In this article, we shall study different modes of vibrations of string, the expression for its fundamental frequency of vibration.

Concept of Overtones:

• Whenever a string or an air column is set into vibrations, the vibrations consist of the fundamental frequency accompanied by certain higher frequencies. These higher frequencies are called overtones.
• They need not necessarily be all the integral multiples of the fundamental frequency.
• All overtones are present. Overtones are called first, second, etc. In case of vibrations of a string, the first overtone is the second harmonic second overtone is the third harmonic and so on.

Concept of Harmonics:

• Harmonics are simply integral multiples of the fundamental frequency. Thus if the fundamental frequency is n, the harmonics are 2n, 3n, 4n, etc.
• They are always the integral multiples of the fundamental frequency.
• They may be or may not be present in the given sound.
• There is nothing like first harmonic. If the fundamental frequency is n, 2n, is called second harmonic, 3n is called third harmonic, etc.
• In case of vibrations of string, the first overtone is the second harmonic second overtone is the third harmonic and so on.
• In case of air column vibrating in a pipe closed at one end only odd harmonics are present. In the case of an air column vibrating in a pipe open at both ends, all harmonics are present.

Vibrations of String:

If ‘T’ is the tension in the string and ‘m’ is its linear density, then the velocity of the transverse wave along its length of a stretched wire is given by the expression

Now let us consider a string of finite length and fixed at its two ends. The wave disturbance, originating at the point at which the string is plucked, travels up to the ends and is reflected back. The incident and reflected waves interfere to produce a stationary wave.  The particles of the string, at the two ends, which are fixed, always remain at rest.  Hence, the two ends of the string always become nodes.  However, the other particles in the string can vibrate in different ways giving rise to what is called the different modes of vibration of the string.

Vibrations of String (Fundamental Mode):

In the following figure, the string is shown to be vibrating in the simplest mode called the fundamental mode.  In this mode, the frequency of vibration is the least. The two ends are nodes and there is an antinode exactly midway between the two ends.  In other words, one complete loop is formed on the string.  Since the distance between the two consecutive nodes is λ/2, the length of the string (l).

This is the frequency of the fundamental mode of vibration of the stretched string.

Vibrations of String (First Overtone):

In the following figure, the string is shown to have broken up into two complete loops, there is a node midway between the two nodes and an antinode at a distance equal to a quarter of the length of the string from each end. This mode of vibration is called the first overtone.  If the wavelength corresponding to this mode is λ₁ and frequency is n₁.

This is the frequency of the first overtone of vibration of the stretched string. Thus the first overtone is the second harmonic.

Vibrations of String (Second Overtone):

In the following figure, the string is shown to have broken up into three complete loops. This is the second overtone. If the wavelength corresponding to this mode is λ₂and frequency is n₂. Then.

This is the frequency of the second overtone of vibration of the stretched string. Thus the second overtone is the third harmonic.

Thus we can conclude that, in the case of vibrating string

• The stretched string vibrates with all harmonics.
• The p^th overtone is (p + 1)^th harmonic.
• Frequency of p^th overtone = (p +1)n
• Frequency of p^th harmonic = pn
• Mode of vibration of string gives frequency n, 2n, 3n, …… so on.

Frequency of the Fundamental Mode in Terms of its Radius of Cross-section and Density of the Material of the String:

The frequency (n) of the fundamental mode of transverse vibration of a stretched string is given by

This is an expression for the fundamental mode of transverse vibration of a string in terms of its radius of cross-section and density of the material.

• For a given material of the wire, having a constant length and same tension, the frequency of transverse vibration is inversely proportional to its radius.
• For a wire having the same radius, same length and same tension, the frequency of transverse vibration is inversely proportional to the square root of the density of its material.

Frequency of the Fundamental Mode in Terms of Young’s Modulus of Elasticity of the Material of the String:

The frequency (n) of the fundamental mode of transverse vibration of a stretched string is given by

Substituting the value of equation (2) and (3) in (1)

This is an expression for the fundamental mode of transverse vibration of a string in terms of Young’s modulus of elasticity of the material.

Frequency of the Fundamental Mode in Terms of Coefficient of Thermal Expansion of the Material of the Wire:

The frequency (n) of the fundamental mode of transverse vibration of a stretched string is given by

m = area of the cross-section of wire  × density

∴   m = π r² ρ   ………… (2)

Now, T = F = tension in the wire

Substituting these values in equation (1)

This is an expression for the fundamental mode of transverse vibration of a string in terms of the coefficient of thermal expansion of the material.

