PHYSICS As LEVEL(FORM FIVE) NOTES – WAVE MOTION-1(3)

END CORRECTION OF A PIPE

• In practice the air just outside the open end of a pipe is set into vibration and the displacement antinode of a stationary wave occurs a distance ―C‖ called end correction beyond the open end.
• The effective length of the air column is therefore slight greater than the length of the pipe.

Example

( a ) For a closed pipe

( b ) For an open pipe

Definition

-The end correction C of a pipe is that small length of a stationary wave which protrudes just outside the open end of a pipe instrument where the air inside it is set into vibration.

RESONANCE IN A CLOSED PIPE

Apparatus

• A resonance jar is used.
• It consists of a glass tube which stands in a tall jar full of water.

• The length of the air column is varied by raising or lowering the glass tube.

WORKING

• Starting with a very short air column, a vibrating fork is held over the mouth of the tube and the length of the column is then gradually increased.
• Strong resonance occurs when the column reaches a certain critical length (say).
• This is called the first position of resonance.
• At this position the air in the tube vibrates at its fundamental note/first harmonic

Where

• If the length of the air column is now increased still further a second position of resonance is obtained when the column is approximately three times as long as ( say).
• At this position the air in the tube vibrates with its harmonic/first overtone.

• Equation (2) – equation (1)

–

2

———————- (3)

• If the frequency f of the fork is known then the speed V of sound in air can be found.

V =

• Substitute equation (3) in this equation

V = 2 (

V = 2f ( ——————— (2)

Problem 21

What are the two successive response lengths of a closed pipe containing air at for a turning

Take the speed of sound in air at

Problem 22

Two open organ pipes of lengths 50cm and 51cm respectively give beats of frequency 6.0HZ when sounding their fundamental notes together. Neglecting end correction, what value does this give for the velocity of sound in air?

Problem 23

A Cylindrical pipe of length 28cm closed at one end is found to be at resonance when a tuning fork of frequency 864HZ is sounded near the open end. Find the mode of vibration of the air in the pipe and calculate the value of the end correction (speed of sound in air = 340m)

BEATS

Definition

Beats are the periodic increase and decrease in loudness heard when two notes of slightly different frequency is sounded at the same time.

• The two notes producing beats must be of the same amplitude.
• The periodic increase and decrease in loudness is a result of successive occurrence between the two notes, as the repeatedly become in phase and then out in phase with each other.

MATHEMATICAL TREATMENT OF BEATS

be the individual displacement two notes whose frequencies are respectively.

• Let ―a‖ be the amplitude of each note.

And

• Applying the principle of superposition of waves, the resultant displacement y is given by:

Y =

Y = sin (2

.

.

.

But 2a cos = A = Amplitude of the resultant wave.

• This equation shows that resultant wave has an effective frequency equal to the average frequency of the two sources.

i.e.

• The amplitude A of the resultant wave is

A =

• This equation shows that the amplitude varies in time with a frequency given by:-

BEAT FREQUENCY (B.f)

• This is the number of beats heard per second when two notes of slightly different frequency are sounded at the same time.
• It is given by difference between the frequencies of the two notes.

B.f =

OR

B.f =

NOTE

Beat period T =

APPLICATION OF THE PHENOMENON OF BEATS MEASUREMENT OF THE UNKNOWN FREQUENCY

• The phenomenon of beats can be used to determine the unknown frequency of some wave motion by causing the wave to beat with a wave of the same kind whose frequency is known.
• Suppose a note of unknown frequency is made
  to produce beats with a note of known frequency.
• If is not very different from then it is possible to count the number of beats that occur in some given time, and hence determine the beat frequency (B.f).

B.f = if

OR

• In order to discover which of is the higher frequency, one of the frequencies, is changed slightly.
• This can be done by loading a small pieces of plasticine / wax and the effect on the beat frequency is noted.

Problem 24

The beat frequency of two notes of nearly equal frequency is 6HZ. If one is loaded with wax, the beat frequency becomes 4HZ. What is the frequency of the other note of the one note loaded is 250HZ?

Problem 25

Two turning forks are sounded together and their beats frequency is 4HZ. One of the turning forks has a frequency of 320HZ. When the other turning fork is loaded with plasticine, the beat frequency becomes 6HZ. What is the frequency of the other tuning fork before being loaded with plasticine?

Problem 26

Calculate the speed of sound in a gas in which two waves of wavelength 1.00m and 1.01m produce 30 beats in 10 seconds.

Problem 27

Two similar sonometer wires of the same materials produce2 beats per second. The length of one is 50cm and that of the other is 50.1cm. calculate the frequency of the two wires.

DOPPLER EFFECT

Definition

Doppler Effect is the apparent change in the observed frequency of a wave as a result of relative motion between the source and the observer.

Example

– There is sudden decrease in pitch (frequency) heard by a personal standing in a railway station as a sounding train siren passes by him.

CASE 1

SOURCE MOVING

• Motion of the source affect the wavelength of the wave and hence the apparent wavelength ( is given by:

.

