(Waves, Waves, Waves... All the way the waves !!)
If we tickle the water in a tray with the tip of our fore finger, we find concentric circles of water running away from the point of disturbance called 'ripples'. When a pebble is dropped in still water of a pond, concentric circles from the place of disturbance (where the pebble is dropped) called 'waves' will travel to the edge of the pond. We all are familiar with waves in a sea. All these are water waves.
One end of a thin rope is tied to a nail stuck in a wall and the other end is pulled a little and jerked up and down a hump like shape of the rope will be moving to and fro along the rope. This is the wave in the rope.
Water waves and the waves in ropes and strings are visible.
Sound travels in the form of a wave. How can we say that? When some one is talking in a room, a person standing beside the wall can hear that sound (people say walls will have ears!!) because the sound can bend around the corners. Only a 'wave' can bend around the corners of an obstacle.
Earlier, Light energy is considered to be a propagation of fast moving particles in a straight path called corpuscles by Isaac Newton. But, in order to explain the phenomenon exhibited later by light Christian Huygens a dutch physicist proposed that light should be a wave. Light is a form of an electro magnetic wave. All the other invisible components of electro magnetic spectrum i.e., γ rays, X - rays, Ultra violet rays, Infrared rays are having a wave nature. Even radio waves, cosmic rays and micro waves are waves. 'Gravity' is associated with gravity waves. Earth tremors are associated with 'sesimic waves'.
Later French Physicist Louis de Broglie proposed that not only energy but also matter will be in the form of waves called 'matter waves'.
Thus the whole of the universe is full of waves. Why? Even the human thoughts are depicted as 'thought waves'! In Cinemas, the past of a hero or heroine recollect in the form of concentric circles i.e., 'waves'!
What is a Wave?
When a small stone is dropped in a still water of a lake, we find concentric circles running from the point of disturbance to the edge of the lake which we call 'waves'.
When the stone is dropped, the amount of water at the point of disturbance will go down because of the weight of the stone. Once the stone reaches the bottom of the lake, the water which went down will come up little above the surface of the lake because of 'capillarity' which is the effect of 'surface tension' (otherwise, wherever a stone is dropped in a lake, there will be a narrow tunnel). The water which has come up will go down again under the influence of its own weight i.e., the gravitational force.
Thus when a stone is dropped in a still water, the water that is disturbed will be moving up and down under the influence of gravitational and surface tension forces along a straight path between two fixed points i.e., the particles of disturbed water will be making simple harmonic motion (SHM). This motion of disturbance is propagated to the next amount of the particles which in turn make SHM which in turn is propagated to the next particles of water and so on. So what is propagated in the medium (water) in the form of concentric circles (waves) is not the particles of the medium but the disturbance. This can be proved by placing a small 'leaf' on the surface of water before dropping the small stone in water. When concentric circles (waves) are spreading on water, the floating leaf moves 'up and down' at the same place which is due to the motion of water particles. Thus the particles of the medium are making SHM, about their mean position, what that is spreading is the 'disturbance' in the form of waves!
Thus, a wave is a propagation of disturbance in an elastic material medium.
Water waves, waves in a rope sesimic waves and sound waves require a material medium for their propagation and hence they are called 'mechanical waves'. They cannot travel through vacuum.
Light waves, radio waves, X − rays etc., (electro magnetic waves) do not require a material medium. They can travel through vacuum with a velocity of 300 ms−1.
Matter waves are assoicated with constituents of matter: Electrons, protons, neutrons, atoms and molecules.
SOUND - A PRIMARY ENERGY IS A WAVE
Our universe emerged with 'Big Bang' (Any doubt?) which is associated with 'Sound'. When we approach an Ocean even before we see the Sea, The mystic 'Sound' of the waves is heard. Why all this? Even the Religious Scriptures. Say in the beginning there is a 'word' i.e., sound.
A word is a Sound. The Sound of the word forms all things. And man listens and hears the word of all things.
'Sound' is a form of energy which is produced by a vibrating body travels through a 'material medium' in the form of a 'wave' and appeals to the sense of hearing 'music' and 'noise'.
Sound is classified as 'music' and 'noise'. Musical Sound is that which is pleasant to hear. Noise is that which is unpleasant to hear. No one says that Pandit Ravishankar makes 'noise' while playing on Sitar. No teacher in class room shouts at students 'Dont make music'!
Source of Sound - A Vibrating Body
The source of sound is a vibrating body. A vibrating string in a Veena, Violin and Piano, vibrating air column in a Flute and Harmonium; vibrating stretched membranes in Tabla and Mridangam, a Bell struck by a hammer and a vibrating Tuning fork in laboratory produce sound (Of course music if properly played).
Not all Vibrating Bodies....
Why a vibrating Simple Pendulum can not produce sound? Vibrations whose frequencies lie between 20 hertz (Hz) i.e., 20 vibrations per second and 20,000 Hz can only be detected by human ear. These are called audible sounds.
A person delivering a public speech for the first time was shivering (Vibrating) due to stage fear and a another person in the audience murmering with his friend ''see, how sound is produced by a vibrating body", do you agree with this statement?
Requires a medium
Sound cannot be propagated through vacuum.
The sound of a gun fired by a standing person on the ground of moon cannot be heard by another person standing little away from the former. Becuase there is no atmosphere (medium) on the moon.
Sound travels in the form of a wave for a sound wave to propagate continuously the medium should be elastic should possess the property of 'inertia' and should be 'frictionless'.
