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POLARIZATION OF LIGHT WAVES
Light is an electromagnetic wave whose electric and magnetic vibrations are perpendicular to each other and to the direction of propagation.
Polarization of light is the process of confining the vibrations of the electric vector of light waves to one direction.
- In unpolarized light the electric field vibrates in all directions perpendicular to the direction of a wave .
- The commonly used pictorial representation of an unpolarized light wave is as shown below
Unpolarized light beam is equivalent to two equally intense beams whose planes of vibration are perpendicular to each other.
After reflection or transmission through certain substances the electric field is confined to the direction and the radiation is said to be plane – polarized light.
POLAROID
This is a device used to produce plane polarized light.
In a polarizer, there is characteristic direction called transmission axis which is indicated by the dotted line.
If a polarizer is placed in front of unpolarized light source, then the transmitted light is plane – polarized in specific direction.
Since the human eye is unable to detect polarized light it is necessary to use an analyzer to detect the direction of polarization.
If the plane of polarization of the polarizer and the plane of the analyzer are perpendicular then no light is transmitted when the polarizer and the analyzer are combined.
METHODS / WAYS OF PRODUCTION PLANE POLARIZED LIGHT
(1) By Polaroid
(2 )By reflection
(3)By double refraction
(4)By using Nicol prism
POLARIZATION BY POLAROIDS
Polaroid is an artificial crystalline material which can be made in thin sheets.
It has the property of allowing light vibrations only of a particular polarization to pass through.
Uses of Polaroid (i) They are used in sunglasses to reduce the intensity of light and to eliminate glare.
- They are used to control the intensity of light entering trains and aeroplanes.
- They are used in wind shields of automobiles.
POLARIZATION BY REFLECTION
The reflecting surface of a transparent medium can be able to produce plane polarized light.
This happens when unpolarized light is incident to any transparent medium e.g. glass.
Where, AO – Is an incident natural light
OB – Is a strongly plane – polarized reflected ray
OC – Is a partially plane-polarized reflected ray.
BREWSTER’S ANGLE
- Polarization by reflection occurs at a certain special angle of incidence at which maximum polarization occurs.
Example
- For a glass of refractive index 1.5, Brewster‘s angle is 57
.
BREWSTER’S LAW
- The law states:
The extent of polarization of light reflected from a transparent surface is maximum when the reflected ray is at right angles to the refracted ray. – By Snell‘s law of refraction of light.
è 
è 
- When η = refractive index of the transparent medium
- From the figure above, we have:
- Equation (1) above becomes:
This equation leads to Brewster’s law
- The equation shows that the angle of incident for maximum polarization depends only on the refractive index of the medium.
POLARIZATION BY DOUBLE REFRACTION
Double refraction is the property possessed by certain crystals e.g. calcite, Iceland spar of forming two refracted rays from a single incident ray.
o-ray – ordinary ray
e-ray – extra ordinary
- Where a beam of unpolarized light is incident on one face crystal, its internal molecular structure produces two beams of polarized light E and O whose vibrations are perpendicular to each other.
NICOL PRISM
- Is a device for producing plane-polarized light.
- It consists of two pieces of calcite which are stuck together with Canada balsam material ( a transparent material used to join up the two pieces of calcite together)
- The extraordinary ray, E passes through the prism while the ordinary ray, O undergoes total internal reflection at the interface between the two crystals when the angle of incidence exceeds the critical angle value.
NOTE
If the incidence ray to the Nicol prism does not produce double refracted rays it means that the incident ray is a polarized ray of light.
APPLICATION OF POLARIZED LIGHT
(1) Reducing glare
Glare cause by light reflected from a smooth surface can be reduced by using polarizing materials since the reflected light is partially or completely polarized.
Example
In sunglasses.
In photography as filters. Where they are placed in from of the camera lens.
(2) Optical activity
Certain crystals e.g. sugar solutions rotate the plane of vibration of polarized light passing through them and are said to be optically active.
Definition
Optical activity is the ability of certain substances to rotate the place of vibration of plane-polarized light as it passes through them.
For a solution the angle of rotation depends on its concentration which can be measured by the instrument known as Polarimeter.
