Physics Oscillation and Waves For NDA

Topics Covered

• Wave
• Longitudinal Waves
• Transverse Waves
• Wave Motion
• Sound
• Production of Sound Waves
• Propagation of Sound
• Characteristics of Sound Wave
• Range of Hearing
• Sonic Boom
• Speed of Sound in Different Media
• Effect of Physical Parameters on Speed of Sound
• Echo
• Relation between Speed of Sound, Time of Hearing Echo and Distance of Reflecting Body
• Audible, Infrasonic and Ultrasonic Waves
• Ultrasound
• Sonar
• Speed of Longitudinal Waves (or Sound) in Gases : Newton Formula
• Laplace's Correction
• Superposition Of Waves
• Superposition Of Waves
• Standing Waves in String
• Vibration in Closed Organ Pipe
• Vibration in Open Organ Pipe
• Doppler's Effect in Sound

Wave

A wave is a vibratory disturbance in a medium which carries energy from one point to another point without any actual movement of the medium. There are mainly three types of waves

`"(i) Mechanical Waves"`
The waves which can be propagated or produced only in a material medium, are called mechanical waves. Mechanical waves are of two types- 1. Longitudinal Waves 2. Transverse Waves

`"(ii) Electromagnetic Waves"`
The waves which require no medium for their propagation or production, are called electromagnetic waves.

`"(iii) Matter Waves"`
The waves associated with moving particles like electrons, protons, etc, are called matter wave.

Longitudinal Waves

• A wave in which the particles of the medium vibrate in the same direction of propagation of wave is called longitudinal wave.

• Longitudinal waves can be produced in all the three media such as solids, liquids and gases.

• The waves which are produced in air are always longitudinal.

`"Example"` Wave travel along a spring when it is pushed and pulled at one end, are longitudinal waves.
When coils are closer to each other than normal, compression are observed in the spring. When coils are farther apart than normal, rarefactions are observed. A long feasible spring which can be compressed or extended easily, is called sinky.

`"Note"`
When a longitudinal wave passes through air, the density or air changes continuously and the pressure and energy are being transferred.

Transverse Waves

• A wave in which the particles of the medium vibrate perpendicular to the direction of propagation of wave, is called transverse wave.

• Transverse waves can be produced only in solids and liquids e.g., Light is a transverse wave but it is not a mechanical wave.

• The waves produced by moving one end of long spring or rope up and down rapidly and whose other end is fixed, are transverse waves.

• The water waves (or ripples) formed on the surface of water in a pond (when a stone is dropped in the pond of water), are transverse waves.

• A transverse wave travels horizontally in a medium and the particles of the medium vibrate up and down in the vertical direction. In transverse waves, crest and trough are formed.

• A crest is that part of the transverse wave which is above the line of zero disturbance of the medium. A trough is that part of the transverse wave which is below the line of zero disturbance.

• A transverse wave has been represented by a displacement-distance graph as shown below

Comparison between Transverse waves and Longitudinal waves

Wave Motion

When a large number of particles vibrates simultaneously in a medium, then disturbance propagates in the medium. The notion of disturbance is called wave motion. Energy of momentum is transferred to the neighbouring particles of the medium as wave proceeds. The transfer of energy in the form of waves is known as "wave motion".

Some definitions are given below related to wave motion.

`text(Amplitude )`
It is the maximum displacement suffered by the particles of the medium about their mean positions. It is denoted by `A.`

`text(Time Period)`
The time period of a wave is the time in which a particle of medium completes one vibration to and fro about its mean position. It is denoted by `T.`

`text(Frequency )`
The frequency of a wave is the number of waves produced per unit time in the given medium. It is equal to the reciprocal of the time period T of the Particle and is denoted by n. Thus
`n = 1/T`

S.I unit of n is `S^-1` or hertz (Hz)

`text(Angular Frequency)`
The rate of change of phase with time is called angular frequency of the wave. It is denoted by `omega`. Thus
`omega = (2 pi ) /T = 2 pi n`

SI unit of `omega` is `"rad"` `s^-1`

`text(Wavelength )`
The distance between two nearest particles of the medium which are vibrating in the same phase. It is denoted by `lamda`.