Effect of dipping Stretching Load in a Liquid on Frequency of Vibrating Wire:

The specific gravity of a liquid is defined as the ratio of the weight of the body in the air to the loss of the weight of the body when immersed in the water. Thus, the specific gravity of a liquid is given by

This relation gives the new frequency of vibration of wire on immersing the load in a liquid of specific gravity σ.

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Science > Physics > Stationary Waves > Vibration of String

The post Vibration of String appeared first on The Fact Factor.

In this article, we shall study construction and working of sonometer, and its use to verify the laws of string.

Laws of Vibrating String:

Law of Length:

If the tension in the string and its mass per unit length of wire remains constant, then the frequency of transverse vibration of a stretched string is inversely proportional to the vibrating length.

Law of Tension:

If the vibrating length and mass per unit length of wire remain constant then, the frequency of transverse vibration of a stretched string is directly proportional to the square root of the tension in the string.

Law of Mass:

If the vibrating length and tension in the string remain constant then, the frequency of transverse vibration of a stretched string is inversely proportional to the square root of its mass per unit length

Sonometer:

Construction:

A sonometer consists of a hollow rectangular wooden box to which a uniform wire is attached at one end. The other end of the wire is passed over two horizontal knife edges or bridges and then over a pulley. A weight hanger is suspended from the free end of the wire. By placing different weights in the weight hanger, the tension in the wire can be suitably adjusted.

The points at which the wire rests on the knife edges cannot vibrate at all. Hence, when the wire is set up into vibrations, these two points become nodes and the wire vibrates in the fundamental mode. The frequency of vibration of the wire can be varied by either changing the positions of the knife edges by changing the vibrating length or by placing different weights in the pan by changing the tension.

Use of Sonometer to Determine the Unknown Frequency of a Tuning Fork:

To determine the unknown frequency of a tuning fork, the tension T in the wire is kept constant and the vibrating length between the knife edges is so adjusted, that the fundamental frequency of the wire becomes the same as that of the fork. To test this a small paper rider is placed on the wire midway between the knife edges where an antinode is formed.

The fork is set up into vibration and its stem is placed on the wooden box.  The length of the wire is adjusted till it vibrates in unison with the fork.  When this happens, the centre of the vibrating wire vibrates with maximum amplitude due to resonance, and the paper rider is thrown off. Then the frequency of the tuning fork which is the same as the fundamental frequency of the wire is given by

Where ‘m’ is the mass per unit length of the wire. The frequency of the fork is determined, knowing l, T, and m.

Use of Sonometer to Verify the Law of Length:

If the tension in the string and its mass per unit length of wire remains constant, then the frequency of transverse vibration of a stretched string is inversely proportional to the vibrating length.

To verify , the given wire (m = constant) is kept under constant tension (T = constant). A set of tuning forks having different frequencies n₁, n₂, n₃, n_4, etc. is taken. The length of the wire, vibrating in unison with each fork, is determined in turn using a paper rider or hearing beats. Let the lengths corresponding to the frequencies be l1, l2, l3, l4, etc.

Then, it is found that, within the limits of experimental error, n₁l₁ = n₂l₂ = n₃l₃ = n₄l₄ = constant. Thus in general n l = constant or n ∝ 1/l. If a graph of n  against  1/l  is plotted, it comes out as a straight line.

Use of Sonometer to Verify the Law of Tension:

If the vibrating length and mass per unit length of wire remain constant then, the frequency of transverse vibration of a stretched string is directly proportional to the square root of the tension in the string. To verify the law, the vibrating length of the given wire and linear density ‘m’ is constant.  A set of tuning forks having different frequencies n₁, n₂, n₃, n₄ etc. is taken.

By adjusting the tension T, each fork is made to vibrate in unison with the fixed length of the wire, one after the other. Let the tensions corresponding to the frequencies n1, n2, n3, n4, etc. be T1, T2, T3, T4, etc. respectively. Then it is found that, within limits of experimental error.

A graph of n² against T comes out as a straight line.