(3) Stress analysis
When glass, Perspex, polythene and some other plastic materials are under stress (e.g. by bending,or uneven heating) they become doubly refracting.
The effect is called photo elasticity and is used to analyze stresses in plastic models of various structures.
Problem 71
A point P is situated at 20.1cm and 20.28cm from two coherent sources. Find the nature of illumination at the point P if the wavelength of light is 6000
.
.Problem 72
The path difference between the two identical waves arriving at a point is 85.5
. Is the point bright or dark? If the path difference is 42.5 micrometer, calculate the wavel
ength of light.
. Is the point bright or dark? If the path difference is 42.5 micrometer, calculate the wavelength of light.
Problem 73
In young’s experiment, the distance between the two slits is 0.8mm and the distance of the screen from the slits is 80cm. If the fringe width is 0.6mm, find the wavelength of light.
Problem 74
In young’s experiment, interference bands were produced on a screen placed at 1.5m from two slits
0.15mm part and illuminated by a light of wavelength 6500.U. find
.U. find- Fringe width and
- Change in fringe width, if the screen is move away from the slit by 50cm
Problem 75
Two parallel slits 1.2mm apart are illustrated with light of wavelength 5200 from a single slit. A screen is placed at 1.0 m from the slits. Find the distance between the fifth dark band on one side and the seventh bright band on the other side of the central bright band.
from a single slit. A screen is placed at 1.0 m from the slits. Find the distance between the fifth dark band on one side and the seventh bright band on the other side of the central bright band.Problem 76
In a biprism experiment, the distance between the slit and the eyepiece is 80 cm and the separation between the two virtual images of the slit is 0.25 mm. if the slit is illuminated by a light of wavelength 6000 Å, find the distance of the second bright and from the central bright band.
Problem 77
In a bi prism experiment, with the distance between the slit and the screen as 0.5m and the separation between the two virtual images of the slit as 0.4 cm an interference pattern obtained with a light of wavelength = 5500 Å. Find the distance between the 3rd and the 8th bands on the same side of the central band.
= 5500 Å. Find the distance between the 3rd and the 8th bands on the same side of the central band.Problem 78
In a biprism experiment, the distance of the 10th bright band from the center of the interference pattern is 6 mm. Find the distance of the 15th bright band from the center.
Problem 79
The fringe separation in a biprism experiment is 3.2 
10-4 m when red light of wavelength 6.4 10-7 m is used. By how much will this change if blue light of wavelength 4 
m is used with the same setting?
10-4 m when red light of wavelength 6.4 10-7 m is used. By how much will this change if blue light of wavelength 4 m is used with the same setting?Problem 80
In a biprism experiment, the fringe width is 0.4mm, when the eyepiece is at a distance of 1m from the slit. Find the change in the fringe width, if the eyepiece is moved through a distance of 25cm towards the biprism, without changing any other arrangement.
Problem 81
In a biprism experiment, the distance between the slit and the screen is 1.0m and the distance between the images of the slit is 2.7mm. If the fringe width is 0.2mm, find the wavelength of light used.
Problem 82
In a biprism experiment, the distance between the slit and the screen is 0.8m and the two virtual sources formed by the biprism are 0.4mm apart. The wavelength of light used is 6000. Find the band width.
. Find the band width.Problem 83
Calculate the distance between the second dark band and the fifth bright band on the same side of the central bright band of an interference pattern produced by coherent sources separated by 1.2mm from each other. The screen is placed at one metre from the coherent sources and the wavelength of light used is 6000.
.Problem 84
In a biprism experiment, the slit is illuminated by a light of wavelength 5000. The distance between the slit and the biprism is 20cm and the distance between the biprism and the eyepiece is 80cm. If the distance between the two virtual source is 0.25cm, calculate the distance between the fifth bright band on one side of the central bright band and the sixth dark band on the other side.
. The distance between the slit and the biprism is 20cm and the distance between the biprism and the eyepiece is 80cm. If the distance between the two virtual source is 0.25cm, calculate the distance between the fifth bright band on one side of the central bright band and the sixth dark band on the other side.Problem 85
In a biprism experiment, the distance between the two virtual images of the slit is 1.5mm and the distance between the slit and the focal plane of the eyepiece is 1metre. Find the distance between the second and the eighth dark fringe on the same side, if the wavelength of the light used is 5000.