`text(Wave Number)`
The number of waves present in a unit distance of the medium is called wave number. It is equal to the reciprocal of wavelength `lamda.`
Thus, Wave number `bar v = 1/lamda`

SI unit of wave number is `m^-1`

`text(Angular wave number of propagation constant)`
The quantity `(2pi)/ lamda` is called angular wave number or propagation constant of a wave. It represents phase change per unit path difference. It is denoted by K.
Thus , `K = (2 pi) /lamda`

The SI unit of K is radian/metre or `"rad"` `m^-1`

`text(Wave Velocity or Phase Velocity )`
The distance covered by a wave per unit time in its directions of propagation is called its wave velocity or phase velocity. It is denoted by v.
`"Wave Velocity" = "Wavelength of the wave"/"Time period of the wave"`
`v=lamda/T` or `v=n xx lamda`

Sound

• Sound is a form of energy which effects our sensation of hearing through the ear. The sensation is produced by longitudinal waves in· an elastic medium, where the vibrations (oscillations) of the particles are in the same direction in which the wave propagates. The waves strikes our eardrum and makes it vibrates as a result of which we are able to hear.

• Sound waves need a medium to travel.

• The speed of sound in a medium depends on the the elasticity and density of the medium. Both elasticity and density of the solids are very large as compared to that of the liquids and gases.

Production of Sound Waves

In laboratory sound is produced by a tuning fork by striking its one prong on a soft rubber pad. Sound can also be produced by plucking a stretched string (violin), by blowing flute, by striking table and many other ways.

Propagation of Sound

Take a tuning fork (a source of standard frequency). It is set into vibrations and its prongs A and B are kept vertical. The prongs move in and out from their mean positions and have a transverse vibratory motion. When the prongs are in mean positions, the air in their surrounding has normal density. (It is shown in figure 3.10(a) with equidistant lines).

As the right prong moves out towards right, it pushes the air layers to the right. This produces a compression (It is shown in figure 3.10(b) with closer lines).

The prong returns inward to mean position. The compression moves to the right. The air near the prong again has normal density as shown in figure 3.10(c).

As the prong continues moving towards extreme left, vacating the space, density of air falls in the region and a rarefaction is produced (It is shown in figure 3.10(d) with spread lines).

As the prong moves back to right extreme, it competes one vibration. Also the motion of the prong produces a new compression. This completes one wave.

Since one vibration of the prong has generated one wave in the medium (air), in one second and many waves will be generated equal to the number of vibrations that the tuning fork will make in one second. This number is called frequency of the tuning fork (This number is engraved on the tuning fork near the bend). Hence we conclude that the wave frequency (the number of waves being generated per second) is equal to the frequency of the tuning fork.

Characteristics of Sound Wave

`"Speed"`
The speed of sound basically depends upon elasticity and density of medium speed of sound in air is 332 m/s, in water is 1483 m/s and in iron is 5130 m/s. When sound enters from one medium to another medium, its speed and wavelength changes but frequency remains unchanged.

`"Pitch"`
• Pitch is the sensation (brain interpretation) of the frequency of an emitted sound.

• Faster the vibration of the source, higher is the frequency and higher is the pitch. Similarly low pitch sound corresponds to low frequency.

• A high pitch sound is called a shrill sound (E.g. a female voice, humming of a bee, sound of guitar etc.)

• A low pitch sound is called a hoarse sound (E.g. a male voice, roar of a lion, car horn etc.)

`"Loudness or Softness"`
• Loudness or softness of sound wave is the sensation that depends upon its amplitude.

• When we strike a table top with more force, it vibrates and produces loud sound waves which have more amplitude. When struck with smaller force, vibrating table top produces soft sound waves which have less amplitude.

• A loud sound wave carries more energy and can be heard at large distance.

• Reduction in amplitude at large distance, makes the sound soft.

• Loudness can be compared with the intensity of sound. Higher the intensity, louder the sound.