1. SOURCE MOVING TOWARDS A STATIONARY OBSERVER
• Consider a source S of sound wave to be moving towards a stationary observer O

• Where velocity of the source

V = velocity of sound

• Velocity of the wave relative to observer at O = V –
• The apparent wavelength reaching the observer at O is

——————— (1)

• Where f = true frequency of the source
• Let be apparent frequency
• 
  From V =

—————- (2)

• Substitute equation (1) in equation (2)

1. SOURCE MOVING AWAY FROM A STATIONARY OBSERVER
• Consider a source S of sound wave to be moving with velocity away from a stationary observer O

-Where V = Velocity of sound wave

• Velocity of wave relative to observer at 0 = V +
• The apparent wavelength reaching the observer at 0 is :

——————- (4)

• The apparent frequency ——————- (5)
• Substitute equation (4) in equation (5)

——————- (6)

CASE 2

OBSERVER MOVING

• The motion of the observer affects the velocity of the waves he receives.
• In this case the wavelength is unchanged and is given by:

————————— (7)

• The apparent frequency is given by

(i) OBSERVER MOVING TOWARDS A STATIONARY SOURCE

• Where Velocity of observer

V = Velocity of wave

• Velocity of wave relative to observer = V +
• The apparent frequency of the wave is

———————— (8)

(ii) OBSERVER MOVING AWAY FROM STATIONARY SOURCE

• Velocity of waves relative to observer

= V –

• The apparent frequency is given by:

———————- (9)

CASE 3

SOURCE AND OBSERVER ARE MOVING

(i) SOURCE AND OBSERVER ARE APPROACHING

• Velocity of wave relative to observer is:

Vo = V + Uo

• Apparent wavelength is

• The apparent frequency is given by

—————– (10)

(ii) SOURCE AND OBSERVER ARE MOVING AWAY FROM EACH OTHER

• Velocity of wave relative to observer is:

• Apparent wavelength

• Apparent frequency

———————— (11)

Problem 28

Calculate the frequency of the beats heard by a stationary observer when a source of sound of frequency 100HZ moves directly away from him with a speed of 10.0m towards a vertical wall. Given that speed of sound in air =340m.

Problem 29

Two whistles A and B each has a frequency of 500HZ. “A” is stationary and “B” is moving towards the right (away from A) at a velocity of 200ft . An observer is between the two whistles. Moving towards the right with a velocity of 100ft . Take the velocity of sound in air as 1100ft.

( a) What is the frequency from A as heard by the observer?

( b)What is the frequency from B as heard by the observer?

( c)What is the beat frequency heard by the observer?

Problem 30

A source of sound waves “S” emitting waves of frequency 1000HZ, is traveling towards the right in still air with a velocity of 100ft . At the right of the source is large, smooth, reflecting surface moving towards the left at a velocity of 400ft

(a) How far does the emitted wave in 0.01 second?

( b) What is the wave length of the emitted waves in front of (i.e at the right of) the source?

( c) How many waves strike the reflecting surface in 0.01sec? Take the velocity in air as 1100ft

Problem 31

A car sounding a horn produce a note of 500HZ, approaches and then passes a stationary observer O at a steady speed of 20m. Calculate the apparent frequency in each case. Velocity of sound = 340m/s.

Problem 32

A cyclist and railway train are approaching each other. The cyclist is moving at 10m/s an the train at 20m m. The engine driver sounds a warning siren at a frequency of 480HZ. Calculate the frequency of the note heard by the cyclist.

1. Before and
2. After the train has passed by

Given that speed of the sound in air = 340m

Problem 33

An observer standing by a railway track notices that the pitches of an engine whistle change in the ration of 5:4 on passing him. What is speed of the engine?

THE RADAR SPEED TRAP

• This is an instrument used to determine the speed of a moving car.
• The instrument sent out microwaves of frequency f (about 10.7GHZ) towards a moving car.
• The speed of moving car can be found by measuring the shift in frequency of microwaves reflected by it.

Consider a car moving with speed V towards a stationary source of microwave of frequency f.

• The car act as an observer moving towards a stationary source and the wave as received by the car have a frequency f.

• f= is given by

=

=

But

—————— (1)

• Where C = speed of microwaves in free space
• Wave of this frequency () are reflected back to the source, so that the car is now acting as a source moving with velocity V and the radar set is acting as a stationary observer.

• Let be frequency of waves on reaching the radar set.

=

But

• Substitute equation (1) in equation (2)

. =

. = ——————— (3)

• The fractional change in frequency is given by

—————— (4)

• Substitute equation (3) in equation (4)

• Since V << C, therefore C – V ≈ C

• Where ∆f = beat frequency of the waves transmitted and received by the radar set.
• Thus, by causing the incoming signal to beat with the transmitted signal and knowing the frequency of the transmitted signal, we can find V from equation (5) above:

Problem 34

Calculate the beat frequency produced if car travels a radar speed trap at 30m , the operating frequency of the speed trap being 10.7GHZ. Take the velocity of light to be 3 .