The velocity of sound in liquids is more than its velocity in gases and its velocity in solids is more than that in the liquids.
The particles of the medium when a sound wave propagates through it executes 'Simple Harmonic Motion'.
When sound is propagated through a medium, the particles of the medium vibrate about their mean position and never move away from each other permanently. Each vibrating particle transmits energy from one point to the other in the medium.
Frequencies below 20 Hz are called infrasonic sounds (e.g.: Earthquake waves).
Frequencies above 20,000 Hz are called ultrasonic sounds (e.g.: Vibration of Quartz crystal when alternating current is applied to them).
The sounds emitted by bats are 'ultrasonic' in nature. That is why we cannot hear those sounds.
Progressive waves
In the process of wave propagation energy is transferred (progressed) from particle to particle of the medium and hence the waves are called progressive waves.
Progressive waves are classified as
i) Transverse waves
ii) Longitudinal waves
TRANSVERSE WAVES
It is a wave where the particles of the medium vibrate perpendicular to the direction of motion of the propagation of energy (disturbance).
for example i) Waves in water
When a stone is dropped in water (at rest), the water particles disturbed move up and down where as the disturbance is transferred to the next amount of particles which in turn move up and down. The concentric circles which we see on water surface is the propagation of disturbance and not the propagation of the water particles which are disturbed initially.
ii) Waves in ropes and strings
When a rope tied at one end to a fixed support and the other end is moved up and down, we find a kink will be travelling from one end to the other. Perpendicular to the direction of motion of the rope. The 'kink' which is moving along the rope is the propagation of disturbance which is perpendicular to the direction in which the rope is moved.
iii) Electro magnetic waves (Light Waves) are considered to be transverse waves.
Transverse waves are associated with crests and troughs.
LONGITUDINAL WAVES
It is a wave where the particles of the medium vibrate in the direction of motion of the propagation of the energy (disturbance).
Let us consider a collection of springs connected to each other (fig 3). If the spring at one end is pulled suddenly and left, the disturbance travels to the other end. What happens is the first spring is disturbed from its equilibrium position. As the second spring is connected to the first, it is also stretched or compressed and the disturbance travels from one end to the other. Each spring performs small vibrations about its equilibrium position what that is propagated in the system is the disturbance.
Same situation for a Railway Train
Let us consider a railway train at rest. Different compartments of train are connected to each other through a spring coupling.
When an engine is suddenly attached at one end of the connected compartments, it gives a push to the compartment next to it. This push is propagated one compartment to another without the entire train being displaced from its original position.
What happens is same when a sound wave propagates through a gaseous medium?
When a sound wave propagates through a gaseous medium the layers of the gas (air) makes to and fro motion about their mean position just as springs in the above examples (fig 3) move without their displacement from their original position. What that is propagated is the sound energy and not the particles of the medium.
Where the particles (layers) of the medium are coming closer (Compressing or Condensing) are called "Compressions" (C) and those regions where the particles (layers) of the medium are moving away from each other are called 'rarefactions' (R) (fig 4)
'Compressions' are the regions where the values of density, pressure and temperature of the medium are above the normal values where as at 'rarefactions' their values will be below the normal values.
Characteristics of Sound Waves
i) Displacement: It is the extent through which a vibrating particle moves from the mean position.
ii) Amplitude: It is the maximum displacement of a particle.
iii) Period (T): It is the time taken for one complete vibration.
iv) Frequency (υ): Number of Vibrations made per second.
v) Wavelength (λ): It is the distance between two consecutive crests or troughs in case of a transverse wave.
(or)
It is the distance between two consecutive compressions or rarefactions in case of a longitudinal wave.
(or)
It is the distance between two successive particles which are in the same phase (state of vibration).
It is the distance travelled by the wave by the time the vibrating particle (body) completes one vibration.
Relation between Frequency and Period
Consider a vibrating body executing simple harmonic motion having a frequency (υ) and period of oscillation (T) i.e. for υ vibrations time taken is 1 second.
Time taken for one vibration is 
seconds
But time taken for one vibration is T seconds (Period)
Relation between frequency and angular frequency
To arrive at this relation one should recollect about simple harmonic motion (SHM) and how it is sponsored by a particle executing uniform circular motion (A geometrical feat indeed to arrive at the concept of SHM!!)
If a particle 'P' is moving on the circumference of a circle of radius 'a' uniformly i.e., with a constant angular velocity 'ω', then the foot of the perpendicular PN dropped from P on one of the diameters AB i.e., the point N will be executing SHM along AB (fig - 5).
Thus, SHM is the projection of uniform circular motion on any one of the diameters of the circle.
In the fig - 5 'P' is making uniform circular motion and its projection 'N' is making SHM.
The angular displacement of P at any instant 't' is given by θ = ωt
PN and PN' are the perpendiculars from P on the diameters AB and CD respectively. These perpendiculars meet AB and CD at N and N'. As P moves once along the circumference of the circle, N or N' complete one vibration along their respective diameters.
(Please observe how one motion in nature is sponsoring another kind of motion and how elegently the mathematical models devised by human brains in geometrical form explaining them!)