`"Timbre or Quality"`
• We differentiate between the sound from a Tabla and Gitar by saying that they have different qualities. Quality or timbre is characteristic of a sound which enables us to distinguish between the sound of same loudness and pitch.

• This characteristic of sound helps us to recognize our friend from his voice without seeing him.

• The quality of two sounds of same loudness and pitch produced by two different sources are distinguishable because of different wave forms produced by them. The wave forms produced by a vibrating tuning fork, violin and flute (Bansuri) are shown in figure 3.14.

`"Intensity"`
• Intensity of a sound is defined as the sound energy transferred per unit area placed perpendicular to the direction of the propagation of sound.

• `"Intensity of sound" = "sound energy"/"Time x Area"`

• Intensity of a sound is an objective physical quantity. It does not depend on the response of our ears.

• The S.I. unit of intensity of sound is `"joule" s(-1)m(-2)` or `"watt" m(-2)` (`.: 1 Js(-1)=W`)

`"Note"`
Frequency of the wave depends on the source from where the wave originates. In reflection and transmission, since source does not change, frequency and hence time period does not change. As sound is transmitted from one medium to another, the speed and wavelength of the sound change, but not the frequency.


Range of Hearing

• The human ear is able to hear sound in a frequency range of about 20 Hz to 20 kHz.

• These limits vary from persons to person and with age. Children can her sounds of somewhat higher frequencies, say up to 30 kHz. With age, our ability to hear high frequency sound diminishes. For the elder, the upper limit often falls to 10-12 kHz. We take 20 Hz-20 kHz as the audible range for an average person.

• Even in the audible range the human ear is not equally sensitive for all frequency. It is most sensitive to frequencies around 2000-3000 Hz.

• Sound of frequencies less than 20 Hz is known as infrasonic sound or infra sound. Sound of frequency greater than 20 kHz is known as ultrasonic or ultrasound.

• Different animals have different ranges of audible frequencies.
A dog can hear sound of frequencies up to about 50 kHz and a bat up to about 100 kHz.
Dolphins can hear sounds of even higher frequencies.
Animals such as elephants and whales can hear sounds of frequencies less than 20 Hz. Some fishes can hear sounds off frequencies as low as 1-25 Hz.

Sonic Boom

• When a body moves with a speed which is greater than the speed of sound in air, it is said to be traveling at supersonic speed. Jet fighters, bullets, etc, often travel at supersonic speed. And when they produce sound, they produce a sharp, loud sound called a sonic boom.

The source moves at a speed greater than that of sound waves. The sound waves traveling at the speed of sound, are left behind. The high-pressure layers due to sound waves originating at different points bunch together as shown in figure 3.15. Actually, these layers fall on the surface of an imaginary cone of which OA, OB is a part. The total pressure on the surface of this cone is very high.

• The source is at the apex of this cone. As the source moves ahead, it drags the cone together with it. When the surface of the cone reaches a person, the ears experience a sudden increase in pressure. After the surface crosses him, the pressure is suddenly reduced. This causes the person to hear a sharp, loud sound-the sonic boom.

• A region consisting of a very-high-pressure layer followed by a lower-pressure layer travels through the space together with the cone. This is called a shock wave. This shock wave give rise to the sonic boom when it reaches a person.

• The shock waves produced by supersonic aircraft have enough energy to shatter glass and even damage weak buildings.

`"Note"`
In a stationary wave, all particles of the medium have the same phase at a given instant but have different amplitudes.

Reflection of Sound

When sound waves strike a surface, they return back into the same medium. This phenomenon is called reflection.

The reflection of sound waves is similar to that of light rays. 'The only difference is that sound waves being larger in length, require bigger surfaces for reflection

`"Laws of Reflection"`
(i) Angle of incidence is equal to the angle of reflection.
(ii) 'The incident wave, the reflected wave and the normal, all lie in the same plane.