Use of Sonometer to Verify the Law of Mass:

If the vibrating length and tension in the string remain constant then, the frequency of transverse vibration of a stretched string is inversely proportional to the square root of its mass per unit length. This law cannot be verified directly, as either ‘n’ nor ‘m’ can be varied continuously as in the case of l or T. Therefore, this law is verified indirectly as follows.  The relation can be written as

Then, to verify the law, we must show that when n and T are kept constant.  A number of wires having linear densities m₁, m₂, m₃, m_4, etc. are taken.  Each one of them is subjected to the same tension T. Then, using a given tuning fork (n = constant) each wire is made to vibrate in unison with the fork, by adjusting its length. Let l1, l2, l3, l4, etc. be the vibrating length corresponding to linear densities m1, m2, m3, m4, etc. respectively.

Then it is found that, within the limits of experimental error.

Hence the law is indirectly verified.

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Next Topic: Melde’s Experiment

Science > Physics > Stationary Waves > Sonometer Experiment

The post Sonometer Experiment appeared first on The Fact Factor.

In this article, we shall study Melde’s experiment of vibrating string and find the relation between the number of loops and the tension in the string.

Melde’s Experiment for Longitudinal or Parallel Position:

Melde’s experiment set up consists of a light string is tied to one of the prongs of a tuning fork which is mounted on a sounding board. The other end of the string is passed over a horizontal pulley and a light pan is suspended from the free end. The tension in the string can be adjusted by changing the weights placed in the pan, while the vibrating length can be altered by moving the pulley towards or away from the fork.

Experimental Arrangement:

The fork is adjusted so that its arms vibrate parallel to (longitudinal position) the length of the string. The fork is set into vibrations by gently hammering a prong, and the vibrating the length or the tension in the string is adjusted to obtain a number of clear loops on the string. The loops are formed due to interference between the wave starting from the fork and traveling towards the pulley and the wave reflected back from the pulley.

When the prongs vibrate parallel to the length of the string, the frequency of the fork is twice the frequency of the string. Since the string is vibrating in the fundamental mode, its frequency ‘n’ is given by

Therefore the frequency of the fork N will be

Hence by measuring the length of the string, the tension T, the mass per unit length m, and by counting the number of loops p, the frequency N of the tuning fork can be found out.

To Prove p²T = Constant:

Frequency of a tuning fork is given by

Melde’s Experiment for Transverse or Perpendicular Position:

In Melde’s experiment setup a light string is tied to one of the prongs of a tuning fork which is mounted on a sounding board. The other end of the string is passed over a horizontal pulley and a light pan is suspended from the free end. The tension in the string can be adjusted by changing the weights placed in the pan, while the vibrating length can be altered by moving the pulley towards or away from the fork.

The fork is adjusted so that its arms vibrate perpendicular to (transverse position) the length of the string. The fork is set into vibrations by gently hammering a prong, and the vibrating the length or the tension in the string is adjusted to obtain a number of clear loops on the string. The loops are formed due to interference between the wave starting from the fork and traveling towards the pulley and the wave reflected back from the pulley.

When the prongs are arranged so that they vibrate at a right angle to the length of the string, the frequency of fork is the same as the frequency of the string. Since the string is vibrating in the fundamental mode, its frequency ‘n’ is given by

Hence by measuring the length of the string, the tension T, the mass per unit length m, and by counting the number of loops p, the frequency N of the tuning fork can be determined.

To Prove p²T = Constant:

Frequency of a tuning fork is given by

To Show that the number of loops is doubled when parallel position is changed to the perpendicular position:

Suppose that the fork is adjusted in the longitudinal position and p₁ loops are obtained on the string. Then,

Next, without changing the value of or T, the fork is adjusted in the transverse position and suppose that p₂ loops are obtained on the string.

From equation (1) and (2) we get

Thus, the number of loops in the transverse position is twice the number of loops in the longitudinal position for the same values of l and T.

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Next Topic: Vibrations of Air Column

Science > Physics > Stationary Waves > Melde’s Experiment

The post Melde’s Experiment appeared first on The Fact Factor.

In this article, we shall study the vibrations of air columns in a pipe closed at one end, opened at other and in a pipe open at both ends.

Vibrations of Air Columns in a Pipe Closed at One End: 

Consider a pipe closed at one end and open at the other. If a vibrating tuning fork of proper frequency is held near its open end, it sends longitudinal waves in the pipe. They are reflected from the closed end. The incident and reflected waves interfere and stationary waves are formed.

