A function which repeats its value after a fix interval of time is called a periodic function.

`y(t) = y (t +T )`

where, T is the period of the function.

Trigonometric functions `sin theta` and `cos theta` are simplest periodic functions having period of `2 pi`.

A periodic to and fro motion of a body about a fixed point, is called an oscillatory or vibratory motion.

e.g.,

• The motion of the pendulum of a wall clock.

• The motion of a loaded spring.

• The motion of a bar magnet suspended in the earth's magnetic field.

• Simple pendulum, spring pendulum, etc.

`text (Types of Oscillatory Motion)`

There are two types of oscillatory motion

(i) Harmonic Oscillation :

When a body repeats its motion about a fixed point after a regular time interval is known as the harmonic oscillation.

e.g., sine or cosine function is called harmonic function.

(ii) Non-harmonic Oscillation :

A non-harmonic oscillation is the combination of two or more than two harmonic oscillations.

It is a special kind of oscillatory motion in which particle moves to and fro about a mean position (on a straight line) under a restoring force which is directed towards mean position and its magnitude (of restoring force) is directly proportional to the displacement of particle (at all instants). For simple harmonic motion, displacement should be very small.

Simple harmonic motion is a special form of oscillatory or vibratory motion.

It is of two types

1 . Linear simple harmonic motion

2. Angular simple harmonic motion

In linear SHM, a particle moves to and fro (on a straight line) about fixed point under a restoring force (always directed towards mean position) whose magnitude is directly proportional to displacement of the particle (at all instants). e.g. Motion of a block connected to a spring on a smooth surface.

Restoring force `prop` Displacement

`F prop -x` (if displacement is along X -axis )

`:.` Acceleration , `a prop -x`

or `(d^2x)/(dt^2) prop -x`

`=> a = - omega^2 x`

where, `a` is acceleration and `omega` is angular frequency.
Negative sign indicates direction of restoring force and acceleration is towards equilibrium position, but in opposite direction of displacement.

In angular SHM, the restoring torque (or angular acceleration) acting on the particle is proportional to the angular displacement of the particle and directed towards equilibrium position.

Restoring torque `prop` Angular displacement

`tau prop - theta`

`=> alpha prop - theta`

or `(d^2 theta)/(dt^2) prop - theta`

Some important terms related to SHM are as follow

`(i)text( Mean position :)` It is a point at which restoring force on the particle is zero.

`(ii) text(Restoring force :)` The force which is acting on the particle tends to bring the particle towards its mean position is known as restoring force.

Restoring force, `F =- kx`,

`(iii) text(Force constant)` (Spring constant)

We know that, defining equation of SHM is

`a =- omega^2 x`, also `a = F/m`

`:. F/m = - omega^2 x`

`F = m omega^2 x => F = kx`

k is called spring constant or force constant

Also , `k = m omega^2`

`(iv) text(Amplitude :)` Maximum displacement of particle from mean position on either side is defined as amplitude. It is a scalar quantity.

Amplitude ` =1/2 xx` (Distance between extreme point or position )

`(v) text(Instantaneous displacement)` Displacement of particle ti·om mean position in a particular direction (at any instant of time ) is definedd as instantaneous displacement

It is given by `x = Asin(omega t + phi)`

where, A = amplitude
`phi` = initial phase or phase constant

Simple harmonic motion can be defined as the straight line motion of the foot of perpendicular drawn from the particle on the diameter of the circle.

`(i)text( Displacement)`

The displacement of a particle executing SHM at any instant is given by

`y = a sin omega t = a sin\ \(2 pi)/T t = a sin 2 pi nt` ................................(i)

`y = acos omega t = a cos \ \(2 pi)/T t = a cos 2 pi n t ` ............................(ii)

`y = a sin ( omega t pm phi )` ......................(iii)

First relation is valid when time is noted from the instant when the vibrating particle is at mean position.

Second relation is valid when the time is noted from the instant when the vibrating particle is at extreme position.