`ul"Applications of Reflection of Sound"`

`"Mega phone or speaking tube"`
When we have to call someone at a far off distance (say 100 m), we cup our hands and call the person with maximum sound we can produce. The hands prevent the sound energy from spreading in all directions. In the same way, the people use horn shaped metal tubes, commonly called megaphones. The loud speakers have horn shaped openings. In all these devices, the sound energy is prevented from spreading out by successive reflections from the horn shaped tubes. Horns, musical instruments such as trumpets and shanghais, loudhailers (megaphones), loudspeakers etc. are all designed similarly to keep sound from spreading in all directions. In these instruments a tube followed by a conical opening reflects sound successively to guide most of the sound waves from the source in the forward direction, towards the audience. The sound wave add up and the loudness of sound increases.

`"Stethoscope"`
It is an instrument used by the doctors for listening sound produced within the body, especially in the heart and lungs. In the stethoscope, the sound produced within the body of a patient is picked up by a sensitive diaphragm and then reaches the doctor's ears by multiple reflection. There is a little loss in energy.

`"Sound board"`
The sound waves obey the laws of reflection on the plane as well as curved reflecting surfaces. In order to spread sound evenly in big halls or auditoriums, the speaker (S) is fixed at the principle focus of the concave reflector. This concave reflector is commonly called sounding board. The sound waves striking the sound board get reflected parallel to the principal axis.

`"Ceilings of Auditoriums"`
Generally the ceilings of concert halls, conference halls and cinema halls are curved so that sound after reflection reaches all corners of the hall, as shown in figure 3.20.

Speed of Sound in Different Media

Sound travels with different speeds in different media like solid, liquid and gas. This is because, sound travels in a medium due to the transfer of energy from one particle to another particle of the medium.

`"Solid"` Since the particles of solid are close to each other, so transfer of energy from one particle to another takes place in less time (i.e. faster). Hence speed of sound in solids is large.

`"Liquid"` Speed of sound in liquids in less than in solids since the particles are away from each other as compared to solids.

`"Gas"` Speed of sound in gases is less than the speed in liquids and solids as the particles are far away from each other as compared to solids and liquids.

Effect of Physical Parameters on Speed of Sound

`"Effect of Temperature"`
• Sound travels faster as the temperature of the medium increases and vice-versa. This happens because as temperate increases, the particles of the medium collide more frequently and hence the disturbance spreads faster.

• Speed of sound in air increases by 0.61 m/s with every 1"C increases in temperature. For example if speed of sound in air at 0"C is 330 m/s, then its speed at 25°C will be 345 m/s.

`"Effect of Pressure"`
If temperature remains constant, then there is no effect of change in pressure on the velocity of sound.

`"Effect of Humidity"`
In humid air, velocity of sound increases as compared to the dry air.

`"Effect of Frequency"`
There is no effect of frequency on the velocity of sound.

`"Effect of Wind "`
If wind is blowing, then the speed of sound changes. The speed of sound is increased, if wind is blowing in the direction of propagation of sound wave.

Echo

When a person shouts in a big empty hall, we first hear his original sound, after that we hear the reflected sound of that shout. This reflected sound is known as echo. An echo is nothing but just the reflected sound. So, the repetition of sound caused by reflection of sound waves is called an echo. It is of three types:
(a) Instantaneous echo
(b) Syllabic echo
(c) Successive echo

Relation between Speed of Sound, Time of Hearing Echo and Distance of Reflecting Body

If t is the time at which an echo is heard, d is the distance between the source of sound and the reflecting body and v is the speed of sound. The total distance traveled by the sound is 2d.

Speed of sound, `v= (2d)/(t) \ \ \ \or d = (vt)/(2)`

`(a) "Calculation of Minimum Distance of Hearing Echo"` d is minimum distance required for hearing an echo when persistence of hearing is `1/15` second The velocity of sound (at room temperature) is `340 m//s`

So, `d = (vt)/(2) = (340)/(2) xx1/15= (22.67)/(2)`

So 11 meters is the minimum distance of hearing echo.

`(b) "Conditions for Formation of an Echo"`
• The minimum distance between the source of sound and the reflecting body should be 11 metres.

• The wavelength of the sound should be less than the height of the reflecting body.

• The intensity of sound should be sufficient so that it can be heard after reflection.

