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 slightly 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 = 2f ()

V = 2f () ——————— (2)

Problem 21

What are the two successive response lengths of a closed pipe containing air at for a tuning fork? Take the speed of sound in air at .

Problem 22

Two open organ pipes of lengths 50 cm and 51 cm respectively give beats of frequency 6.0 Hz 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 28 cm closed at one end is found to be at resonance when a tuning fork of frequency 864 Hz 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 = 340 m/s).

Beats

Definition

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

• The two notes producing beats must be of the same amplitude.
• The periodic increase and decrease in loudness result from the two notes repeatedly becoming in phase and then out of phase with each other.

Mathematical Treatment of Beats

Let y₁ and y₂ be the individual displacements of two notes whose frequencies are f₁ and f₂ respectively.

• Let “a” be the amplitude of each note.

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

Y = 2a cos(π(f₁ – f₂)t) sin(2π(f₁ + f₂)t/2)

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

i.e. f = (f₁ + f₂)/2

• The amplitude A of the resultant wave is

A = 2a cos(π(f₁ – f₂)t)

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

Beat frequency (B.f) = |f₁ – f₂|

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 the difference between the frequencies of the two notes.

B.f = |f₁ – f₂|

Note

Beat period T = 1 / B.f

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 f_x is made to produce beats with a note of known frequency f_k.
• If f_x is not very different from f_k, 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 = |f_x – f_k|

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

Problem 24

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

Problem 25

Two tuning forks are sounded together and their beat frequency is 4 Hz. One of the tuning forks has a frequency of 320 Hz. When the other tuning fork is loaded with plasticine, the beat frequency becomes 6 Hz. 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 1.00 m and 1.01 m produce 30 beats in 10 seconds.

Problem 27

Two similar sonometer wires of the same material produce 2 beats per second. The length of one is 50 cm and that of the other is 50.1 cm. 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 a sudden decrease in pitch (frequency) heard by a person standing in a railway station as a sounding train siren passes by him.

Case 1: Source Moving

• Motion of the source affects 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 moving towards a stationary observer O.

• Where is 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:

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

• Substitute equation (1) in equation (2).

2. Source Moving Away from a Stationary Observer

• Consider a source S of sound wave moving with velocity away from a stationary observer O.

• Where V = velocity of sound wave.
• Velocity of wave relative to observer at O = V + .
• The apparent wavelength reaching the observer at O is:

• The apparent frequency is given by:

• Substitute equation (4) in equation (5).

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:

• The apparent frequency is given by:

(i) Observer Moving Towards a Stationary Source

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

(ii) Observer Moving Away from Stationary Source

• Velocity of waves relative to observer = V – .
• The apparent frequency is given by:

Case 3: Source and Observer Are Moving

(i) Source and Observer Are Approaching

• Velocity of wave relative to observer is: V_o = V + U_o
• Apparent wavelength is .

(ii) Source and Observer Are Moving Away from Each Other

• Velocity of wave relative to observer is: V – U_o
• Apparent frequency is given by:

Problem 28

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

Problem 29

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

• (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 1000 Hz is traveling towards the right in still air with a velocity of 100 ft/s. At the right of the source is a large, smooth, reflecting surface moving towards the left at a velocity of 400 ft/s.

• (a) How far does the emitted wave travel in 0.01 second?
• (b) What is the wavelength of the emitted waves in front of (i.e., at the right of) the source?
• (c) How many waves strike the reflecting surface in 0.01 sec? Take the velocity in air as 1100 ft/s.

Problem 31

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

Problem 32

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

1. Before
2. After the train has passed by

Given that speed of sound in air = 340 m/s.

Problem 33

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

The Radar Speed Trap

• This is an instrument used to determine the speed of a moving car.
• The instrument sends out microwaves of frequency f (about 10.7 GHz) towards a moving car.
• The speed of the 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 acts 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

• Where Δf = 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 a car travels a radar speed trap at 30 m/s, the operating frequency of the speed trap being 10.7 GHz. Take the velocity of light to be 3 × 10⁸ m/s.

Problem 35

Calculate the change in frequency of the radar echo received from an aeroplane moving at 250 m/s 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 of light in free space then:

Expressions of Apparent Wavelength and Frequency

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

• Consider a source of light (e.g., a star) 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

Substitute equation (3) in equation (2)

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

• 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)

Substitute equation (3) in equation (2)

• It is clear from equation (8) that the apparent wavelength is less than the original wavelength.
• 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)

Applications of Doppler Effect

1. It is used in radar speed traps to determine the speed of a moving car.
2. It is used for measurement of speed of stars.
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 including:

1. Interference
2. Diffraction
3. Polarization

The Nature of Light

• Light is a form of energy which stimulates the sense of vision.
• It can be transmitted in air or vacuum with a speed of 3 × 10⁸ m/s.

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 consists of a stream of particles carrying energy with them called photons.
• This is like corpuscular theory but with a 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 the 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 waves 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 homogeneous medium, the wave fronts are spherical.

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

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 acts as a 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 envelope of the secondary wavelets at that time.

Where AA’ = original wave front

BB‘ = wave front after time 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 gives the portion of the new wave front OB‘.
• Those wavelets originating near the lower end of AA‘ strike the reflecting surface and get reflected 180° 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.