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11.1 Standing (stationary) waves
11.1.1 Describe the nature of standing (stationary) waves.
Standing or stationary waves are the result of the superposition of
two or more waves travelling in opposite directions. With the
same speed, frequency and amplitude.
There are a few main points to note:




At places where there is no motion we see a node.
At places where maximum motion we see an antinode.
Standing wave patterns can only be established at certain
frequencies of vibration.
The point of reflection must always be a node.
Question:
For sound waves of frequency 2800Hz, it is found that two nodes are separated by
23.0cm, with three antinodes between them. What is the wavelength of these sound
waves? What is the speed of sound in air?
11.1.2 Explain the formation of one-dimensional standing waves.
Standing waves don't form under just any circumstances. They require energy be put into a
system at an appropriate frequency. That is, when the driving frequency applied to a system
equals its natural frequency. This condition is known as resonance. Standing waves are always
associated with resonance. Resonance can be identified by a dramatic increase in amplitude of
the resultant vibrations.
11.1.3 Discuss the modes of vibration of strings and air in open and in closed pipes.
We can have different patterns of standing waves, the simplest is called the fundamental.
We can also have progressively more complicated waveforms as the frequency of
vibration of the wave increases, such as the second and third harmonic.
.
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For a standing wave in a pipe with a closed end, the closed end must be a node, since it
can't move. This means that harmonics are set up for frequencies of waves for which the
end of the pipe is a node. So as to hear the maximum effect the open end of a pipe must
be an antinode for a harmonic to be established. The opposite condition is true for open
ended pipes, both ends must be antinodes.
11.1.4 Compare standing waves and travelling waves.
A mechanical wave is a disturbance that is created by a vibrating object and subsequently
travels through a medium from one location to another, transporting energy as it moves. The
mechanism by which a mechanical wave propagates itself through a medium involves particle
interaction; one particle applies a push or pull on its adjacent
neighbor, causing a displacement of that neighbor from the
equilibrium or rest position. As a wave is observed traveling through a
medium, a crest is seen moving along from particle to particle. This
crest is followed by a trough that is in turn followed by the next crest.
In fact, one would observe a distinct wave pattern (in the form of a
sine wave) traveling through the medium. This sine wave pattern
continues to move in uninterrupted fashion until it encounters another
wave along the medium or until it encounters a boundary with another medium. This type of
wave pattern that is seen traveling through a medium is sometimes referred to as a traveling
wave.
It is however possible to have wave confined to a given space in a medium; and still produce a
regular wave pattern that is readily discernible amidst the motion of the medium. For instance,
if an elastic rope is held end-to-end and vibrated at just the right frequency, a wave pattern
would be produced that assumes the shape of a sine wave and is seen to change over time. The
wave pattern is only produced when one end of the rope is vibrated at just the right frequency.
When the proper frequency is used, the interference of the incident wave and the reflected
wave occur in such a manner that there are specific points along the medium that appear to be
standing still. Because the observed wave pattern is characterized by points that appear to be
standing still, the pattern is often called a standing wave pattern. There are other points along
the medium whose displacement changes over time, but in a regular manner. These points
vibrate back and forth from a positive displacement to a negative displacement; the vibrations
occur at regular time intervals such that the motion of the medium is regular and repeating. A
pattern is readily observable
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11.1.5 Solve problems involving standing waves.
Determine the length of guitar string required to produce a fundamental frequency (1st
harmonic) of 256 Hz. The speed of waves in a particular guitar string is known to be 405 m/s.
Given:
v = 405 m/s
f1 = 256 Hz
L=?
Finding the length of the string is from knowledge of the wavelength. But the wavelength is not
known. However, the frequency and speed are given, so we can use the wave equation (speed
= frequency x wavelength) and knowledge of the speed and frequency to determine the
wavelength.