Audible, Infrasonic and Ultrasonic Waves

`"Audible Range"`
The human ear is sensitive to sound waves of frequency between 20 Hz to 20 kHz. This range is known as audible range. E.g. By vibrating sitar, guitar, organ pipes, flutes, shehnai etc.

`"Infrasonic Wave"`
A longitudinal elastic wave whose frequency is below the audible range i.e. 20 Hz, is called an infrasonic wave. It is generally generated by a large source. E.g. Earthquake.

`"Ultrasonic Wave"`
A longitudinal wave whose frequency is above the upper limit of audible range i.e. 20 kHz, is called ultrasonic wave. It is generated by very small sources. E.g. Quartz crystal.

Ultrasound

Sound of very high frequency (greater than 20 kHz) is called ultrasound.

`"Properties"`
Sound wave of all frequencies carry energy with them, with increase in frequency, vibration becomes faster and also energy content and force increase. When ultrasound travels in solid, liquid and gas it subjects the particles of matter to face large force and energy.

`"Good directionality"`
Ultrasonic waves are able to travel along well defined straight paths, even in the presence of obstacles. Therefore, they are used for imaging objects.

`"Applications of ultrasound"`
• `"Welding metal"` They are used for welding metals like tungsten which cannot be welded by conventional methods.

• `"Medical purposes"` The ultrasonic vibrations can be reflected from the boundaries between the materials of nearly same density. The technique is used in scanning the internal organs of human body. It is superior to the X-ray scanning, as it does not cause any harm to human cells, unlike X-rays.

• The instrument which uses ultrasonic waves for getting the images of internal organs of human body is called ultrasound scanner. In this technique, the ultrasound waves travel through the tissues of the body and get reflected from the region where there is change in density. These reflected waves are then converted into electrical signals. 'These signals are then displayed on TV monitor or can be printed on a film. This technique is called ultrasonically and help doctors to detect abnormalities, such as stone in gall bladder and kidney or tumors in different organs.

• Ultrasound waves of high intensity are employed to break small stones in the kidney into fine grains. The fine grains then get flushed out with urine.

• Echocardiography is a technique in which ultrasonic waves, reflected from various parts of heart form an image of the heart.

• `"Drilling holes or making cuts of desired shape"` We can use a hammer and a steel punch to make holes in metal plates, plastic sheets or other solid materials. Such holes can also be made using ultrasonic vibrations produced in a metallic rod, called a horn.

• `"Ultrasonic cleaning"` For small parts such as used in watches, electronic components, odd-shaped parts such as a spiral tube and parts located in hard to reach places, this method is used. Such objects are placed in a cleaning solution and ultrasonic waves are sent into the solution.

• `"Ultrasonic detection of defects in metals"` If there are cracks or holes inside the metal, such defects are not visible from the outside. Ultrasonic waves can be used to detect such defects. Ultrasonic waves are sent through the metallic object under study. If there is no crack or cavity in its path, it goes through the object. A detector placed on the other side detects the transmitted wave. A defect present in the path of the wave reflects the wave.

• `"Bats"` fly in the darkness of night without colliding with other objects by the method of echolocation. Bats emit high frequency ultrasonic squeaks while flying and listen to the echoes produced by the reflection of their squeaks from the objects in their path. From the time taken by the echo to be heard, bats can judge the distance of the object in their path and hence avoid it by changing the direction. Bats search their prey at night by the method of echolocation.

Sonar

• The word 'SONAR' stands for 'Sound Navigation and Ranging'.

• Sonar is an apparatus which is used to find the depth of a sea or to locate the underwater things like shoals of fish, enemy submarines etc.

`"Principle of Sonar"`
Sonar works by sending short bursts of ultrasonic sound from a ship down into sea-water and then picking up the echo produced by the reflection of ultrasonic sound from under-water objects like bottom of sea, shoal of fish, and a submarine. The SONAR employs ultrasonic waves for working. Due to its very high frequency, ultrasonic sound has a greater penetrating power than ordinary sound. These waves have frequency more than 20,000 Hz.