The closed-end acts as a rigid wall. The air at the closed end is not free to vibrate. Hence a node is formed at the closed end. Air at the open end is free to vibrate with maximum amplitude and antinode is formed at the open end. This is the simplest mode of vibration of an air column closed at one end and is called the fundamental mode.

If v is the velocity of sound in air at room temperature, then

v = n λ

In the fundamental mode     L = λ/4, hence λ = 4 l

∴ V = nλ = 4 n l

∴ n = v /4 l

This is the frequency of the fundamental mode. Where ‘l’ is length of vibrating air column

Now consider the following figure. The same air column can also be made to vibrate so that there is a node at the closed end, an antinode at the open end, with one more node and antinode in between. In this case, the air column is said to be emitted the first overtone. If n₁ is the frequency and λ₁ the wavelength for the first overtone,

Thus the first overtone is the third harmonic.

Now consider the following figure. In this case, the air column is shown emitting the second overtone. If n₂ is the frequency and λ₂­ the wavelength for the second overtone, then

Therefore, the second overtone is the fifth harmonic.

Thus, for an air column closed at one end, only the odd harmonics are present as overtones.

For air column in a pipe closed at one end p^th overtone is (2p + 1) th harmonic and the frequency of p^th overtone = (2p+1)n.

Vibrations of Air Column in a Pipe Open at Both Ends:  

Consider a tube open at both ends. A vibrating tuning fork held near anyone end produces longitudinal waves in the air column. They are reflected from the other end. The incident and reflected waves interface with each other and stationary waves are formed.

When an air column open at both ends is set up into vibrating, both the ends become antinodes. In the fundamental mode, as shown in the following figure, there is only one node midway between the two antinodes. Therefore, the length of the air column,

This is the fundamental frequency.

Now consider the following figure. When the air column is emitting the first overtone, there are two nodes and one antinode in between the two antinodes at the ends. Therefore, if n₁ is the frequency and λ₁ is the wavelength for the first overtone,

Hence, in this case, the first overtone is the second harmonic.

In the following figure, the air column is shown emitting the second overtone. If n₂ is the frequency and λ₂ the wavelength for the second overtone, then

The second overtone is the third harmonic.

Thus for an air column open at both ends, both odd, as well as even harmonics, are present as overtones i.e. all harmonics are present.

For air column in a pipe open at both ends p^th overtone is (p + 1)^th harmonic and the frequency of p^th overtone = (p+1)n.

Note:

The frequency of the fundamental note of an open pipe is double that for a closed pipe of the same length.

End Correction:

It was shown by Regnault, that the antinode is not formed exactly at the open end but at a distance 0.3 d above the open end where d is the internal diameter of the tube. This additional distance called the end correction,

End correction is necessary, as the wave spreads out slightly, just above the open end, and the air particles just outside the open end are also set into vibrations. For a pipe open at one end and closed at the other

Corrected length = l + 0.3d

Hence, the velocity of sound in air at room temperature is

∴   v = nλ = 4n (l + 0.3d)

Hence, knowing n, l and d, the velocity of sound in air is determined.

For a pipe open at both ends, end correction = 2 × 0.3 d = 0.6 d

Corrected length = l + 0.6 d

Hence, the velocity of sound in air at room temperature is

∴   v = nλ = 2n ( l  + 0.6d)

Elimination of End Correction:

By the usual method, we find the first resonating length (l₁). Similarly, we find the second resonating length (l₂) for which a loud sound is heard once again. If “e” is the end correction, then

l₁ + e = λ/4    …………. (1)

l₂ + e = 3λ/4    …………. (1)

Subtracting equation (1) from (2)

l₂ –  l₁ = 3λ/4 – λ/4

∴    l₂ –  l₁ = λ/2

∴  λ = 2( l₂ –  l₁ )

Now, v = n λ = 2n( l₂ –  l₁ )

Using this formula velocity of sound can be found without considering end correction.

End Correction for a Pipe Closed at One End:

If v is the velocity of sound in air, λ is the wavelength of sound. ‘l’  is the vibrating length of a pipe closed at one end, n is the frequency of vibration in the fundamental mode which is equal to the frequency of the tuning fork at time of resonance. Let ‘e’ be the end correction. This end correction depends on the diameter of the tube. we have

V = nλ=  4n ( l + e)

For the first case

V = 4n₁ ( l₁+ e)    ………… (1)

Where l₁ is a length of vibrating air column vibrating in unison with a tuning fork of frequency n₁.