.Problem 86
In biprism experiment, fringes were obtained with a monochromatic source of light. The eyepiece was kept at a distance of 1.2m from the slit and fringe width was measured. When another monochromatic source of light was used without disturbing slit and biprism, the same fringe width was obtained when the eyepiece was at 0.8m from the slit. Find the ratio of wavelength of light emitted by the two sources.
Problem 87
The distance between two consecutive dark bands in a bi prism experiment is 0.32mm when red light of wavelength 6400 is used. By how much will this distance change if yellow light of wavelength 5900 is used with the same setting?
is used. By how much will this distance change if yellow light of wavelength 5900 is used with the same setting?Problem 88
Newton’s rings are observed with a plane convex lens in constant with a glass plate. The radius of the first bright ring is 1mm. If the radius of the convex surface is 4metres, what is the wavelength of light used.
Problem 89
The diameter
of 10th dark ring in a Newton’s ring system viewed normally by reflected light of = 5900Å is 0.5cm, calculate the thickness of the air film and radius of curvature of the lens.
of 10th dark ring in a Newton’s ring system viewed normally by reflected light of
= 5900Å is 0.5cm, calculate the thickness of the air film and radius of curvature of the lens.Problem 90
Newton’s rings formed with sodium light between a flat glass plate and a convex lens are viewed normally. What will be the order of the dark ring which will have double the diameter of that of the 40th dark ring?
Problem 91
If the diameter of the two consecutive Newton’s rings in reflected light of wavelength 5890A.U. are 2.00 and 2.20cm respectively, what is the radius of curvature of the lens surface in contact with the plane glass surface?
Problem 92
Newton’s rings formed with sodium light (
cm) between a plane glass plate and convex lens surface. The diameters of two successive dark rings are 2mm and 2.236mm. what is the radius of curvature of the lens surface.
cm) between a plane glass plate and convex lens surface. The diameters of two successive dark rings are 2mm and 2.236mm. what is the radius of curvature of the lens surface.Problem 93
Newton’s rings are formed by placing lens on a glass surface. If the 10th bright ring of sodium light by reflection( 
be 5mm in diameter. What is the radius curvature of the lens.
be 5mm in diameter. What is the radius curvature of the lens.Problem 94
In a Newton’s ring experiment, the plane convex lens and the glass plate are in optical contact and the thickness of film at that point is zero. Find out the thickness of the air wedge at the fourth bright ring for light of λ = 500Å
Problem 95
In a young’s double-slit experiment, sodium light of wavelength 0.59 
m was used to illuminate a double slit with separation 0.36mm. If the fri
nges are observed at a distance of 30cm from the double slits, calculate the fringe separation.
m was used to illuminate a double slit with separation 0.36mm. If the fringes are observed at a distance of 30cm from the double slits, calculate the fringe separation.
Problem 96
In an experiment using young’s slit, fringes were found to occupy 3.0mm when viewed at a distance of
36mm from the double slits. If the wavelength of the light used is 0.59
, calculate the separation of the double slits.
, calculate the separation of the double slits.Problem 97
When red monochromatic light of wavelength 0.70m is used in a Young’s double-slit arrangement, fringes with separation 0.60mm are observed. The slit separation is 0.40mm. Find the fringe spacing if (independently)
m is used in a Young’s double-slit arrangement, fringes with separation 0.60mm are observed. The slit separation is 0.40mm. Find the fringe spacing if (independently)(a ) Yellow light of wavelength 0.60 is used
is used(b) The slit separation becomes 0.30mm;
(c)The slit separation is 0.30mm and the slits fringe distance is doubled.
Problem 98
Interference fringe are formed in an air wedge using monochromatic light of wavelength 0.60
m. The fringes are formed parallel to the line contact, and a dark fringe is observed along the line of contact. Calculate the thickness of the air wedge at position where:
(a) The twentieth dark fringe and
( b) The thirtieth bring fringe from the line of contact are observed.