Speed = frequency • wavelength
Wavelength = speed / frequency
Wavelength = (405 m/s) / (256 Hz)
Wavelength = 1.58 m
Length = (1/2) x Wavelength
Length = 0.791 m
11.2 Doppler Effect
11.2.1 Describe what is meant by the Doppler Effect.
The Doppler effect (or Doppler shift), named after Austrian physicist Christian Doppler
who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer
moving relative to the source of the wave. It is commonly heard when a vehicle
sounding a siren or horn approaches, passes, and recedes from an observer. The received
frequency is higher (compared to the emitted frequency) during the approach, it is
identical at the instant of passing by, and it is lower during the recession.
The relative increase in frequency can be explained as follows. When the source of the
waves is moving toward the observer, each successive wave crest is emitted from a
position closer to the observer than the previous wave. Therefore each wave takes
slightly less time to reach the observer than the previous wave. Therefore the time
between the arrivals of successive wave crests at the observer is reduced, causing an
increase in the frequency. While they are traveling, the distance between successive
wave fronts is reduced; so the waves "bunch together". Conversely, if the source of
waves is moving away from the observer, each wave is emitted from a position farther
from the observer than the previous wave, so the arrival time between successive waves
is increased, reducing the frequency. The distance between successive wave fronts is
increased, so the waves "spread out".
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11.2.2 Explain the Doppler Effect by reference to wave front diagrams for moving-detector and movingsource situations.
First we should look at the case of a moving source and stationary observer.
The wavelength due to a stationary source is:
Where v is the velocity of sound in the medium and fs is the frequency of the sound. If
the source is moving to the right at a speed of vs, then the distance between the peaks
(the wavelength) is shortened and can be described by:
Now the frequency measured by the observer is:
We get the formula given:
The plus/minus has been added to compensate for the direct of the source. The sign
should be negative if the source is approaching the observer and positive if the source is
moving away from the observer.
Now for a stationary source and a moving observer:
You may ask why would it be different if the
observer or the source moves? After all motion is
relative, and it is, but the speed of sound is fixed
relative to the medium (air) that it is traveling in,
this causes differences.
In the case of the moving observer the wavelength of the sound does not change, but the
frequency as measured by the observer does change. This happens because the observer
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encounters a wave front more frequently… The frequency as measured by the observer
is:
Where v0 is the velocity of the observer. The wavelength is speed of sound divided by
the frequency, we can then rewrite the equation as:
The plus or minus is added to compensate for the direction of the observer. The sign
should be negative if the observer is approaching the source and positive if the observer
is moving away from the source.
11.2.3 Apply the Doppler Effect equations for sound.
The pitch of the sound you hear from a moving source will be either higher or lower than the
emitted frequency, depending on the direction the source is moving. This is called the Doppler
effect. Knowing the initial frequency, the velocity of the source and the speed of sound,
equations are available that allow you to calculate the new frequency. The angle between the
source and the line-of-sight adds another factor to the equations.
General expression for apparent frequency n' =
Here n = Actual frequency; vL = Velocity of listener; vS =
Velocity of source
vm = Velocity of medium and v = Velocity of sound wave
Sign convention : All velocities along the direction S to L are taken as positive and all velocities
along the direction L to S are taken as negative. If the medium is stationary vm= 0 then n' =
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11.2.4 Solve problems on the Doppler Effect for sound.
If a vehicle is coming toward you at 96 km/hr (60 miles per hour) and sounds its horn
that blares at 8000 Hz, what is the frequency of the sound you hear when the speed of
sound is 340 m/s (1115 ft/s)?
Convert kilometers per hour to meters per second:
96 km/hr = 96000 m/hr
Since 1 hour = 3600 seconds,
96000 m/hr = 96000/3600 = 26.7 m/s
Calculate the frequency:
fo = fv/(v − vt) = 8000*340/(340 − 26.7)
fo = 8682 Hz
In other words, the frequency you hear is about 682 Hz higher than the actual sound of
the horn.