`"Working of Sonar"`
A sonar apparatus consists of two parts: (i) A transmitter (for emitting ultrasonic waves) and (ii) a receiver (for detecting ultrasonic waves).
Now suppose a sonar device is attached to the underside of a ship and we want to measure the depth of sea (blow the ship). To do this, the transmitter of sonar is made to emit a pulse of ultrasonic sound with a very high frequency of about 50,000 hertz. This pulse of ultrasonic sound travels down in the sea-water towards the bottom of the sea. When the ultrasonic sound pulse strikes the bottom of the sea, it is reflected back to the ship in the form of an echo. This echo produces an electrical signal in the receiver part of the sonar device. The sonar device measures the time taken by the ultrasonic sound pulse to travel from the ship to the bottom of the sea and back to the ship. Half of this time gives the time taken by the ultrasonic sound to travel from the ship to the bottom of the sea.

`"Depth of sea" = text(Velocity of sound in sea water x time recorded by the recorder)/(2)`
`d = (vxxt)/(2)`

Speed of Longitudinal Waves (or Sound) in Gases : Newton Formula

• Newton gave a relation to calculate the velocity of sound in a gas. According to Newton, the velocity of sound
`v = sqrt(B/d)`
where, B is volume coefficient of elasticity (also called bulk modulus of elasticity) of the gas and `d` is density

• Newton assumed that the changes in pressure and volume of a gas when sound waves are propagated through it, are isothermal. Hence, in the above formula, B is isothermal bulk modulus of the gas whose value is equal to the initial pressure (p) of the gas.
Therefore, according to Newton, the speed of sound in a gas `v = sqrt(p/d)`

`text(Laplace's Correction)`
Laplace pointed out that Newton's assumption was wrong. According to Laplace, the changes in pressure and volume of a gas when a gas propagates through the air, are not isothermal but should be adiabatic. Because when sound waves are propagated through air, these are accompanied by the change of temperature of gas. Hence, changes are adiabatic and not isothermal.

Hence, in Newton's formula, B should represent the adiabatic bulk modulus of the gas whose value is equal to `gamma p`
i.e. `B = gamma p`
where `gamma = C_p/C_v =` ratio of two principal specific heat of gas

Thus, Laplace's formula for the speed of sound in a gas is `v = sqrt ((gamma p)/d )`

Superposition Of Waves

• The principle of superposition of waves states that when a number of waves travel through a medium simultaneously, the resultant displacement of any particle of the medium at any given time is equal to the algebaric sum of the displacement due to the individual waves.

• If `y_1, y_2, y_3, ... y_n` are the displacements due to waves acting separately, then according to the principle of superposition the resultant displacement, when all the waves act together is given by the algebraic sum `y = y_1 +y_2 +y_3 +.... y_n`.

Standing or Stationary Waves

• When two identical waves of same amplitude and frequency travelling in opposite directions with the same speed along the same path superpose each other, the resultant wave does not travel in the either direction and is called stationary or standing wave.

• On the path of stationary wave, there are some points where the amplitude is zero. These points are known as nodes.

• On the other hand, there are some points where the amplitude is maximum. These points are known as antinodes

Standing Waves in String

When a wave is set up on a string of length L fixed at two ends, then this wave gets reflected from the two fixed ends of the string continuously and as a result of superimposition of these waves, transverse standing waves are formed on the string.

Consider a suing of length L and mass m per unit length stretched with tension T. The fundamental modes of vibration setup in a string fixed at both ends are shown below.

• Fundamental frequency or frequency in first normal mode of vibration as Shown in Fig 1

`n_1 = v/(2L)= 1/(2L) sqrt(T/m)`

where, v = speed of wave in spring
This is called normal or fundamental mode of vibration. The sound or note so produced, is called fundamental note or first harmonic.

• Frequency in second normal mode of vibration as shown in Fig 2

`n_2 = v/L = (2v)/(2L)`

`n_2 = 2n_1`

Frequency of vibrating string becomes twice the fundamental frequency. The note or sound so produced, is called second harmonic or first overtone.

• Frequency in third normal mode of vibration as shown in fig 3

`n_3 =3( v/(2L))=3n_1`

Frequency of vibration of string becomes three times the fundamental frequency. The note or sound so produced, is called third harmonic or second overtone.