For the second case

V = 4n₂ ( l₂+ e)    ………… (2)

Where l₂ is a length of vibrating air column vibrating in unison with a tuning fork of frequency n₂.

From (1) and (2)

4n₁ ( l₁+ e) =  4n₂ ( l₂+ e)

∴    l₁n₁  +  en₁ =  l₂n₂  +  en₂

∴    l₁n₁  – l₂n₂ =  en₂ – en₁

∴    l₁n₁  – l₂n₂ =  e(n₂ – n₁)

End Correction for a Pipe Open at Both Ends:

If v is the velocity of sound in air, λ is the wavelength of sound.  ‘l’ is the vibrating length of a pipe open at both the ends, n is the frequency of vibration in the fundamental mode which is equal to the frequency of the tuning fork at time of resonance. Let ‘e’ be the end correction. This end correction depends on the diameter of the tube. we have

V = nλ=  2n ( l + 2e)

For the first case

V = 2n₁ ( l₁+ 2e)    ………… (1)

Where l1 is a length of vibrating air column vibrating in unison with a tuning fork of frequency n₁.

For the second case

V = 2n₂ ( l₂+ 2e)    ………… (2)

Where l2 is a  of vibrating air column vibrating in unison with a tuning fork of frequency n₂.

From (1) and (2)

2n₁ ( l₁+ 2e) =  2n₂ ( l₂+ 2e)

∴    l₁n₁ +  2en₁ =  l₂n₂  +  2en₂

∴    l₁n₁  – l₂n₂ =  2en₂ – 2en₁

∴    l₁n₁  – l₂n₂ =  2e(n₂ – n₁)

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Science > Physics > Stationary Waves > Vibrations of Air Columns

The post Vibrations of Air Columns appeared first on The Fact Factor.

In this article, we shall study the phenomenon of resonance, its characteristics, advantages, and disadvantages.

Free Vibrations:

A body or a system capable of vibrating, when displaced from its position of rest, vibrates with a certain definite frequency. This frequency is characteristic of the body or the system. Such oscillations are called free oscillations or free vibrations and the frequency of such oscillations is called the natural frequency of the body or the system.

• Example – 1: When a wire under tension, which is fixed at its ends, is plucked and released, it vibrates with a frequency which depends on the length of the string, its mass per unit length and tension in the string.
• Example – 2: When the bob of a simple pendulum oscillates, its frequency of oscillation depends on the length of the pendulum.

Due to frictional force, the amplitude of oscillation decreases continuously and finally, the body stops vibrating.

Characteristics of Free Vibrations:

• They are produced when a body capable of vibrating is disturbed from its normal equilibrium position and then released.
• The frequency of vibration depends on the body and is called natural frequency.
• The frequency of vibration is the same as the natural frequency of the body.
• The amplitude of vibration is large.
• Vibration continues for a little more time after the external force is removed.
• Example: Oscillations of bob of the pendulum

Forced Vibrations:

Forced vibrations are the vibrations produced in a body by applying an external periodic force having a frequency, normally different from the natural frequency of the body.

A body or a system, capable of vibrating can also be made to vibrate at any desired frequency. The body can be made to vibrate with the same frequency as the frequency of the applied periodic force. Suppose that the natural frequency of a metal vessel is 200 Hz.  If a tuning fork of frequency 256 Hz is set up into vibrations and its stem is placed in contact with the vessel, then the vessel will be forced to vibrate at a frequency of 256 Hz.  In such a case, the vessel is said to perform forced vibrations.

Initially, the body tends to vibrate with its natural frequency. But very soon, the natural vibrations die out and it begins to vibrate with the frequency of the applied periodic force.

The amplitude of the forced vibrations depends on:

• The difference in frequencies of the external force and the natural frequency of the body.
  • The amplitude of the applied force.
  • damping.

Characteristics of  Forced Vibrations:

• They are produced when an external periodic force acts on the body.
• The frequency of vibration is the same as the frequency of external periodic force.
• The frequency of vibration is different from the natural frequency of the body.
• The amplitude of vibration is small.
• Amplitude becomes zero as soon as the external force is removed.
• Example: A vibrating tuning fork on a wooden box, a musical instrument having a soundboard or box.