Time taken by P to make one complete rotation is 
Hence time taken by N or N' to make one vibration = seconds.
i.e., the peirod of a particle moving simple harmonically is
... Angular velocity or angular frequency of a particle moving simple harmonically
Displacement of N = ON = x = a cos ωt
Displacement of N' = ON' = y = a sin ωt
(Where 'a' is radius of the circle)
The above equations can be written as
x = a cos ωt = a cos 
t = a cos 2π υt
y = a sin ωt = a sin t = a sin 2π υt
Expression for the velocity of a wave in terms of 'frequency' and 'wavelength'
Let 'C' be the velocity of a wave whose frequency is 'υ' and wavelength is 'λ' for one complete vibration of the particle executing SHM the distance advanced by the disturbance (or) wave is 'λ'.
Time taken for one vibration is T seconds
PHASE
It is defined as the state or condition with regards to the position and direction of motion of a vibrating particle from its mean position and expressed in 'radians'.
(for e.g.: When a building is under construction, we usually ask the builder at what phase (stage) the construction is?)
Phase of a particle executing SHM is the angle traced by the radius vector (radius connected to a particle moving in specific direction) since the particle has last passed its mean position in the positive direction.
for example 
= θ = Phase angle (shown in fig. 5)
(or)
It can also be expressed in terms of the fraction of time period which has elapsed since the particle left its mean position in the positive direction of motion.
Phase Difference
In the fig − 6, Q1 and Q2 are two generating particles moving in the same direction with the same speed along the circumference of a circle whose centre is 'O' and radius 'a'.
Q1N1 and Q2N2 are two perpendiculars on AB. The points N1 and N2 are executing simple harmonic motion (vibration).
Taking LK as the reference axis, the first vibration crosses P1 in the same direction when the second vibration crosses K. It means that the first vibration has a 'phase lead' over the second one or the second vibration 'lags' behind the first by phase angle (This small angle of phase or change in phase is called 'Epoch')
Hence displacement equations of N1 and N2 are given by
y1 = a sin ωt
y2 = a sin (ωt − 
)
The angle is known as phase difference.
If the phase difference between the first and second particles is 2Π radians, then the second particle will cross the reference axis after one complete period T seconds later than the first. In other words, the second particle will behave similar to the first one, provided the phase difference is 2Π. Such particles, which differ in phase by 2Π, are said to be 'in phase'. In case the phase difference between the two particles is Π, then they are said to be 'out of phase'.
Relation between 'Phase difference' and 'Path difference'
Wavelength (λ) is defined as the distance between any two successive particles vibrating in the same manner (or) in the same phase (or) differing in phase by 2Π. In other words, for path difference of λ between two particles, the phase difference is '2Π' radians.
In case, the path difference is x, the corresponding phase difference is, 
x
Thus, phase difference = × path difference
Equation of a progressive wave
The simplest type of wave motion is simple harmonic motion. In wave motion all the particles of the elastic medium virbrate simple harmonically with different phases. In such a medium if one particle is displaced due to a periodic vibration (disturbance) other particles also get displaced. Therefore a comprehensive equation for displacement connecting all particles can be obtained.
For a particle executing simple harmonic motion with an amplitude 'a' and angular frequency 'ω' the displacement 'y1' at any instant 't' is given by
y1 = a sin wt
= a sin t
= a sin 2Π υt
Consider any particle of the medium executing simple harmonic motion with a phase difference '' or path difference 'x' from the first particle. The displacement of second particle 'y2' is given by y2 = a sin (ωt − ), a and ω being same as in the first particle.
Thus we have
y2 = a sin(ωt − Kx) for a wave moving towards right
similarly Y'2 = a sin(ωt + Kx) for a wave movinig towards left.
Characteristics of a progressive wave
1. Each particle of the medium vibrates about its mean position of rest and is not carried away by the wave.
2. Each particle of the medium reaches the maximum displacement (amplitude) a little later than the previous one.
3. The waves do not die down i.e., there is no damping.
4. If the paritcles of the medium vibrate parallel to the direction of propagation of the wave, it is said to be longitudinal wave. On the other hand if the vibration is normal to the direction of wave propagation, it is known as transverse wave.
5. The particles of the medium vibrate 'Simple harmonically'.
6. The wave motion resembles a sine curve. 
7. The distance between any two successive particles which are in same state of vibration is 'wavelength' and is denoted by 'λ'.
8. Phase difference between two particles separated by a distance 'λ' is 2Π radians.
9. The velocity of the particle is 
10. The velocity of the wave 
where E is the modulus of elasticity of the medium, 'ρ' is the density of the medium.
Stationary waves (or) Standing waves
Suppose two persons walking on a very narrow plank (over which only one can walk at a time) stretched across a stream in opposite directions, when they came face to face with each other, the progress of their march will come to a halt or stationary. Similarly, when two progressive waves with similar characteristics travelling along a straight line but in opposite directions interfere their progress and energy come to a stand still and such a wave pattern is called Stationary or Standing Wave.
When a progressive wave moving forward, meets an obstacle it gets reflected and travels in the opposite direction and if these forward and backward waves interfere, stationary (or) standing wave is formed (fig - 7).
Thus, standing waves are formed due to superposition of two progressive waves of same characteristics moving in opposite directions.
Definition: When two progressive waves of same frequency, amplitude and velocity travelling along same straight line but in opposite directions interfere, the crests and troughs in case of transverse waves and compressions and rarefactions in case of longitudinal waves come to a stand still such that the energy is not propagated in the medium but confines to certain region.
