thousands of small dots called pixels. In an LCD, each pixel is made of a tiny liquid crystal. Liquid crystals have a very useful property; normally they rotate the plane of polarization through 900, but when a voltage is applied across them, they don’t. So if a liquid crystal is placed between two crossed polarizers the crystal goes dark when the voltage is applied.

Back-lit display: The light source is behind the liquid crystal, as in televisions and computer screens.

Front-lit display: Uses light from the room to illuminate the display by placing a mirror behind the analyzer to reflect the room light if the voltage is off, as in calculators and watches.

Applications of polarization

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