Standing Waves in Organ Pipes

Organ pipes are musical instruments which are used for producing musical sound by blowing air into the pipe.
There are two types of organ pipes

(i) Vibration in Closed Organ Pipe
(ii) Vibration in Open Organ Pipe

Vibration in Closed Organ Pipe

• Closed organ pipe is closed at one end and open at the other end. Sound wave is sent by a source vibrating near the open end. The wave is reflected from the fixed end. This inverted wave is again reflected at the open end. After two reflections, it moves towards the fixed end and interferes with the new wave sent by the source in that direction.

• In an organ pipe, the closed end is essentially a node point of minimum amplitude of vibration and the open end is antinode point of maximum amplitude of vibration. The fundamental modes of vibration are shown below, when there is a node at the closed end and an antinode at the open end.

• Fundamental frequency or frequency in first normal mode of vibration on, is shown in Fig 1
`n_1 = v/(4L)`
This is the lowest frequency of vibration and is called the fundamental frequency. The note or sound so produced, is called fundamental note or first harmonic.

• Frequency in second normal mode of vibration as shown in Fig 2
`n_2 =3( v/(4L))=3n_1`
Thus, the frequency of vibration in 2nd normal mode is thrice the fundamental frequency. The note so produced, is called third harmonic or first overtone.

• Frequency in third normal mode of vibration as shown in Fig 3
`n_3 =5( v/(4L))=5n_1`

• The frequency of vibration in 3rd normal mode is five times the fundamental frequency. The note or sound so produced, is called fifth harmonic or second overtone.

`:. n_1 : n_2 : n_3 .. = 1 : 3 : 5 : ...`

Vibration in Open Organ Pipe

• An open organ pipe is a cylindrical tube of which both ends are open. A source of sound near one of the ends sends the wave in the pipe. The wave is reflected by the other open end and travels towards the source. It suffers second reflection at the open end near the source and then interferes with the new wave sent by the source. The fundamental modes of vibration are shown below, when there are antinodes at both ends.

• Fundamental frequency or frequency in first normal mode of vibration as shown in Fig 1
`n_1 = v/(2L)`
This is the lowest frequency of vibration and is called fundamental frequency. The note or sound so produced, is called fundamental note or first harmonic.

• Frequency in second normal mode of vibration as shown in Fig 2
`n_2=2( v/(2L))2n_1`
Frequency in vibration in second normal mode is twice the fundamental frequency. The note so produced, is called second harmonic or first overtone.

• Frequency in third normal mode of vibration as shown in Fig 3
`n_3 = 3(v/(2L))=3n_1`


• Frequency of vibration in third normal mode is thrice the fundamental frequency. The note so produced, is called third harmonic or second overtone.

`:. n_1 : n_2 : n_3 .... = 1 : 2 : 3 ...`

• Therefore, even and odd harmonics are produced by an open organ pipe.

Doppler's Effect in Sound

When there is a relative motion between source and observer of the sound, a variation in the frequency (pitch) of sound is observed by the observer. This phenomenon is called Doppler's effect. Here, change in frequency is called Doppler's shift.

The variation in frequency (pitch) of sound depends on the three different relative motions between source and observer.

`"1. If only source S is in motion towards the observer"`
Then `v_o = 0` and `v_s` is +ve
Hence, `n = n_o[ v/(v - v_s)]`

`"2. If only observer O is in the motion towards the source"`
Then `v_s = 0` and `v_0` is: -ve.
Hence `n = n_0 [ (v -(-v_0) )/v ] = n_0[ (v + v_0)/v ]`

But if observer O is the motion away from the source, then `v _0` is +ve
Hence , `n = n_0 [(v -v_0)/v ]`

`"3. If both source S and observe O are in motion and approaching each other"`
Then `v_s` is +ve, but `v_0` is -ve.
Hence `n = n_o [ (v + v_0)/( v + v_s) ]`

`"4. If both source S and observe O are in motion such that they are receding from each other"`
Then v, is -ve but `v_0` is +ve.
Hence , `n = n_0[ (v - v_0)/( v + v_s)]`

 
SiteLock