11.2.5 Solve problems on the Doppler Effect for electromagnetic waves
Suppose a source of radiation moves at constant speed. Over the course of one full period of
oscillation an interval of time will have elapsed and an observer will see on full wavelength of
the light emitted. The start of the wave will have travelled a distance
while the source will
have moved
during that period so the wavelength an observer sees will be
Classically,
is just the period of oscillation or
so
which gives you the basic idea behind the Doppler shift. However, Einstein tells us that since
the source and observer are moving relative to each other, their measures of time are different
so we have to adjust for that and replace
with
from which
Note that this is 1-dimensional and a simple geometric adjustment is necessary if the source is
approaching or receding at an angle with respect to the observer.
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11.2.6 Outline an example in which the Doppler Effect is used to measure speed.
The Doppler effect is used in some types of radar, to measure the velocity of detected objects.
A radar beam is fired at a moving target — e.g. a motor car, as police use radar to detect
speeding motorists — as it approaches or recedes from the radar source. Each successive radar
wave has to travel farther to reach the car, before being reflected and re-detected near the
source. As each wave has to move farther, the gap between each wave increases, increasing
the wavelength. In some situations, the radar beam is fired at the moving car as it approaches,
in which case each successive wave travels a lesser distance, decreasing the wavelength. In
either situation, calculations from the Doppler Effect accurately determine the car's velocity.
Moreover, the proximity fuse, developed during World War II, relies upon Doppler radar to
explode at the correct time, height, distance, etc.
11.3 Diffraction
Diffraction at a single slit
11.3.1 Sketch the variation with angle of diffraction of the relative intensity of light diffracted at a single slit.
The path difference is given by
angle θmin given by
so that the minimum intensity occurs at an
Where d is the width of the slit.
A similar argument can be used to show that if we imagine the slit to be divided into
four, six, eight parts, etc., minima are obtained at angles θn given by
where n is an integer other than
zero.
There is no such simple argument
to enable us to find the maxima of
the diffraction pattern. The
intensity profile can be calculated
using the Fresnel diffraction
integral as
where the sin function is given by
sin(x) = sin(πx)/(πx) if x ≠ 0, and
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sin(0) = 1.
This analysis applies only to the far field, that is, at a distance much larger than the
width of the slit.
11.3.2 Derive the formula for the position of the first minimum of the diffraction pattern produced at a single
slit.
To explain the positions of the diffraction patterns, we must consider all the other waves
going through the gap. Going straight to the halfway point in the gap, a wave from here
and the wave from the top edge must be out of phase by half a wavelength when the
meet at the first order causing deconstructive interference, as will a wave from just
below the midpoint and one just below the upper edge. This argument can be continued
until you see that the wave from above the midpoint cancel out those from below it.
Therefore to determine the angular position of the first minima in a single slit diffraction
pattern (given the small angle approximation of q = sinq )
11.3.3 Solve problems involving single-slit diffraction.
Find the angular width of the central bright maximum in the Fraunhofer pattern of a slit
of width 12 X 10~5 cm when the slit is illuminated by monochromatic light of wavelength
6000A.
Ans. Width of the slit = a = 12 x 10"5 cm = 12 x 10" 7 m
Wavelength of light = A, = 6000 A = 6000 x 10"10 m
Angular width of the central maximum - 20 = ?
. X 6QOOxlOr10 „<,
sin 0 = = 0.50 = 30 , 20 = 6IT.
a 12 X 10
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11.4 Resolution
11.4.1 Sketch the variation with angle of diffraction of the relative intensity of light emitted by two point
sources that has been diffracted at a single slit.
Through experiment of Young on
Interference it can be stated that a
narrow single slit acts as a new source
of light and can spread out. Newton also
had noticed that light spreads out from
a narrow hole or slit. The light ray bends
near the edges of obstacles. It seems to
turn around the corner and enter
regions where we would expect a
shadow. This effect is called Diffraction.
Now let us discuss on diffraction
intensity.