This wave pattern is called a 'Stationary Wave'. The positions of the particles where there is no displacement are called 'NODES' (No Displacement) N1, N2, N3, ......... etc. and the positions of the particles having maximum displacement A1, A2, A3, ...... etc. are called 'ANTINODES.'
Distance between two successive nodes or two successive antinodes is equal to half the wavelength.
N1N2 = A1A2 = 
Distance between a node and antinode is 
N1A1 = A1N2 = N2A2 =
The rate of transfer of energy across any section of stationary wave is 'Zero'.
Equation of standing wave:
Suppose the two wave trains are represented by equations
(since there is a phase change of Π on reflection + is taken)
Resultant displacement after interference
y = y1 + y2
= a sin (ωt − ) + a sin (ωt + )
= 2a cos sin ωt
The resultant wave is also 'simple harmonic' with amplitude 2a cos 
x
When 
etc., amplitude is maximum.
These points are 'antinodes'. Where values of '' are Π, 2Π, 3Π etc.
When 
etc., amplitude is zero and these points are 'nodes'.
Where values of are 
etc.
In stationary waves:
The amplitude of different particles is different and energy associated with a stationary wave is ρa2ω2, that is energy of a stationary wave is twice the energy of a progressive wave.
In a stationary wave the free end of the vibrating system is an 'antinode' and the fixed end corresponds to a 'node'.
In a standing wave, energy is not transported across the nodes. In between the nodes the energy alternates between vibrational KE and elastic PE.
Do you know that the air columns in the "flute" and "sehnai" are stationary waves when they are in action!
They are called organ (music) pipes. "Organ" is the term associated with musical effect, for example the musical instrument "mouth organ".
ORGAN PIPES:
Organ Pipes are of two types.
1) Open Organ Pipe: Here both the ends of pipe are open and hence at both the ends, "anti nodes" are formed.
Ex: Flute
2) Closed Organ Pipe: Only one end is open (If both the ends are closed, it is no more a pipe!). Hence at the open end an "antinode" and at the closed end "node" is formed.
Ex: Sehnai.
Vibrating Air Columns
Case 1: Closed Pipe (Pipe closed at one end and open at the other end)
In Fig-8 nodes of vibration are shown indicating the formation of stationary waves of different frequencies for the same length of air column. In pipes closed at one end, free vibrations at closed end occur and the particles of the air are kept static. At the open end, maximum vibrations are possible without any pressure changes, because any variations in pressure will be neutralised by atmospheric pressure. referring to fig 8(i) mouth of the tube always corresponds to antinode "A" and closed end a node "N".
Hence the length of the pipe (l) is related to the wavelength (λ) by
Therefore, the frequency of the air column 
This frequency is called fundamental frequency.
If the external periodic force is changed, the mode of vibrations also change, as shown in the figures 8 (ii), (iii), (iv) and (v).
In fig 8 (ii) there is another antinode besides a node in between the closed end and open end for the same length 'l'.
Hence, the frequnecy ratio in the above mode of vibrations in closed pipes is 1 : 3 : 5 : 7
Except the first frequency, the other frequencies are called OVER TONES.
Case 2: Pipe open at both ends
In fig-9, a pipe open at both ends is shown. The mode of vibration of air column in this pipe is such that there are two antinodes one at each end with a node in the middle (fig 9(i)).
Frequnecy 
; this frequency is the fundamental frequency.
In fig 9(ii), length of air column 'l' corresponds to one wavelength λ.
This frequency is first overtone.
Similarly in fig 9(iii)
Thus the frequency ratio in case of pipe open at both ends is 1 : 2 : 3 : 4
In case of closed pipe, the frequency ratio is 1 : 3 : 5 : 7 i.e., odd multiples of fundamental frequency.
In case of open pipe, the frequency ratio is 1 : 2 : 3 : 4 i.e., both odd and even multiples of fundamental frequency.
Thus the number of "overtones" produced in open pipe are more than that produced in closed pipe. Where the number of overtones are more, better the musical effect.
Thus open organ pipes produce better musical effect.
BEATS
If one stands besides a sea, it is found that the sea waves are "rising" and "falling" along with that one hears the sound that is varying between "maximum" and minimum. Whenever the waves are rising (waxing) the sound heard is maximum and when they are falling (waning) the sound heard is minimum.
The same "waxing" and "waning" of sound occurs when two sound waves interfere (in what way?) which is called the phenomenon of BEATS.
If two waves of same frequency travelling along the same straight line but in opposite direction superpose standing or stationary waves are produced. This is called interference in space.
When two sound waves of slightly differnt frequencies travelling in the same medium along the same direction when superimposed on each other give rise to a resultant sound whose intensity will not be uniform. A listener would hear a regular rise and fall of the sound i.e. the intensity of sound successively waxes to a maximum and minimum just as the sound of rise and fall of sea waves varies between maximum and minimum successively.
This is called the phenomenon of "Beats".
One waxing and the subsequent waning of sound constitute a 'Beat'.
In one wave the compressions and rarefactions will be closer where as in another wave they are spaced little apart. At certain instant two compressions or two rarefactions arrive together to the ear of the listener and the sound heard is maximum (louder). At a later instant the compression of one wave reaches with rarefaction of the other wave to the ear of the listener and the sound heard is minimum (feeble) i.e., when two waves of slightly different frequencies are superimposed, when a compression of one wave comes over the compression of the other, two amplitudes reinforce each other and the sound waxes to a maximum. But when a compression comes over a rarefaction, the amplitude of one cancels the amplitude of the other such that the resultant amplitude is zero and the sound waves to a minimum.