Problem 99
When interference fringes are formed using an air wedge, it is found that the twentieth bright fringe is formed at an air thickness of 6.8. Calculate
. Calculate(a)The wavelength of the light used Problem 100
( b) The diameter of the wire
Problem 101
Interference fringes of separation 0.40mm are with yellow of wavelength 0.60. Calculate the fringe spacing if the blue light of wave length 0.45 is used.
. Calculate the fringe spacing if the blue light of wave length 0.45 is used.Problem 102
Interference fringes of spacing 1.0mm are obtained using the light of wavelength incident on an air wedge of angle . The angle of the wedge is now double and the light replaced by one of the wavelength 1.5 . Calculate the new fringe separation. 
incident on an air wedge of angle . The angle of the wedge is now double and the light replaced by one of the wavelength 1.5 . Calculate the new fringe separation. Problem 103
A loudspeaker emits a note which gives a beat frequency of 4Hz when sounded with a standard turning fork of frequency 280Hz. The beat frequency decreases when the fork is loaded by adding a small piece of plastic to its prongs. Calculate the frequency of the note emitted by the loudspeaker.
Problem 104
A note from a loudspeaker gives a beat frequency of 10Hz when sounded with a turning fork of frequency 440Hz. Calculate
( a) The beat period.
b)Two possible values for the frequency of the note emitted by the loudspeaker
Problem 105
A vibrating sonometer wire emits a note which gives a beat frequency of 6.0Hz when sounded in unison with a standard tuning fork of frequency 256 Hz. When the fork is loaded the beat frequency increases. What is the frequency of the note emitted by the sonometer?
Problem 106
A beam of microwaves of wavelength 3.1cm is directed normally through a double slit in a metal screen and interference effect are detected in a plane parallel to the slit and at a distance of 40cm from them. It is found that the distance between the centers of the first maximum in the interference pattern is 70cm. Calculate an approximate value for the slit separation.
Problem 107
What is the wavelength of light which gives a first order maximum at an angle of 22 when incident normally on a grating with 600 lines mm-1.
when incident normally on a grating with 600 lines mm-1.Problem 108
Light of wavelength 600nm is incident normally on a diffraction grating of width 20.0mm on which 10.0 
lines have been ruled. Calculate the angular positions of the various orders.
lines have been ruled. Calculate the angular positions of the various orders.Problem 109
A source emits special lines of wavelength 589nm and 615nm. This light is incident normally on a diffraction grating having 600 lines per nm. Calculate the angular separation between the first-order diffracted waves. Find the maximum order for each of the wavelengths
Problem 110
When a certain grating is illuminated normally by monochromatic light of wavelength 600 the first-order maximum is observed at an angle of 21.1. If the same grating is now illuminated with light with wavelength from 500nm to 700nm, Find the angular spread of the first-order spectrum.
. If the same grating is now illuminated with light with wavelength from 500nm to 700nm, Find the angular spread of the first-order spectrum.Problem 111
When monochromatic light of wavelength 5.0 m is incident normally on a plane diffraction lines are formed at angle of 30. What is the number of lines per millimeter of the grating?
m is incident normally on a plane diffraction lines are formed at angle of 30. What is the number of lines per millimeter of the grating?Problem 112
A spectral line of known wavelength (5.792 m) emitted from the mercury vapour lamp is used to determine the spacing between the lines ruled on a plane diffraction grating. When the light is incident normally in the grating, the third-order spectrum, measured using a spectrometer, occurs at an angle of 60 19′ to normal. Calculate the grating spacing.
m) emitted from the mercury vapour lamp is used to determine the spacing between the lines ruled on a plane diffraction grating. When the light is incident normally in the grating, the third-order spectrum, measured using a spectrometer, occurs at an angle of 60 19′ to normal. Calculate the grating spacing.Problem 113
Light from a cadmium discharge lamp can be used to determine the spacing of the lines on a plane diffraction grating. This is done by measuring the angle between the diffracted beams either side of the normal in the first order spectrum for light incident normally on the grating.
between the diffracted beams either side of the normal in the first order spectrum for light incident normally on the grating.( a) If the measured value of is 46 43′ and the red line used in the cadmium spectrum is of wavelength 644nm, calculate the number lines per metre on the grating.
is 46 43′ and the red line used in the cadmium spectrum is of wavelength 644nm, calculate the number lines per metre on the grating.( b) Make a suitable calculation to the whether the second order spectrum of this line will be visible.