`(ii) text(Velocity)`

The velocity of the particle executing SHM at any instant, is defined as the time rate of change of its displacement at that instant.

velocity, `v = omega sqrt(a^2 - y^2)`

(a) when the particle is at mean position i.e. y = 0, then its velocity is maximum

`:. v_(max) = omega a`

(b) When the particle is at extreme position, i.e, `y = pm a`, then its velocity is zero.

`:. v = omega sqrt(a^2 -a^2) = 0`

`(iii) text(Acceleration)`

The acceleration of the particle executing SHM at any instant, is defined as the rate of change of its velocity at that instant.

`A = (dv)/(dt) = d/(dt) ( a omega cos omega t)`

`= - omega^2 a sin omega t = - omega^2 y` [ As , `y = a sin omega t `]

(a) When the particle is at the mean position i.e. `y = 0,` then acceleration is zero.

`:. a _text(mean position ) = 0`

(b) When the particle is at the extra position i.e. `x = a,` then acceleration is maximum.

`:. a_text(extreme position) = - omega^2a`

A particle executing SHM possesses two types of energy.

(i) Potential Energy

When a body is displaced from its equilibrium position by doing work upon it, acquires potential energy.

Potential energy `(U) =1/2 m omega^2 y^2`

Potential energy is maximum at extreme position i.e. at

`y = pm a`

`U_(max) = 1/2 m omega^2 a^2`

Potential energy is minimum (zero) at mean position i.e , at

`y =0`

`U_(max) = 0`

(ii) Kinetic Energy :

When a body is released, it begins to move back with a velocity, thus acquiring kinetic energy,

Kinetic energy `(K) = 1/2 m omega^2 (a^2 - y^2)`

K. E is maximum at mean position i.e `y =0`

`:. K_(max) = 1/2 m omega^2 a^2`

Kinetic energy is minimum (zero) at extreme position i.e,

y = a

`:. ` At `y pm a`

`K =0`

(iii) Total Energy :

Total energy can be obtained by adding potential and kinetic energies. Therefore,

`E = K + U = 1/2 m omega^2 ( a^2 - y^2) + 1/2 m omega^2 y^2 = 1/2 m omega^2 a^2`

A simple pendulum, in practice, consists of a heavy but small sized metallic bob suspended by a light, inextensible and flexible string. The motion of a simple pendulum is simple harmonic for very small angular displacement `(alpha)` whose time period and frequency are given by

`T = 2 pi sqrt (l/g)` and `v = 1/( 2 pi ) sqrt (g/l)`

where, `l` is the effective length of the string and `g` is acceleration due to gravity.

(i) If a pendulum of length `l` at temperature `theta^oC` has a time period T, then on increasing the temperature by `Delta theta °C` its time period changes to `T xx Delta T`

where , `(Delta T)/T = 1/2 alpha Delta theta`

where `alpha` is the temperature coefficient of expansion of the string.

(ii) A second's pendulum is a pendulum whose time period is 2 s. At a place where g = 9.8 m/s², the length of a second's pendulum is 0.9929 m (or 1 m approx).

(iii) If the bob of a pendulum (having density `rho`) is made to oscillate in a non -viscous fluid of density a, then it can be shown that the new period is

`T = 2 pi sqrt(l/(g ( 1 - sigma/rho)))`

(iv) If a pendulum is in a lift or in some other carriage moving vertically with an acceleration a, then the effective value of the acceleration due to gravity becomes (g ± a) and hence,

`T = 2 pi sqrt ( l/(g pm a))`

Here, positive sign is taken for an upward accelerated motion and negative sign for a downward accelerated motion.

(v) If a pendulum is made to oscillate in a freely falling lift or an orbiting satellite then the effective value of g is zero and hence, the time period of the pendulum will be infinity and therefore pendulum will not oscillate at all.

If the pendulum is suspended from the ceiling of the lift.

(i) If the lift is at rest or moving upward with constant velocity.

`T = 2 pisqrt(l/g)` and `n = 1/(2pi ) sqrt (g/l)`

(ii) If the lift is moving upward with constant acceleration `a`

`T = 2 pi sqrt(l/(g +a) ) ` and `n = 1/(2 pi sqrt ((g +a ) /l)`

Time period decreases and frequency increases

(iii) If the lift is moving downward with constant acceleration `a`

`T = 2 pi sqrt (l/( g -a))` and `n = 1/(2 pi) sqrt ((g -a )/l )`

Time period increases and frequency decreases.