Resonance:

The amplitude of the forced vibrations depends on the difference between the natural frequency of the body and the frequency of the applied periodic force. When the difference between the two frequencies is large, the response of the body is poor or the forced vibrations are of small amplitude. When the frequency difference becomes smaller, the body vibrates more readily or the amplitude of the forced vibrations increases. Finally, when the frequency (f) of the applied periodic force becomes the same as the natural frequency fo of the body, the amplitude of the forced vibrations becomes maximum and the phenomenon is known as resonance.

If anybody is made to vibrate, by an external periodic force, with a frequency which is the same as the natural frequency of the body, the body begins to vibrate with a very large amplitude. This phenomenon is called resonance.

Distinguishing Between Forced Vibrations and Resonance:

Advantages of Resonance:

• Resonance is useful to determine an unknown frequency.
• Resonance is useful to increase the intensity of sound in musical instruments.
• Resonance is useful to tune a radio receiver to any desired frequency.
• Resonance is useful to analyze musical notes.

Disadvantages of Resonance:

Soldiers are asked to break steps when crossing a bridge. It can be explained as follows

Soldiers marching on a bridge take steps with definite frequency and force the bridge to vibrate with the frequency of the steps.

If the forced frequency on the bridge is equal to the natural frequency of vibration of the bridge, the bridge is set into resonant vibrations.

Due to the resonance, the bridge vibrates with higher amplitude and due to this, it may collapse.

Due to the rhythmic clapping of the audience, the roof of the stadium may collapse. It can be explained as follows

When the audience claps rhythmically they do so with a certain frequency and force the roof of a stadium to vibrate with the frequency of the clap.

If the forced frequency on the roof of a stadium is equal to the natural frequency of vibration of the roof of a stadium, the roof of a stadium is set into resonant vibrations.

Due to the resonance, the roof of a stadium vibrate with higher amplitude and due to this, it may collapse.

When the speed of an aircraft increases, different parts are forced to vibrate. which is dangerous for the structure of the aircraft.

Resonance Tube Experiment to Determine Velocity of Sound in Air:

A metal tube, open at both the ends is immersed in a tall glass jar filled with water. An air column is thus formed between the open end of the metal tube and the surface of the water. The length of the air column, which is closed by the water surface at the lower end, can be varied by raising or lowering the metal tube. A tuning fork is set up into vibrations and held near the mouth of the tube so that its arms vibrate parallel to the axis of the tube. The longitudinal wave, starting from the tuning fork, travels along the length of the air column and is reflected back from the surface of the water. The incident and reflected waves interfere to produce a stationary wave.

The molecules of air, in contact with the water surface, remain at rest. Therefore, the closed-end becomes a node. The molecules of air, near the mouth of the tube, vibrate with maximum amplitude. Therefore, the open end becomes an antinode. The frequency of the air column can be changed by adjusting its length. When its frequency becomes the same as the frequency of the fork, resonance takes place, and a loud sound is heard.

If the length of the air column is increased from a small value, the first resonance occurs when there is a node at the closed end and an antinode at the open end, with no other nodes or antinodes in between. Therefore, the length of the air column  l = λ/4

∴   λ = 4 l

Due to resonance, the frequency of the air column is the same as that of the fork. Now velocity of sound is given by v = nλ

∴  v =   4 n l

End correction :

It was shown by Regnault, that the antinode is not formed exactly at the open end but at a distance 0.3 d above the open end where d is the internal diameter of the tube. This additional distance, called the end correction, is necessary, as the wave spreads out slightly, just above the open end, and the air particles just outside the open end are also set into vibrations.

∴ corrected length = l + 0.3d

Hence, the velocity of sound in air at room temperature is

V = nλ = 4n (l + 0.3d)

Hence, knowing n, l, and d, the velocity of sound in air is determined.

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Science > Physics > Stationary Waves > Resonance

The post Resonance appeared first on The Fact Factor.

In this article, we shall study the Doppler effect in case of sound waves and a brief idea of the Doppler effect in case of light.

Doppler Effect:

The apparent change in the frequency of the sound heard by an observer, due to relative motion between the source of the sound and the observer, is called the Doppler Effect.

Example:

Case – I: When a moving train blowing whistles approaches the observer standing on the railway platform, the observer hears the sound of higher frequency than the actual frequency of the whistle.