Problem 114
A light source emits two distinct wavelengths, one of which is 450nm. When light from the source is incident normally on a diffraction grating, it is observed that the fourth order image formed by the same angle of diffraction as the third order image for the other wavelength. If the angle of diffraction for each image is 46, calculate
, calculate( a)The second wavelength emitted by the source,
(b)The number of lines per meter of the grating
Problem 115
A horizontal string is stretched between two point points a distance 0.80m apart. The tension in the string is 90N and its mass is 4.5g. Calculate
(a)The speed of transverse waves along the string and
(b)The wavelength and frequencies of the three lowest frequency modes of vibration of the string.
Problem 116
The fundamental frequency of vibration of a stretched wire is 120Hz. Calculate the new fundamental frequency if
( a)The tension in the wire is doubled, the length remaining constant
(b)The length of the wire is doubled the tension remaining constant
(c)The tension is doubled and the length of the wire is doubled.
(d)The wavelength of waves with frequency 120Hz.
(e)The length of wire which when fixed at its end, gives a fundamental frequency of 120Hz.
Problem 117
The fundamental frequency of vibration of a stretched wire is 150Hz. Calculate the new fundamental frequency if;
- )The tension in the wire is tripled, the length remaining constant.
- )The length of wire is halved, the tension remaining constant.
(c)The tension is tripled and the length of wire is halved.
Problem 118
A wire having a diameter of 0.80mm is fixed in a sonometer and has a fundamental frequency of 256Hz alongside it’s wire is made of the same material but of diameter 0.60mm. Both wires are stretched over the same bridges on the sonometer but the thinner wire is subjected to only half the tension of the thicker wire. Calculate the fundamental frequency of vibration of the thinner wire.
Problem 119
A closed organ pipe is of length of 0.60 m. Calculate the wavelengths and frequencies of the tree lowest frequency modes of vibration. Take the speed of sound to be 345 m
and neglect any end correction of the pipe.
and neglect any end correction of the pipe.Problem 120
Two open organ pipes are sounding together and produce a beat frequency of 12.0Hz. If the longer pipe has length of 0.400 m. Calculate the length of the pipe. Take the speed of sound as 340 m and ignore end corrections.
and ignore end corrections.Problem 121
A piece of
glass tubing is closed at one end by covering it with a sheet of metal. The fundamental frequency is found to be 280 Hz. If the metal sheet is now removed, calculate
glass tubing is closed at one end by covering it with a sheet of metal. The fundamental frequency is found to be 280 Hz. If the metal sheet is now removed, calculate
(a)What length the tube is
(b)The wavelength and frequencies of the fundamental and the first overtone 280 Hz. Ignore end corrections.
Problem 122
A tall vertical cylinder is filled with water and a tuning fork of frequency 512 Hz is held over its open end. The water is slowly run out and the first resonance of the air column is heard when the water level is 15.6 cm below the open end. Calculate
- The end correction of the tube.
- The position of the water level when the second resonance is heard.
Problem 129
An open tube of length 30.0 cm has an end correction of 0.60 cm. calculate its fundamental frequency.
Problem 130
Two open pipes of length 0.700 m and 0.750 m are sounded together and vibrate in their fundamental frequencies. Find the beat frequency, assuming that end corrections can be ignored.
Problem 131
Two identical closed pipes of length 0.322 m are each vibrating with their fundamental frequency. If one pipe is held at 00C and the other at 17oC, calculate the beat frequency which is observed. Take the speed of sound at 0oC to be 331 m-1 and ignore end corrections.
Problem 132
A closed pipe is of length 0.300 m. calculate:
- Its fundamental frequency at 0oC, given that the speed of sound at 0oC is 331 m s-1,
- The temperature, in oC, at which it will be in unison with a tuning fork of frequency 288 Hz.