The phenomenon of 'Beats' is the interference of sound waves in time.
The number of 'Beats' produced are equal to the difference in frequencies of the sound waves interfering.
If υ1 and υ2 are the frequencies of interfering sound waves then the number of beats produced are n = υ1 ~ υ2
for example, if there are two sound waves of frequencies 49 Hz and 56 Hz (fig. 10) the number of beats produced are 56 − 49 = 7
Explanation
The sound wave of frequency 56 Hz in one second produces 56 compressions and 56 rarefactions, the total being 112.
Similarly, the total number of compressions and rarefactions produced per second by the sound wave of frequency 49 Hz are equal to 98.
The difference of the number of compressions and rarefactions between the two waves is 112 − 98 = 14
In 
seconds the number of vibrations produced by the wave of frequency 56 Hz = 56 × = 4
The number of vibrations produced by the wave of frequency 49 Hz = 49 × = 3.5
When both these wave interfere, in 1/14 seconds, the compression of one wave falls on the rarefaction of the other (the difference in the number of vibrations being 0.5) producing minimum amplitude i.e., minimum intensity of sound.
Similarly in 
seconds, the number of vibrations produced by the wave of frequency 56 Hz = 56 × = 8
The number of vibrations produced by the wave of frequency 49 Hz = 49 × = 7
i.e., when both the waves, interfere the compression of one wave falls over the compressions of the other wave, producing maximum amplitude i.e., maximum intensity of sound.
Thus in seconds, one maximum and one minimum sound i.e., one 'Beat' is produced.
The number of 'Beats' produced in one second is equal to 
= 7 which is equal to the difference of frequencies of interfering waves 56 − 49 = 7.
Applications (uses) of Beats
1. To determine the unknown frequency of sound note or a tuning fork.
2. In tuning musical instruments.
3. To detect the presence of harmful gases in mines.
Note: (i) For human being to distinguish between two sounds, the interval between them should not be less than 1/10th of a second i.e. the human ear cannot hear more than 10 beats for second.
(ii) When a tuning fork is failed, its frequency increases.
(iii) When a tuning fork is loaded with wax its frequency decreases.
Tune it man!
Beats in musical instruments produce pleasant musical effect. That is why in musical concerts especially 'Violin concerts', the players will 'tune' their Violins to nearly same frequency!
DOPPLER EFFECT
Is the Universe expanding?
Astronomers in earlier days were not able to decide whether the Universe is steady (static) state or in motion (dynamic). An Austrian scientist C.J.Doppler in 1842 found change in colour of stars in the galaxy and their colours are shifting towards red (Red shift). As colour of light depends on frequency and as red colour has lesser frequency, Doppler concluded that light waves of stars are moving away from the observer i.e., the universe is not only in motion but expanding. This has been established later by Hubble through his 'Hubble Telescope'.
Doppler extended the same argument to the sound energy also and showed that there is an apparent change in frequency (pitch) of a sound wave whenever there is a change in relative motion between the source of sound and the listener.
To a listener standing near a railway track, the pitch of the sound from the whistle (shrillness or cuteness) of an approaching train appears to be increasing. Similarly, the pitch of the sound of the whistle appears to fall when the train is moving away from the listener. The same effect is found when the listener moves towards the source of sound, the number of waves that fall on his ear for every second is greater than number of waves that fall when he is at rest. Thus, the pitch of the sound which he hears is greater than when he is at rest. Similarly, when he moves away from the source, the number of waves per second he receives will be lesser and thus the pitch appears to fall. Similar changes in pitch occur when the listener is stationary and the source moves.
This phenomenon of apparent change in frequency of a sound wave due to relative motion between the source and the observer is called the ''Doppler effect''.
Different cases of apparent change in frequency.
Let 'S' be a source of sound, and 'O' be the observer (Listener). Let 'υ' be the frequency of the sound wave emitted by the source and 'λ' be its wavelength and 'C' be the velocity of sound in the medium.
Let 'Vs' and 'Vo' be the velocities of source and observer (listener) respectively.
Case 1: Source at rest, observer in motion
i) Source at rest, observer moving towards the source
X 
X
S V0 O
When the source (S) and the observer (O) are at rest, the number of vibrations received by the listener per second are υ = 
When the observer is moving towards the source at rest with velocity Vo in addition to the number of vibrations υ, he will receive 
vibrations per second.
The total number of vibrations received (falls on his ear) per second i.e., apparent frequency
This, apparent frequency υ' > ν i.e., the apparent frequency increases.
If λ' is the apparent wavelength, and since λ' = 
i.e., λ' < λ i.e., the apparent wavelength decreases.
ii) Source at rest, observer moving away from the source
When the source is at rest and the observer is moving away from the source with velocity V0, substituting the value of velocity of the observer (−V0) in equation (i). (negative sign indicates the opposite direction of motion of the observer)
X X 
V0
S O
Since υ' < υ, apparent frequency decreases.
The apparent wavelength from equation (ii)
λ' > λ i.e., apparent wavelength increases.
Case 2: Observer at rest, and source in motion
i) Observer at rest and source is moving towards the observer
X 
Vs X
S O
Let the source move towards the observer at rest with a velocity Vs.
