Chapter 15 Wave motion

Transverse and longitudinal waves, speed of wave motion, displacement relation for a progressive wave, principle of superposition of waves, reflection of waves, standing waves in strings and organ pipes, fundamental mode and harmonics, Beats, Doppler effect.

Unit-10(B): Waves

21 Waves Motion:

It is a disturbance which travels through a medium due to vibrations of the particles of the medium about their mean position. The disturbance is handed from one particle to the next particle.

 In a wave motion both information and energy is transferred in form of signal from one place to another place.

 For example when a stone is dropped in pond of water then a circular pattern of alternating crest and trough is formed due to vibration of particles of water. These particle transfers energy from one layers to another layer and when a piece of paper is placed on water surface then it does not moves, show that during wave motion the particle of the medium does not moves.

Characteristics of Wave Motion:

In wave motion disturbance moves in a medium due to vibration of the particle of the medium.
The energy is transferred during wave motion.
There is a little phase difference between vibrations of the particle.
The velocity of wave motion is different from velocity of vibrations of the particles.
The velocity of particles is maximum at mean position and zero at extreme positions. While wave velocity remains constant in medium.
There should be elasticity, properties of inertia and minimum friction for propagation of a wave.

22 Types of Wave:

Waves are mainly of three types:

Mechanical Wave:

The wave which requires a material medium for propagation is called M.W.

e.g: Water waves, sound waves, seismic waves (waves produced during earth quake).

 The propagation of M.W. depends upon elastic properties of medium so also called elastic waves.

Electromagnetic Waves:

The waves which travel in from of oscillating electric and magnetic field and do not require any material medium for their propagation is called E.M. waves e.g: Visible and ultraviolet lights, radio waves, micro waves, infrared waves, rays and rays.

Matter Waves:

The waves associated with material particles, when they are in motion are called matter waves. These waves have significance in microscopic particles.

e.g: Electron microscope uses matter wave associated with fat moving.

23 Spring Model for Propagation of a Wave through an Elastic Medium:

Propagation of Sound Waves through Air:

If we suppose a small region of air as spring, connected with other small regions or spring. When sound waves moves through air then it compresses or expand a small region. When there is compression in one region then there is rarefaction in other region. Thus compression and rarefaction moves from one region to another region which causes the propagation of wave in a medium.

Propagation of Sound Waves in Solids:

In solids, we can assume that each atom is connected with other neighboring atom with a elastic spring. When sound wave propagates through solid then atoms displaces from their equilibrium produced by this force travel to next atom and so on. In this way a wave propagate in solids.

24 Transverse Waves:

The wave in which the particle of the medium vibrates perpendicular to the propagation of waves is called transverse waves.

Suppose a string whose one end is fixed and when a jerk is given to the other end held in hand, then a crest or trough is forms in string which moves toward other fixed end of the string.

Each part of the string vibrate up and down while wave along the string so the wave in the string are transverse in nature.

The point of maximum displacement in upward direction are called crest and the points of maximum displacement in downward direction are called trough. One crest and one trough together from a wave.

25 Longitudinal Waves:

The waves in which the particle of medium vibrates along the propagation of waves are called longitudinal waves for example.

Suppose if we push or pull a piston in a small hollow cylinder filled with some air then sound wave travel in from of alternating Compressions and rarefactions. The oscillations of particle are parallel to propagation of wave. Thus sound waves are longitudinal waves.

 A transverses wave can be produced in solids or string and not in fluids as it moves in from of crest and trough.

 But due to surface tension, the free surface of liquid maintains its level so transverse waves can be formed on liquid surface.

 Longitudinal waves can be propagating in all three mediums solid, liquid and gas.

26 Some Definitions in Connection with Wave Motion:

Amplitude(A):

The maximum displacement of the particle of medium about their mean position is called amplitude.

Time Period(T):

The time in which a particle complete its one vibration to and fro about mean position is called time period.

Frequency( ):

The number of waves produced in a medium per unit time is called frequency of the wave

i.e. Unit: or Hertz (Hz).

Angular Frequency(w):

The rate of change of phase of wave is called angular frequency of the wave

i.e.

Unit: .

Wave Length ():

The distance between two successive crest or trough is called wave length.

The distance of a crest and trough is called wave length.

Wave number ():

The number of wave present in unit distance is called wave number.

It is equal to reciprocal of wave length.

Thus

S.I. Unit:

Angular Wave Number or Propagation Constant(K):

The wave number multiplied by phase 2π is called angular wave number.

i.e.

Wave Velocity or Phase Velocity:

The distance travelled by wave per unit time in its direction of propagation is called its wave velocity or phase velocity.

Relation between Wave Velocity, Frequency and Wavelength:

⟹

27 Speed of Transverse Wave:

It is found that speed of transverse wave is directly proportional to tension in the string and inversely proportional to mass per unit length of the string

So where

In case of solids the speed of transverse wave is

Where 𝛈 is coefficient of in rigidity and is the density of the solid.