(iv) If the lift is moving downward with acceleration `a = g`

`T = 2 pi sqrt (l/( g -g )) = oo` and ` n = 1/(2p i ) sqrt (( g -g )/l) = 0`

It means there will be no oscillation in a pendulum.

There are five types of simple pendulum

(i) Second's Pendulum

A simple pendulum having time period of 2 s, is called second's pendulum. The effective length of second's pendulum is 99.992 cm i.e., approximately 1 m on the earth.

(ii) Conical Pendulum

If a simple pendulum is fixed at one end and the bob is rotating in a horizontal circle, then it is called conical pendulum.

(iii) Compound Pendulum

Any rigid body mounted, so that it is capable of swinging in a vertical plane about some axis passing through it is called a compound pendulum.

(iv) Physical Pendulum When a rigid body of any shape is capable of oscillating about an axis, is called a physical pendulum.

(v) Spring Pendulum A point mass suspended from a mass less (or light) spring constitutes a spring pendulum. If the mass is once pulled downwards, so as to stretch the spring and then released, the system oscillates up and down about its mean position simple harmonically.

A spring pendulum consists of a point (small sized) mass m either suspended from a massless (or light) spring or placed on a smooth horizontal plane attached with a spring. If the mass is once pulled so as to stretch the spring and is then released, then a restoring force acts on it which continuously tries to restore its mean position, restoring force F =- kl , where k is force constant and I is the change in length of the spring under the restoring force. The spring pendulum oscillates simple harmonically having time period and frequency given by

`T = 2pi sqrt (m/k)` and `n = 1/(2 pi ) sqrt( k/m)`

where, k is the force constant of the spring and it is numerically equal to the force required to increase the length of the spring by unity. if the spring is not light but has a mass `m_s`, then

`T = 2 pi sqrt (((m +1)// 3m_s)/k)`

If two masses `m_1` and `m_2` , connected by a spring, are made to oscillate on a horizontal surface, then its period will be

`T = 2 pi sqrt (mu/k)`

where , `mu = (m_1m_2)/( m_1 + m_2)` = reduced mass of the system.

There are two types of spring combination

1. Series Combination of Springs

2. Parallel Combination of Springs

If two springs of spring constants `k_1` and `k_2` are joined in series (horizontally and vertically), then their equivalent spring constant k, is given by

`1/k_s = 1/k_1 + 1/k_2`

`=> k_s = (k_1k_2)/( k_1 + k_2)`

`:. T = 2 pi sqrt (m/k_s) = 2 pi sqrt ((m( k_1 + k_2 ))/(k_1k_2))`

If the two springs of spring constants `k_1` and `k_2` are joined in parallel as shown in fig., then their equivalent spring constant `k_p = k_1 + k_2` hence,

`T = 2 pi sqrt (m/k_p) = 2 pi sqrt (m/((k_1 + k_2))`

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.

(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.

Mechanical waves are of two types

1. Longitudinal Waves

2. Transverse 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.

e.g., those wave:; which 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.

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

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.

Some definitions related to wave motion
`(i) text(Amplitude )`

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

`(ii)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. lt is denoted by `T.`

`(iii) 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)

`(iv)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`

`(v) text(Wavelength )`

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

`(vi) 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`

`(vii) 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 per metre or rad `m^-1`

`(viii) 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.

Sound is a form of energy, which produces the sensation of hearing. These are longitudinal mechanical waves. Sound waves have low frequency and high wavelength. Sound waves cannot travel in vacuum.. The rebouncing back of sound, when it strikes a hard surface is called reflection of sound. The repetition of sound due to reflection of sound wave is called an echo.

According to their frequency range, waves are divided into the following categories

`1. text(Audible or Sound Waves )` The longitudinal mechanical waves, which lie in the frequency range 20Hz to 20000 Hz are called audible or sound waves. These waves are sensitive to human ear.

`2.text( Infrasonic Waves)` The longitudinal mechanical waves having frequencies less than 20 Hz are called infrasonic waves. These waves are produced by sources of bigger size such as earthquakes, volcanic erruptions, ocean waves, elephants and whales.