11.4.2 State the Rayleigh criterion for images of two sources to be just resolved.
The interplay between diffraction and aberration can be characterized by the point
spread function (PSF). The narrower the aperture of a lens the more likely the PSF is
dominated by diffraction. In that case, the angular resolution of an optical system can be
estimated (from the diameter of the aperture and the wavelength of the light) by the
Rayleigh criterion invented by Lord Rayleigh:
Two point sources are regarded as just resolved when the principal diffraction maximum
of one image coincides with the first minimum of the other. If the distance is greater, the
two points are well resolved and if it is smaller; they are regarded as not resolved. If one
considers diffraction through a circular aperture, this translates into:
where
θ is the angular resolution in radians,
λ is the wavelength of light,
and D is the diameter of the lens' aperture.
11.4.3 Describe the significance of resolution in the development of devices such as CDs and DVDs, the
electron microscope and radio telescopes.
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An electron microscope is a type of microscope that uses a particle beam of electrons to
illuminate the specimen and produce a magnified image. Electron microscopes (EM)
have a greater resolving power than a light-powered optical microscope, because
electrons have wavelengths about 100,000 times shorter than visible light (photons), and
can achieve better than 0.2 nm resolution and magnifications of up to 2,000,000x,
whereas ordinary, non-confocal light microscopes are limited by diffraction to about
200 nm resolution and useful magnifications below 2000x.
The electron microscope uses electrostatic and electromagnetic "lenses" to control the
electron beam and focus it to form an image. These lenses are analogous to, but different
from the glass lenses of an optical microscope that forms a magnified image by focusing
light on or through the specimen. In transmission, the electron beam is first diffracted by
the specimen, and then, the electron microscope “lenses" re-focus the beam into a
Fourier-transformed image of the diffraction pattern for the selected area of
investigation. The real image thus formed is magnified by a factor ranging from a few
hundred to many hundred thousand times, and can be viewed on a detecting screen or
recorded using photographic film or plates or with a digital camera. Electron
microscopes are used to observe a wide range of biological and inorganic specimens
including microorganisms, cells, large molecules, biopsy samples, metals, and crystals.
Industrially, the electron microscope is primarily used for quality control and failure
analysis in semiconductor device fabrication.
11.4.4 Solve problems involving resolution.
An American standard television picture is composed of about 485 horizontal lines of varying
light intensity. Assume that your ability to resolve the lines is limited only by the Rayleigh
criterion and that the pupils of your eyes are 5.13 mm in diameter. Calculate the ratio of
minimum viewing distance to the vertical dimension of the picture such that you will not be
able to resolve the lines. Assume that the average wavelength of the light coming from the
screen is 570 nm
diameter of pupils = d = 5.13*10^-3 m
wavelength = r = 570 * 10^-9 m
(theta) = 1.22 * (wavelength)/(Diameter)
s = (radius)*(theta)
theta = (D/485)/d = 1.22(wavelength/diameter)
11.5 Polarization
11.5.1 Describe what is meant by polarized light.
By convention, the polarization of light is described by specifying the orientation of the wave's
electric field at a point in space over one period of the oscillation. When light travels in free
space, in most cases it propagates as a transverse wave—the polarization is perpendicular to
the wave's direction of travel. In this case, the electric field may be oriented in a single
direction (linear polarization), or it may rotate as the wave travels (circular or elliptical
polarization). In the latter cases, the oscillations can rotate either towards the right or towards
the left in the direction of travel. Depending on which rotation is present in a given wave it is
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called the wave's chirality or handedness. In general the polarization of an electromagnetic
(EM) wave is a complex issue. For instance in a waveguide such as an optical fiber, or for
radically polarized beams in free space, the description of the wave's polarization is more
complicated, as the fields can have longitudinal as well as transverse components. Such EM
waves are either TM or hybrid modes.