Problem 133
Two open pipes of length 0.500 m and 0.550 m are sounded together and vibrate in their fundamental frequencies at 7 oC. Calculate:
- The beat frequency, given that the speed or sound at 7 oC is 335 ms-1
If the temperature of the longer pipe is now allowed to change whilst the shorter pipe stays at 7 oC, calculate.
- The value of the temperature of the air in the longer pipe at which the two pipes will be in unison.
Problem 134
Stationery waves are set up in the space between a microwave transmitter and plane reflector. Successive minima are spaced 15mm apart. What is the frequency of the microwave oscillator? Take the speed electromagnetic waves as 3.0 x 108 m s-1.
Problem 135
A system of stationary waves in which the nodes are 2m apart are produced from progressive waves of frequency 200 Hz. Calculate the speed of the progressive waves.
Problem 136
A stretched wire of length 0.7 m vibrates in its fundamental mode with a frequency of 320 Hz. Calculate the velocity of waves along the wire. Why does such a vibration not continue indefinitely?
Problem 137
A wire of mass per unit length 5.0 gm-1 is stretched between two points 30 cm apart. The tension in the wire is 70 N. calculate the frequency of the sound emitted by the wire when it oscillates in its fundamental mode.
Problem 138
- A string of unstretched length 2.0 m and mass 0.15 kg has a force constant of 25 Nm-1.for the experiment, the string is stretched to a total length of 3.0 m. calculate the velocity of propagation of transverse waves along the string.
- The same string in (a) above is clamped between two rigid supports 3.0 m apart, and set in vibration. Calculate the wavelengths and frequencies of the five lowest frequency modes of vibration which can be excited on the string. When the vibrating string is held lightly at the centre, in which of these modes does the string continue to vibrate? Explain your reasoning.
Problem 139
A vertical steel wire is kept in tension by a piece of iron attached to one end. The wire is set in transverse vibration and emits a note of frequency 200 Hz. The iron is now completely immersed in water and the frequency of the note changes to 187 Hz. If the density of the water is 1000 kg m-3 calculate the density of the iron.
Problem 140
A resonance tube is held vertically in water and can be raised or lowered. A tuning fork of frequency 384 Hz is struck and held above open end of the tube on a day when the speed of sound in air is 344 ms-1. The shortest tube length at which resonance occurs is 21.6 cm and the corresponding length when the tube is filled with carbon dioxide is 16.7 cm. calculate:
- The end correction for the tube,
- The speed of sound in carbon dioxide.
Problem 141
A turning fork is sounded at the open end of a tube, containing air, which is closed at the other end. Two successive positions of resonance are obtained when the length 49.0 cm and 82.0 cm. calculate:
- The wavelength of the sound waves in the tube,
- The end correction of the tube.
Problem 142
If the speed of sound in air is 340 m s-1 at a given temperature, calculate the length of an open pipe having a fundamental frequency of 192 Hz. If this pipe were sounded together with another open pipe of length 0.850 m at the same temperature, calculate the beat frequency. Ignore any end corrections.
Problem 143
An open ended pipe, of length 0.50 m, is sounded at 20oC together with a tuning fork of slightly lower frequency. Five beats per second are heard. Calculate the change in temperature required to bring the pipe and fork back into unison. Neglect end corrections and assume that changes in temperature affect only the speed of sound in air, which is 340 ms-1 at 20oC.
Problem 144
A source of sound, when stationary, generates waves of frequency 500 Hz. The speed of sound is 340 ms-1. Find from first principles the wavelength of the waves detected by the observer and the frequency observed when:
- The source is stationary and the observer moves towards it with speed 20.0 ms-1,
- The source moves with away from stationary observer with speed 30.0 ms-1.
- The sources moves with speed 30.0 ms-1 in a direction away from, the observed and the observer moves with speed 20.0 ms-1
Problem 145
A train sounding its whistle (frequency 580 Hz) is traveling at 40.0ms-1 a long straight section of track, and passes an observer standing closed to the track. Calculate the maximum change in frequency which the observer will hear. Take the speed of sound in air as 340 ms-1.
Problem 146
A motorist approaches a road junction at a constant speed of 15 m s-1. A policeman standing at the junction blows on a whistle with frequency 680Hz.