Case – II: When the moving train blowing whistles recedes away from the observer standing on the railway platform, the observer hears the sound of lower frequency than the actual frequency of the whistle. Here we should note that the frequency of sound emitted by the whistle is not changing. It is the observer who is hearing different frequencies. This effect is known as the Doppler Effect.

Explanation:

When the railway engine is at rest, then the number of waves reaching the observer are constant and is equal to the actual frequency of the horn blown by the engine.

Case – I: Consider the case that the observer is stationary on a railway platform and a train blowing whistle approaches him. Due to this the number of waves reaching to observer per second increases. Due to which the apparent frequency heard by the observer is greater than the actual frequency of the sound.

The same effect is observed in all the following cases. Observer Stationary and source is moving towards the observer or Source is stationary and the observer is moving towards the source or both the source and observer are moving towards each other

Case – II: Consider the case that the observer is stationary on a railway platform and a train blowing whistle recedes away from him. Due to this the number of waves reaching to observer per second decreases. Due to which the apparent frequency heard by the observer is less than the actual frequency of the sound.

The same effect is observed in all the following cases. Observer Stationary and source is moving away from the observer or Source is stationary and the observer is moving away from the source or both the source and observer are moving away from each other

Applications of Doppler Effect:

• In colour Doppler sonography, the ultrasonic waves refracted from body tissues can give information about the rate of flow of various fluids including blood.
• RADAR (RAdio Detection And Ranging instrument) is used to locate moving objects like ship, aeroplanes, tank. Using the instrument the distance and speed of moving object can be determined.
• Doppler effect is used to determine the velocity and speed of rotation of astronomical objects like a star.
• It is used to determine the speed of rotation of the sun.
• A traffic police usea speed detection instrument, which works on the principle of the Doppler effect and is used to determine the fast-moving vehicles.

Limitations of the Doppler Effect:

• The Doppler effect is applicable when the velocities of the source of sound and observer are much less than the velocity of sound.
• The motion of both the observer and the source is along the same straight line.
• The medium such as air, in which the observer and source are situated at rest. If the direction of motion are different or wind is blowing, modification in formulae is required.

Different Formulae Used in Doppler Effect:

V_L = Velocity of the listener /observer

V_S = Velocity of source

V = Velocity of sound

n_a  = Apparent frequency

n = True Frequency

(+) sign in the numerator and (-) sign in denominator indicate that the source and listener are moving towards each other.

(-) sign in the numerator and (+) sign in denominator indicates that the source and listener are moving away from each other.

Condition – I: Listener is in motion and source at rest (V_S = 0)

Listener moving towards (approaching) the source:

Listener moving away from (receding) the source:

Condition – II: Source is in motion and Listener at rest (V_L = 0)

Source moving towards (approaching) the Listener:

Source moving away from (receding) the Listener:

Condition – III: Both source and listener Moving:

Source and listener approaching each other:

Source and listening receding from each other:

Compensation For Velocity of Wind:

If the direction of the wind is the same as the direction of the sound, then quantity ‘v’ in the formula should be replaced by (v + v_w)

If the direction of the wind is opposite to that of the direction of the sound, then quantity ‘v’ in the formula should be replaced by (v – v_w)

Where v_w is the velocity of the wind.

If the direction of the wind is making some angle with the direction of the sound then the component of the velocity of wind in the direction of sound should be taken.

Doppler Effect in Light:

The apparent change in the frequency of the light observed by an observer, due to relative motion between the source of the light and the observer, is called the Doppler effect.

One major difference between the Doppler effect exhibited by sound and light is as follows. In the case of sound, the frequency change depends on whether the source is moving or the observer is moving even if their relative velocities are the same. In the case of light, the Doppler effect depends only on the relative velocity of the source and the observer, irrespective of which of the two is moving. Hence the Doppler effect exhibited by light is symmetric.

Red Shift and Blue Shift of Light:

When the source and observer move away from each other, the wavelength in the middle of the spectrum will be shifted towards the red. This phenomenon is called redshift due to the Doppler effect. When the source and observer move away from each other, the observer observes the lower frequency than the actual frequency of the light (towards red).

When the source and observer move towards each other, the wavelength in the middle of the spectrum will be shifted towards blue. This phenomenon is called the blue shift due to the Doppler effect. When the source and observer move towards each other, the observer observes the higher frequency than the actual frequency of the light (towards blue).

The measurement of the Doppler shift helps in the study of motions of stars and galaxies.

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