As shown in fig−11, the effect is to shorten the wavelength. When the source is moving towards, the observer at rest with a velocity Vs, then for each vibration, the source is moving towards the observer, the wavelength decreses by a distance
(distance = velocity × time)
Hence the apparent frequency becomes
i.e., υ' > υ , the apparent frequency increases and also
i.e., λ' < λ, i.e., the apparent wavelength decreases.
(ii) Observer at rest, source is moving away from the observer
Here considering the velocity of the source as (−Vs) in equation (v)
υ' < υ i.e., the apparent frequency decreases and the apparent wavelength is
λ' > λ, i.e., the apparent wavelength increases.
Case 3: Source and Observer both moving along the same straight line
Let the observer be at rest and the source is moving towards the observer with a velocity Vs.
Now, the apparent frequency from equation (v) is
Now, if the observer is also moving in the same direction as source (away from the source with velocity Vo), the apparent frequency
If the observer is moving in opposite direction to the source
Note: The equation (vii) 
can be considered as a general equation (formula) for obtaining apparent frequency for all the cases discussed above taking the +ve or -ve sign for velocities of the source (Vs) and observer (Vo) appropriately.
The treatment of Doppler effect is same in light and sound except in case of sound, the velocity of wind is also to be considered unless it is negligible.
If the wind velocity is W, the apparent frequency
Uses of Doppler effect:
1) It is used in accurate navigation of air craft.
2) In war, for bombing enemy bases accurately.
3) The speed of an automobile is found using this principle by traffic police.
4) For tracking an artificial satellite.
Limitation of Doppler effect
Doppler effect can be applied only when the relative velocity between the source of the sound and the listener is less than the velocity of sound in the medium.
Note: Doppler effect in light finds application in spectroscopy and astrophysics.
i) In the study of 'binary stars' by spectroscopical methods.
ii) In finding the speed of the stars.
iii) In arriving at the theory of expanding universe.
iv) To study the nature of Saturn rings recently, Doppler effect is used effectively to locate new planets away from solar system.
Sound Waves
A word is a sound. The sound of the word forms all things. And man listens and hears the word of all things! Sound is a form of energy which is produced by a vibrating body travels through a material medium in the form of a 'wave' and appeal to the sense of hearing.
Source of sound - a vibrating body
The source of sound is a vibrating body. A vibrating string in a Veena, Violin and Piano; vibrating air column in a Flute and Harmonium; Vibrating stretched membranes in Tabla and Mridangam; A bell struck by a hammer; and a vibrating tuning fork produce sound. When we are talking the vocal cords in our throat vibrate and produce sound.
Audiable - Inaudiable sounds
The number of vibrations made per second by a vibrating body is called 'frequency'. The unit of frequency is 'hertz' (Hz) named after Henrich Rudolf Hertz who produced 'Radio Waves' for the first time in a laboratory.
A normal human being can hear sounds of frequency ranging between 20 Hz to 20,000 Hz. These sounds are called 'Audiable sounds'. Sounds whose frequency is less than 20 Hz are called 'Infra sounds', and sounds of frequency more than 20,000 Hz are called 'Ultrasonic sounds'. We cannot hear these sounds. Bats and dogs can hear Ultrasonic sounds.
Medium is a must
Sound travels through solids, liquids and also gases. It can not travel through vacuum.
For e.g. if a person standing on the surface of moon fires a gun, a person standing at a distance cannot hear that sound as there is no atmosphere there i.e., for sound to propagate there should be an elastic material medium.
Two types of sound
Sound is classified as 'Music' and 'Noise'. Musical sound is that which is pleasant to hear. Noise is that which is unpleasant to hear. No one says that Pandit Ravisankar playing on Sitar is popular in making 'Noise'. No teacher in a class room shouts at students 'Don't make music'!.
Sound travels in a wave form
How can you say that? A person talking in a room can be heard by a person out side beside the wall of the room. Only a wave can bend around the corners of a wall (an obstacle).
Waves progress through a medium
While sound is travelling through a medium energy is transferred from particle to particle of the medium and hence propagating sound waves are called progressive waves.
Transverse and Longitudinal waves
Progressive waves are of two types.
i) Transverse waves: Here the particles of the medium vibrates perpendicular (and hence transverse) to the direction of motion of the propagation of the sound
energy and the wave is associated with 'Crests' and 'Troughs'.
ii) Longitudinal waves: Here the particles of the medium vibrates in the same direction (i.e., along and hence longitudinal) as that of the propagation of sound and this wave is associated with 'Compressions' and 'Rarefactions'.
Transverse wave
* 'Compressions' (on condensations) are the places in the medium where the values pressure, density and temperature are above the normal values and 'rarefactions' are the places in the medium where the above parameters are below the normal values when the longitudinal wave is propagated.
* Sound travels only as a longitudinal wave in a gaseous medium where as it may be either longitudinal or transverse in solids and liquids.
Characteristics of sound waves:
i) Displacement: It is the extent through which a vibrating particle moves from its mean position.
ii) Amplitude: It is the maximum displacement of a vibrating particle.
iii) Period (T): It is the time taken for one complete vibration; unit: second.
iv) Frequency (υ): Number of vibrations made per second; unit: hertz (Hz).
v) Velocity (c): It is the distance travelled by the wave in one second; unit: ms-1
c = υλ
vi) Wave length (λ): It is the distance between two consecutive crests or troughs in case of a transverse wave (or) it is the distance between two consecutive compressions (or) rarefactions in case of a longitudinal wave.