- Find from first principles the frequency observed by the motorist. The motorist now reduces his speed at a rate of 5.0 m s-1.
- Calculate the frequency he observers at subsequent 1 intervals until he stops.
Problem 147
A train sounding its whistle moves at a constant speed of 20 ms-1 a long straight section of track. The train passes under a low bridge on which stands an observer. If the observer records a maximum frequency of 638Hz. Calculate:
- The frequency of the whistle,
- The minimum frequency the observer hears.
Problem 148
A source of sound of frequency 400 Hz moves at steady speed of 15 ms-1 towards an observer. If the observer moves a steady speed of 25 ms-1 towards the source, calculate the frequency he observes.
Problem 149
A loudspeaker which emits a note of frequency 250 Hz is attached to a wire and whirled in a vertical circle of radius 1.00 m at a steady rate of 20.0 revolutions per minute. Calculate:
- The speed of rotation of the loudspeaker in m s-1.
- The maximum and minimum frequency detected by a stationary observer.
Problem 150
A train sounding its whistle travels at constant speed on a long straight section of track. An observer standing closed to the track records a range of frequencies between 551 Hz and 658 Hz. Calculate:
- The speed of the train.
- The frequency of its whistle.
Problem 151
A hooter of frequency 360 Hz is sounded on a train approaching a tunnel in a cliff-face at 25 ms-1, normal to the cliff. Calculate the observed frequency of the echo from the cliff-face, as heard by the train driver. Assume that the speed of sound in air is 330 ms-1.
Problem 152
A car travel at a constant speed of 30 ms-1 towards a tunnel and sounds its horn, which has a frequency of 200 Hz.the sound is reflected from the tunnel entrance. Calculate the frequency of the echo observed by;
- The driver of the car
- A stationary observer standing close to the road
- The driver of the car travelling at 20 ms-1 which is following the first car.
Problem 153
A train emerges from a tunnel at a speed of 20 ms-1 and sound its whistle, which has a frequency of
450Hz, Calculate the frequency of the echo from the tunnel entrance as observed by the train driver, Problem 154
A source of sound which is stationary with respect to air emits a note of frequency 340 Hz. An observer receding at uniform speed, from the source hears an apparent frequency of 300 Hz observer is moving, if the speed of sound in air is 340 ms-1.
Problem 155
(a) Show from first principles that the frequency f of sound in still air, heard by a stationary observer as a source of sound of frequency approaches the observer with a velocity 
, is given by
approaches the observer with a velocity 
, is given by
Where C is the velocity of sound in still air,
(c) When =1.0 103 Hz and c = 300 ms-1, what is the percentage change in the frequency heard by the stationary observer when the source velocity changes from 30 ms-1 to 35 ms-1? 
Problem 156
A model aircraft with an engine producing a note of constant frequency of value 400Hz flies at constant speed in a horizontal circle of radius 12m and completes one revolution in 3.0s. An observer, situated in the plane of the circle and 30m from its centre, monitors the frequency of the sound from the engine.
- Explain why the observed frequency shows periodic variations
- Derive a relation for the minimum observed frequency in terms of f, the true frequency of the engine, V, the speed of the aircraft and C, the speed of sound in air. Write down the corresponding relation for the maximum observed frequency.
- Taking to C be 340 ms-1, calculate the maximum and minimum observed frequencies. Determine the time interval between the occurrence of a maximum frequency and the next minimum frequency.
Problem 157
A ship travelling at 3 ms-1 towards a cliff in still air and is sounding its siren at 1 KHz. Find from first principles the frequency of the echo as measured by an observer on the ship. Give sufficient detail for reasoning to be followed. The speed of sound in air is 330 ms-1.
Problem 158
A railway engine traveling at constant speed emits a whistle of constant frequency. When the engine passes a stationary observer closed to the track, the frequency of the sound heard by the observer changes from 600 Hz whilst approaching to 500 Hz whilst receding. Assuming the speed of sound is 340 ms-1 calculate the speed of the engine.
Calculate the frequencies heard if the same engine passes an observer who is travelling at 2 ms-1 in the same direction as the engine and close to the track.