28 Speed of Longitudinal wave in liquid and Gases:

In longitudinal waves, the particle of the medium oscillates forward and backward in the direction of propagation of the wave. They cause compression and rarefaction of small volume element of the fluid. So speed depends upon Bulk modulus and density of the fluid. So speed can be expressed as

In case of solids

Where 𝛈 is modulus of rigidity.

29 Speed of Sounds: Newton’s Formula and Laplace Correction:

The speed of sound wave at constant temperature can be given as

Also

Differentiating both sides, we get

Now Newton’s formula for speed of sound in a gas

At S.T.P.

Also

⟹

The above value is 15% less then experimental value (331) of speed of sound in air at STP.

Latterly Laplace pointed out that sound travel adiabatically not isothermally by this Newton’s formula modified as

Where

⟹

Which is close to the experimental value, hence Laplace correction is justified.

30 Factor Affecting Speed of Sounds in A Gas:

Effect of Pressure:

At constant T

When pressure a change, then density is also changes in same ratio, thus there is no any effect of pressure on speed of sound waves.

(ii) Effect of Density:

If and are the density of two gases, then speed of sound in them will be

and

Hence at constant pressure, the speed of sound in a gas inversely proportionality to the square root of the densities.

(iii)Effect of Humidity:

As ⟹

As the density of water vapor is less than that of dry air so sound travel faster is moist air then dry air.

(iv)Effect of Temperature:

For one mole of a gas,

If M is molecular mass of the gas and is the density then

⟹

Clearly

Hence speed of sound in a gas is directly proportional to the square root of it absolute temperature.

(v)Effect of wind:

As sound moves with the help of wind or air, if wind in direction of sound then speed of sound increases and when wind flow in opposite to direction of sound then speed of sound decreases.

(vi)Effect of Frequency:

The speed of sound is not affected by the frequency of sound.

(vii)Effect of Amplitude:

To a large extent, speed of sound wave is independent on amplitude of the waves.

31 Displacement Relation for a Progressive Wave:

Progressive Wave:

A wave which travels from one of the medium to another is called a progressive wave. It may be transverse or longitudinal.

32 Plane Progressive Harmonic Wave:

During propagation of progressive wave, if particle of the medium vibrate simple harmonically then motion of wave is called plane progressive harmonic wave.

The displacement travelled by simple harmonic progressive wave is

Phase and Phase Difference:

Phase of Wave:

The phase of a wave is a quantity which gives us the complete information of the wave at any time and position. Suppose a harmonic wave is given by

Where phase of wave is

∅

The path difference ∆ and phase difference ∆∅ of the wave are related

33 Principle of Superposition of Waves:

According to principle of superposition of two waves the resultant amplitude of the wave moving though a medium is equal to the algebraic sum of all the amplitudes of the waves. i.e:

When crest falls over crest and trough fells over trough then resultant amplitude of the wave adds up and when crest falls over trough and trough falls over crest then resultant subtract off. As shown in fig.

5.6 REFLECTION OF WAVES

Suppose a pulse travelling along a stretched string and being reflected by the boundary. the reflected wave has the same shape as the incident pulse but it suffers a phase change of π or 180⁰ on reflection. This is because the boundary is rigid and the disturbance must have zero displacement at all times at the boundary. By the principle of superposition, this is possible only if the reflected and incident waves differ by a phase of π, so that the resultant displacement is zero. This reasoning is based on boundary condition on a rigid wall. The phenomenon of echo is an example of reflection by a rigid boundary.

If the boundary is rigid, the pulse or wave gets reflected. If the boundary is not completely rigid, a part of the incident wave is reflected and a part is transmitted into the second medium. The incident and refracted waves obey Snell’s law of refraction.

To study, a travelling wave or pulse suffers a phase change of π on reflection at a rigid boundary and no phase change on reflection at an open boundary.

let the incident travelling wave be

At a rigid boundary, the reflected wave is given by

At an open boundary, the reflected wave is given by

Clearly, at the rigid boundary, at all times.

34 Stationary Waves:

When two identical waves of same amplitude, frequency travel in opposite direction with same speeds along the same path super impose each other and resultant wave does not travel in the either direction called stationary or standing waves.

35 Standing Waves and Normal Modes

Consider a wave travelling along the positive direction of x-axis and a reflected wave of the same amplitude and wavelength in the negative direction of x-axis. With φ = 0,

We get:

The resultant wave on the string is, according to the principle of superposition:

Using trigonometric identity we get,

……..(1)

Here The terms kx and ωt appear separately. The string as a whole vibrates in phase with differing amplitudes at different points.

The wave pattern is not moving to the right or to the left so called standing or stationary waves. The amplitude is different at different locations. The points at which the amplitude is zero are nodes and points at which the amplitude is the largest are called antinodes.