Problem 35

Calculate the change in frequency of the radar echo received from an aeroplane moving at 250m if the operating wavelength of the radar set is 1 metre.

DOPPLER EFFECT IN LIGHT

• The Doppler effect in light is used to measure the speed of distant objects and planets.
• Suppose a source of light emits waves of frequency f and wavelength.
• If C is the velocity in light in free space then:

———————- (1)

EXPRESSIONS OF APPARENT WAVELENGTH AND FREQUENCY

When the source of light is moving away from the stationary observer

• Consider a source of light example star is moving with a velocity V away from the earth

• The apparent wavelength ‘ to an observer on the earth in line with star‘s motion is

From equation (1) above

———————– (3)

Substitute equation (3) in equation (2)

———————- (*)

• It is clear from equation (4
  ) that ‘ is greater than
• Thus, when a star is moving away from the earth, the apparent wavelength increases. They say that light is RED-SHIFTED i.e. it is shifted to longer wavelengths.

V

I

B

G

Y

O

R

Direction of increasing wavelength

• The apparent frequency f‘ is given by

But from equation *

(iii) When the source of light is moving towards the stationary observer

• The apparent wavelength is given by

-From equation (1)

• Where shift in wavelength.

• It is clear from equation (8) that is less than.
• Thus, when a star is moving towards the earth, the apparent wavelength decreases.
• We say that light is blue shifted it is shifted to shorter wavelengths.
• The apparent frequency is given by:

• But from equation (7)

But C/

————— (10)

APPLICATIONS OF DOPPLER EFFECT

1. It is used in radar speed trap to determine the speed of a moving car.
2. It is used for measurement of speed of star.
3. It is used to determine the direction of motion of a star.
4. It is used for measurement of plasma temperature.

NOTE

– There are some cases in which there is no Doppler effect in sound (i.e no changes in frequency)

1. When the source of sound and the observer are moving in the same direction with the same speed.

2. When either the source or observer is at the center of the circle and the other is moving on the circle with uniform speed.

PHYSICAL OPTICS

– This deals with the study of phenomena that depend on the wave nature of light include:

1. Interference
2. Diffraction
3. Polarization

THE NATURE OF LIGHT

Definition

• Light is a form of energy which stimulates the sense of vision.
• It can be transmitted in air / vacuum with a speed of 3

HISTORICAL BACKGROUND

• Three different theories were put forward to explain the nature of light.

(1) NEWTON’S CORPUSCULAR THEORY OF LIGHT

• This theory suggested that light is a stream of particles emitted from a luminous object.
• This theory ignored the wave nature of light.

(2) HUYGEN’S THEORY

• This theory suggested that light travels from one point to another by wave motion.
• Thomas young produced evidence that light behaves as a wave motion by performing an experiment.

(3) EINSTEIN’S THEORY

• This theory suggested that light consist of a stream of particles carrying energy with them called

PHOTONS.

• This is like corpuscular theory but with small difference.

NOTE

• Both particle theory and wave theory of light are accepted in solving and explaining different cases concerning propagation of light.
• Hence we have dual nature of light.

THE DUAL NATURE OF LIGHT

– Is the behavior of light in which two separate aspects can be isolated i.e wave nature and particle nature.

WAVE FRONTS AND RAYS OF LIGHT

Wave fronts

• Assume S to be a source of light waves in air

• The wave spread out equally in all directions.
• A line joining all adjacent points at which the disturbances are in phase is called a wave front

Definition

• A wave front is a locus of points having the same phase of oscillation.

-For a point source of light in a homogenous medium the wave fronts are spherical.

• A small part of a spherical wave front from a distant source will appear plane and therefore called plane wave front.

Example

– Sunlight reaches the earth with plane wave fronts

Rays of light

• A ray is the direction of the path taken by light.
• At any point along the path of light a ray is perpendicular to the wavefront.

HUYGEN’S PRINCIPLE

• Huygens‘s principle provides a geometrical method to determine the position of the wave front at a later time from its given position at any instant.
• The principle states:

1. Each point on a wave front act as fresh source of secondary wavelets, which spread out with the speed of light in that medium.
2. The new wave front at any later time is given by the forward envelop of the secondary wavelets at that time.

–Where AA’ = original wave front

BB‘ = wave front after time the interval t

DERIVATION OF THE LAW OF REFLECTION OF LIGHT ON THE BASIS OF HUYGEN’S PRINCIPLE

-Consider a plane wave front AA’ which is incident on the reflecting surface MM’

The position of the wave front after a time interval t may be found by applying Huygens’s principle with points on AA’ as centers.

• Those wavelets originating near the upper end of AA‘ spread out unhindered and their envelope given the portion of the new wave front OB‘
• Those wavelets originating near the lower end of AA‘ strike the reflecting surface and get reflected 18 out of phase.
• The envelope of these reflected wavelets is then the portion of the wave front OB.
• A similar construction gives the line CPC‘ for the wave front after another time interval, t.

RELATIONSHIP BETWEEN i and r

– Where i = angle of incidence r = angle of reflection