11.5.2 Describe polarization by reflection.
When light reflects at an angle from an interface between two transparent materials, the
reflectivity is different for light polarized in the plane of incidence and light polarized
perpendicular to it. Light polarized in the plane is said to be p-polarized, while that
polarized perpendicular to it is s-polarized. At a special angle known as Brewster's
angle, no p-polarized light is reflected from the surface, thus all reflected light must be spolarized, with an electric field perpendicular to the plane of incidence.
A simple linear polarizer can be made by tilting a stack of glass plates at Brewster's
angle to the beam. Some of the s-polarized
light is reflected from each surface of each
plate. For a stack of plates, each reflection
depletes the incident beam of s-polarized
light, leaving a greater fraction of ppolarized light in the transmitted beam at
each stage. For visible light in air and
typical glass, Brewster's angle is about 57°,
and about 16% of the s-polarized light
present in the beam is reflected for each airto-glass or glass-to-air transition. It takes many plates to achieve even mediocre
polarization of the transmitted beam with this approach. For a stack of 10 plates (20
reflections), about 3% (= (1-0.16)20) of the s-polarized light is transmitted. The reflected
beam, while fully polarized, is spread out and may not be very useful.
11.5.3 State and apply Brewster’s law.
When light encounters a boundary between two media with different refractive indices,
some of it is usually reflected as shown in the figure above. The fraction that is reflected
is described by the Fresnel equations, and is dependent upon the incoming light's
polarization and angle of incidence.
The Fresnel equations predict that light with the p polarization (electric field polarized in
the same plane as the incident ray and the surface normal) will not be reflected if the
angle of incidence is
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where n1 and n2 are the refractive indices of the two media. This equation is known as
Brewster's law, and the angle defined by it is Brewster's angle.
11.5.4 Explain the terms polarizer and analyzer.
If two polarizers are placed one after another (the second polarizer is generally called an
analyzer), the mutual angle between their polarizing axes gives the value of θ in Malus'
law. If the two axes are orthogonal, the polarizers are crossed and in theory no light is
transmitted, though again practically speaking no polarizer is perfect and the
transmission is not exactly zero (for example, crossed Polaroid sheets appear slightly
blue in color). If a transparent object is placed between the crossed polarizers, any
polarization effects present in the sample (such as birefringence) will be shown as an
increase in transmission. This effect is used in polarimetry to measure the optical
activity of a sample.
Real polarizers are also not perfect blockers of the polarization orthogonal to their
polarization axis; the ratio of the transmission of the unwanted component to the wanted
component is called the extinction ratio, and varies from around 1:500 for Polaroid to
about 1:106 for Glan–Taylor prism polarizers.
11.5.5 Calculate the intensity of a transmitted beam of polarized light using Malus’ law.
A beam of polarized light has an average intensity of 15w/m2and is sent through a
polarizer. The transmission axis makes an angle of 25 degrees with respect tote direction
of polarization. What is the rms value of the electric field of the transmitted beam?
Average intensity of light leaving the analyzer=average intensity of polarized light
falling on analyzer x cos squared of the angle between the transmission axes of the
polarizer and analyzer.
From the theory we have the relation for Erms and average intensity as
S = c εoErms2
According to Malus law
S = Socos2θ
Where θ is the angle between the transmission axis and the direction
11.5.6 Describe what is meant by an optically active substance.
Optically active additive (OAA) is an
organic or inorganic material which,
when added to a coating, makes that
coating react to ultra violet light. This
effect enables quick, non-invasive
inspection of very large coated areas
during the application process allowing
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the coating inspector to identify and concentrate on defective areas, thus reducing inspection
time while assuring the probability of good application and coverage. It works by highlighting
holidays and pin-holes, areas of over and under application as well as giving the opportunity for
crack detection and identification of early coating deterioration through life.