(or)
It is the distance between two consecutive particles which are in the same state of vibration (Phase). (or)
It is the distance travelled by the wave by the time the vibrating body completes one vibration.
vii) Phase: It is the state or the condition with regard to the position and direction of motion of a vibrating particle from its mean position and expressed in 'radians'.
viii) Intensity (I): It is the average rate of transfer of energy per unit area normal to the direction of propagation of a wave; units: watt per meter-2 (Wm-2)
Wave intensity depends upon the amplitude (a) of the wave; I a2 If amplitude of a wave is doubled, wave intensity (2 × 2 = 4) increases by four times.
Equation of a progressive wave
Since particles in a wave are executing Simple Harmonic Motion (SHM), progressive
wave is represented by the equation of a particle making SHM i.e., Y = a sin ωt.
Here Y is displacement; a = amplitude and ωt = phase.
If the wave is having a change in phase 
, in the beginning only.
Y = asin (ωt - ) .......... (i)
angular velocity ω = 
, T - period of vibration λ
But T = 
; ω = 2Πυ
But υ = 
; ω = 2Π ....... (ii)
For a path difference of λ the phase difference is 2Π for a path difference of x, the phase difference = 
x................. (iii)
substituting (ii) and (iii) in (i) Y = a sin (ct ± x) ............... (iv)
+x, indicates that wave is travelling along +X direction and -x, indicates that wave is travelling along -X direction. But c = υλ = 
The equation (iv) can be written as 
............ (v)
(By using the above equations, the values of Y, a, λ, υ, T, c of a wave can be obtained)
Stationary or a standing wave - Waves in a flute
When two progressive waves of same frequency, amplitude and velocity travelling in the same medium along a straight line but in opposite directions interfere, the crests and troughs in case of Transverse waves and compressions and rarefactions in case of Longitudinal waves come to a 'stand still' such that energy is not propagated in the medium but conforms to a certain region. This wave pattern is called 'Stationary wave'.
Nodes, Antinodes
The positions of the particles where there is no displacement (N1, N2, N3, N4...) are called 'Nodes' (no displacement) and the positions of the particles having maximum displacement (A1, A2, A3, A4...) are called 'Anti nodes'.
Distance between two successive nodes or two successive antinodes i.e., equal to half the wave length.
i.e., N1N2 = A1A2 = 
Distance between a node and antinode = 
i.e., N1A1 = A1N2 = N2A2 =
The rate of transfer of energy across any section of a stationary wave is 'zero'.
* Sound waves produced in a 'flute' are stationary waves.
Organ (Musical) Pipes
The vibrations of a cylindrical column of air can be taken as made up of two progressive longitudinal vibrations moving with equal and opposite speed super imposed on each other.
Case i) Closed pipe (Pipe close at one end): In pipes closed at one end, no free vibrations at closed end occur, and the particles of the medium (air) are kept static. At the open end, maximum vibrations are possible without any pressure changes. Mouth of the tube always corresponds to 'Antinode' (A) and the closed end to a 'Node' (N). Let the length of the pipe be 'l'.
In figure (i), (ii), (iii), (iv) and (v),
* The first frequency is called 'fundamental note' (or note), the other frequencies are called 'overtones' and all are called 'harmonics' (harmonic means the musical effect).
υ1 : υ2 : υ3 : υ4 : υ5 = 1 : 3 : 5 : 7 : 9
Thus, the frequency ratio of mode of vibrations in closed pipes is odd multiple of fundamental note.
case ii) Open pipes (pipes open at both ends):
for figures (i), (ii), (iii), (iv) and (v),
υ1 : υ2 : υ3 : υ4 : υ5 = 1 : 2 : 3 : 4 : 5
Thus, the frequency ratio of mode of vibrations in open pipes is both odd and even multiples of fundamental note i.e., in open tubes over tones are more in number. More the number of overtones, better will be the musical effect. Hence, the musical instruments like, Flute, Sehnai, Nadaswaram are open organ pipes ('Organ means musical instrument').
Vibrations in strings
A metalic wire where length is more and radius is very less, is considered to be a string. The musical instruments like Veena, Violin and Guitar are fitted with strings under tension.
If a string of length 'l', stretched between two points is plucked at its middle, a transverse wave travels along the string and reflects at either fixed end and forms a stationary wave, such that at the ends of the string 'Nodes' are formed and at the middle an 'Antinode'.
If T is the tension in the string, and 'm' is the mass per unit length of the wire, the velocity of transverse wave in the string c = 
Laws of transverse vibrations
These Laws can be verified by using an instrument called 'Sonometer'.
Your talk can be understood only when...
When you are talking with others, they can follow your speech only when the time interval between two words is atleast 0.1 seconds otherwise, your speech will be heard as 'noise.'
i.e., if one delivers more than 10 words in a second, that talk cannot be understood.
Free or Natural Vibrations
When a tuning fork is vibrated by striking one of its prongs gently (Why?) with a rubber hammer the prongs dance to and fro about their mean position. These vibrations of the tuning fork are called 'free' or 'natural' vibrations.
The fork frequency is inversely proportional to the square of the length of the prongs and directly proportional to the thickness of the prongs.
Even the vibrations of a stretched string when it is plucked are natural vibrations. Shorter the vibrating length, greater will be its frequency.
Why the frequencies of a standard tuning forks are 256, 480, 512... etc. and not 250, 400 and 500?