`3.text( Ultrasonic Waves)` The longitudinal mechanical waves having frequencies greater than 20000 Hz are called ultrasonic waves. Human ear cannot detect these waves. But certain creatures like dog, cat, bat, mosquito etc., can detect these waves.

`text(Intensity or Loudness )`

Intensity of sound at any point in space is defined as the amount of energy passing normally per unit area held around that point per unit time. Its SI unit is watt/metre. Loudness depends on intensity of sound. Unit of loudness is bel and `1/10 th` of bel is decibel (dB)

Source of Sound	Noise level (dB)
Whisper	20
Ordinary conversation	65
traffic on busy road	70
Amplified rock music	120
Jet aeroplane, 30 m away	140

`text(Quality or Timbre of Sound)` Quality is that characteristic of sound, which enables us t distinguishes between sound produced by two sources having the same intensity and pitch. It depends on harmonics and their relative order and intensity. `text(Pitch or Frequency)` The pitch of a sound is the characteristic which distinguishes between a shrillness or graveness of sound. Pitch depends upon frequency. A till and sharp sound has higher pitch and grave and dull sound has lower pitch.

The acronym SONAR stands for Sound Navigation And Ranging. The SONAR is a device, that uses the ultrasonic waves to measure the distances, directions and speed of objects under water. The ultrasonic sound pulse to travel from the ship to the bottom of the sea and back to the ship. In other words, the SONAR measures the time taken by the echo to return 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.

Some applications of SONAR are given

• Target location for torpedoes.
• Resources location for mines.
• Submarine navigation.
• In aircraft.
• Remotely operated vehicles.
• Detecting the vehicle location.

Shock Waves
A body moving with supersonic speed in air leaves behind it, 1 conical region of disturbance, which spreads continuously. Such a disturbance is called shock wave. These waves carry huge energy and may even make cracks in window panes or even damage a building. Earthquakes have shock waves.

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.

Resonance column method is a method for determination of speed of sound in air. Kundt's tube method is a method for determination of speed of sound in gas.

`text( Effect of Physical Parameters on Speed of Sound)`

Effect of Temperature

The speed of sound in a gas is directly proportional to the square root of absolute temperature of the gas, i.e. `v prop sqrtT`. So, velocity of sound in air increases due to rise in temperature.

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.

`text( Refraction of Sound Waves)`

When a sound wave moves from one mechanical medium to another mechanical. medium, then the waves are refracted or transmitted. This phenomenon is called refraction of sound.

The refracted waves deviated from the original path of the incident waves. The main reason for occurrence of refraction in sound is different speeds of sound, in different media at different temperatures.

`text{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 )`

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.

When a number of waves meet simultaneously at a point in a medium, this is called superposition of waves.

`text(Principle of 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 wave; act together is given by the algebraic sum

`y = y_1 +y_2 +y_3 +.... y_n`

`text( 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.

There are two types of stationary waves

1. Longitudinal Stationary Waves

Longitudinal stationary waves are formed as a result of superimposition of two identical longitudinal waves travelling in opposite directions.
e.g., Stationary waves produced in organ pipes and in air column of resonance tube apparatus are longitudinal stationary waves.

2. Transverse Stationary Waves

Transverse stationary waves are formed a:; a result of superimposition of two identical transverse waves travelling in opposite directions.
e.g., Stationary waves produced on the vibrating string of a Sonometer are transverse stationary waves.

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.

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

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 : ...`

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.

When two sound waves of equal amplitudes and nearly equal frequencies are produced simultaneously, then the intensity of resultant sound wave increases and decreases with time. This change in the intensity of sound, is called phenomenon of beans. Resultant frequency is equal to the difference in frequencies of two sound sources.

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.

Special Cases

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 0 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)]`

There will be no change in observed frequency due to Dorpler's effect when

(i) The source S and observer O are at rest or moving in such a way that distance between them remains constant.

(ii) Source and observer are moving in mutually perpendicular directions.

(iii) The velocity of source S and of observer O is equal to or greater than the velocity of sound in the given medium.