11.5.7 Describe the use of polarization in the determination of the concentration of certain solutions.
Concentration polarization is the polarization component that is caused by concentration
changes in the environment adjacent to the surface as illustrated in the following Figure. When a
chemical species participating in a corrosion process is in short supply, the mass transport of that
species to the corroding surface can become rate controlling. A frequent case of concentration
polarization occurs when the cathodic processes depend on the reduction of dissolved oxygen
since it is usually in low concentration, i.e. in parts per million (ppm). The following Tables contain
respectively data related to the solubility of oxygen in air saturated water at different
temperatures and data on the solubility of oxygen in seawater of different salinity, chlorinity, and
temperatures. In both Tables, the level of dissolved oxygen is seen to increase as the temperature
decreases.
11.5.8 Outline qualitatively how polarization may be used in stress analysis.
The method is based on the property of birefringence, which is exhibited by certain
transparent materials. Birefringence is a property by virtue of which a ray of light
passing through a birefringent material experiences two refractive indices. The property
of birefringence or double refraction is exhibited by many optical crystals. But photo
elastic materials exhibit the property of birefringence only on the application of stress
and the magnitude of the refractive indices at each point in the material is directly
related to the state of stress at that point. Thus, the first task is to develop a model made
out of such materials. The model has a similar geometry to that of the structure on which
stress analysis is to be performed. This ensures that the state of the stress in the model is
similar to the state of the stress in the structure.
When a ray of plane polarized light is passed through a photo elastic material, it gets
resolved along the two principal stress directions and each of these components
experiences different refractive indices. The difference in the refractive indices leads to a
relative phase retardation between the two component waves. The magnitude of the
relative retardation is given by the stress optic law:
Where R is the induced retardation, C is the stress optic coefficient, t is the specimen
thickness, σ11 is the first principal stress, and σ22 is the second principal stress.
The two waves are then brought together in a polar scope. The phenomenon of optical
interference takes place and we get a fringe pattern, which depends on relative
retardation. Thus studying the fringe pattern one can determine the state of stress at
various points in the material.
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11.5.9 Outline qualitatively the action of liquid-crystal displays (LCDs).
A liquid crystal display (LCD) is a thin, flat electronic visual display that uses the light
modulating properties of liquid crystals (LCs). LCs do not emit light directly.
They are used in a wide range of applications, including computer monitors, television,
instrument panels, aircraft cockpit displays, signage, etc. They are common in consumer
devices such as video players, gaming devices, clocks, watches, calculators, and
telephones. LCDs have displaced cathode ray tube (CRT) displays in most applications.
They are usually more compact, lightweight, portable, less expensive, more reliable, and
easier on the eyes. They are available in a wider range of screen sizes than CRT and
plasma displays, and since they do not use phosphors, they cannot suffer image burn-in.
LCDs are more energy efficient and offer safer disposal than CRTs. Its low electrical
power consumption enables it to be used in battery-powered electronic equipment. It is
an electronically-modulated optical device made up of any number of pixels filled with
liquid crystals and arrayed in front of a light source (backlight) or reflector to produce
images in color or monochrome. The earliest discovery leading to the development of
LCD technology, the discovery of liquid crystals, dates from 1888.By 2008, and
worldwide sales of televisions with LCD screens had surpassed the sale of CRT units.
11.5.10 Solve problems involving the polarization of light.
A half wave plate and a quarter wave plate are placed between polarizer P1 and an analyzer P2.
all these are parallel to each other and perpendicular to the direction of incident unpolarized
light. Optic axis of half wave plate makes angle 30 degrees with pass axis of polarizer and the
optic axis of quarter wave plate is parallel to pass axis of polarizer P1.
determine state of polarization of light after passing through 1. Half wave plate 2. Quarter
wave
After half wave plate :
I2 = (Io/2)cos^2(30)
= (Io/2 )(3/4)
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the light intensity will equal to = 3Io/8
After quarter wave plate:
I3 = ( I2)cos^2(-30)
I2 = 3Io/8
I3 = (3Io/8)*(3/4) = (9Io/32)