Forced Vibrations
When the stem of a vibrating tuning fork is kept against the top of a table a much louder sound will be heard than would be the case of the vibrating tuning fork was held in the air. The resulting vibrations of the table are called 'forced vibrations.' Here the particles of the table are forced to vibrate under the influence of external periodic force of the tuning fork. The frequency of the table will be that of the tuning fork.
Here the natural frequency of the external periodic force (vibrating tuning fork) and the vibrating body (table) in which forced vibrations are produced need not be equal that the bodies should be in contact with each other.
In stringed instruments like Veena, Guitar and Violin the strings under tension are stretched over Hollow boxes so that the boxes and the air enclosed in them will undergo forced vibrations when the instrument is played by plucking the strings and the music heard is full of 'melody'.
Sympathetic Vibrations - Resonance
Soldiers marching towards a bridge are asked not to go in step (left, right... left, right...) but to disperse and walk normally. If they march in step, and if the natural frequency of their marching in step is equal to the natural frequency of the bridge which is stretched between two points like a stretched string, sympathetic vibrations may be produced in the bridge and it vibrates with greater and greater amplitude and may collapse.
Here the phenomenon is called 'Resonance' in which two vibrating bodies of nearly equal frequencies without actual contact between them and if one is vibrated, the other picks up those vibration and vibrate with greater and greater amplitude.
Can we change the frequency of a tuning fork?
The frequency of a tuning fork depends on its size. The length of the prongs of tuning fork of frequency 512 Hz will be less that of the tuning fork of frequency 256 Hz.
If a small amount of wax is attached to one of the prongs of a tuning fork, its frequency decreases as the size of the fork has increased.
If one of the prongs of a tuning fork is filed, since its size has decreased, the frequency of the tuning fork increases.
Beats - Here the intensity of sound will not be the same.
When two sound waves of slightly different frequencies travelling in the same medium along the same direction comes one over the other gives rise to a resultant wave whose intensity of sound will not be uniform and the listener will hear a regular rise and fall of the sound i.e., the intensity of sound successively waxes to a maximum and waves to a minimum. One waxing (maximum) and the subsequent waving (minimum) of sound constitute a 'Beat'.
The number of beats produced is equal to the difference of frequencies of interfering waves. If υ1 and υ2 are the frequencies of two sound notes, the no. of beats (n) = υ1 υ2.
When two musicians are playing the Sehanai, if the sound wave of one Sehanai interferes with the sound wave of another Sehanai, beats are produced and the musical effect experienced by the listeners will be very pleasant. Beats give 'rhythemic effect' to music which we find in a 'Western music'!
For the beats to be heard distinctly, they should reach the ear atleast by of a second i.e., the human ear cannot hear more than 10 beats per second.
Intensity of sound
Intensity of sound depends on the 'amplitude' of sound wave. Greater the amplitude, greater will be the intensity. Intensity is proportional to the square of the amplitude. If the amplitude of a sound wave is doubled, the intensity will be quadrupled.
Loudness
It is the hearing experience of the listener which depends on the medium in which the sound is travelling, the condition of the medium like its extent, temperature, humidity and also on the condition of listener like deafness etc.
If intensity is the cause loudness is the effect.
Pitch
It is the shrillness or cuteness of sound which depends on the frequency. Greater the frequency greater will be the pitch.
Pitch of woman's voice is greater that of male's voice. Pitch of the child's voice is greater than that of female voice.
Lion and Mosquito
The intensity of 'roaring of a Lion' is greater than the buzzing of a bee or humming of a Mosquito, but the pitch of the later is greater and irritating than the roaring of a lion.
Quality or Timbre
Though all are stringed instruments, one is able to differentiate the music played on Violin, Sitar, Guitar or Piano because of the mode of vibration of the strings of the instruments i.e., number of 'overtones' produced (overtones are the number of loops produced in the given length of a wire or air column).
One is able to recognize a person's voice without seeing the person because of the quality of the voice ie the number of overtones produced by the voice. More the number of overtones, better will be the musical effect.
Doppler Effect
To a listener standing near a railway track, the pitch (shrillness) of an approaching train's whistle appears to be increasing. Here the number of sound waves that reach his ear per second (i.e., frequency) is greater.
Similarly, the pitch of the sound of the whistle appears to fall when the train is moving away from the listener. Here the number of waves received per second will be less.
This phenomenon of apparent change in the frequency of a wave due to the relative motion between the source and the observer is known as 'Doppler Effect' named after the Austrian Scientist C.J. Doppler.
Doppler's formulae
Let c be the velocity of sound, Vo is the velocity of the observer, Vs is the velocity of source, λ the original wavelength and υ the original frequency.
Doppler Effect - Applications
i) In controlling the speed of aeroplanes:
From the apparent frequency (υ') recorded by the relevant apparatus in the airport, the speed of the aeroplanes can be controlled.
If υ' is greater it means the plane is reaching the airport speedily. Similarly, by knowing the apparent frequency, the war planes can be directed by giving signals from the ground, so that they can bomb the targets.
ii) Regulating the traffic
Traffic police can stand at one place and from the apparent frequency of the sound emitted by the motor vehicles crossing them, their speeds can be found.
iii) Even the speeds of stars - Support to the theory of expanding Universe
In fact, Doppler used his theory to know whether the stars in the galaxy are receding away from the earth (or) moving towards the earth. By applying Doppler's theory it is found that the stars are moving away from the earth and thus Doppler effect has supposed the theory of 'Expanding Universe'.