The most significant feature of stationary waves is that the system cannot oscillate with any arbitrary frequency but is characterized by a set of natural frequencies or normal modes of oscillation.

Normal modes for a stretched string fixed at both ends.

First, from Eq. (1), the positions of nodes (where the amplitude is zero) are given by

Sin kx = 0 . This implies

Since,

Taking one end to be at x = 0, the boundary conditions are that x = 0 and x = L are positions of nodes.

The node condition requires that the length L is related to λ by

In the same way, the positions of antinodes are given by when : = 1

This implies

With ,

we get x = .(4)

Thus, the possible wavelengths of stationary waves are constrained by the relation

With corresponding frequencies

The lowest possible natural frequency of a system of stretched string fixed at either end is called its fundamental mode or the first harmonic corresponding to n = 1 given by,

The n = 2 frequency is called the second harmonic. n = 3 is the third harmonic and so on.

Generally, the vibration of a string will be a superposition of different modes; some modes may be more strongly excited and some less.

Musical instruments like sitar or violin are based on this principle. Where the string is plucked or bowed, determines which modes are more prominent than others.

Normal modes of oscillation of an air column with one end closed and the other open.

A glass tube partially filled with water illustrates this system. The end in contact with water is a node, while the open end is an antinode. Taking the end in contact with water to be x = 0, the node condition (2) is already satisfied.

If the other end x = L is an antinode, Eq. (3) gives

for..(7)

The possible wavelengths are then restricted by the relation: for

The normal modes –the natural frequencies –of the system are

The fundamental frequency corresponds to n = 0, is given by .

The higher frequencies are odd harmonics, i.e., odd multiples of the fundamental frequency:, etc.

Stationary Waves may be of Two Types:

Transverse Stationary Waves: When two identical transverse waves travelling in opposite direction overlaps then transverse stationary wave is formed.
Longitudinal Stationary Waves: When two identical longitudinal waves travelling in opposite direction overlap then longitudinal stationary waves are formed.

Characteristics of Stationary Waves:

In stationary waves, the disturbance does not propagate in forward.
All particle of the medium except at nodes execute S.H.M.
The distance between two nodes or antinodes is.
The maximum velocity is different at different points.
There is no any energy transference in stationary waves:

Nodes and Antinodes

The patterns of standing wave such that there are points along the medium having no displacement are referred to as nodes.

There are other points along the medium that undergo maximum displacement during each vibration cycle are called antinodes,

A standing wave pattern always has nodes and antinodes appearing alternatively in them.

36 STANDING WAVES IN CLOSED ORGAN PIPES: ANALYTICAL TREATMENT

37 ORGAN PIPES:-

An organ pipe is a wind instrument which is used for producing musical sound by setting the air column inside the pipe into vibrations. Practical examples of organ pipes are Bugle, Trumpet, Flute, and Clarinet. Shehnai, Saxophone etc.

 An organ pipe consists of a hollow wooden or metallic cylindrical tube of suitable length and suitable diameter. A small narrow tapering tube M is attached to one end of the pipe is called mouth piece. The other end of the pipe may be open or closed as shown in Fig.

If the other end is open the pipe is called an open organ pipe, and if the other end is closed, the pipe is called closed organ pipe.

Inside the pipe, near the mouth piece, there is a slanting wooden block B. The size of the block is so adjusted that a narrow slit L is left between the surface of the wooden block and the wall of the organ pipe. L is called the lip of the pipe

When air is blown into the pipe from the mouth piece, it strikes the sharp lip L and throws the air on either side of the lip into eddies.

The eddies so generated at regular intervals produce a musical sound of definite frequency called edge tone.

The vibrations in air travel to the other end of the pipe and get reflected, these reflected waves superimpose upon the freshly generated vibrations of air column. When frequency of edge tone becomes equal to frequency of vibration of air column in the pipe, resonance occurs and the sound produced is the loudest.

The frequency of the sound depends upon the size of the pipe and also on pressure with which air is blown into the pipe. The eqn. of longitudinal stationary waves set up in a closed organ pipe is

……………………… (1)

At the closed end of the pipe.

From (1) S= 0, i.e. A node is formed

At the open end of the pipe of length L S=L an antinodes is to be formed, i.e., S=max.

From (1) S will be max.

When

where n = 1, 2, 3…..

………………2

First normal mode of vibration

Let, be the wavelength o corresponding to n = 1.

From (2)

The frequency of vibration in this mode is given by

This is the lowest frequency of vibration called fundamental frequency.

This mode of vibration of one node and one antinodes as shown a. The note or sound so produced is called fundamental note or first harmonic,

Second normal mode of vibration

Let be the wavelength corresponding to n=2.

From (2)

The frequency of vibration in this mode is given by

Thus, the frequency of vibration in 2nd normal mode is thrice the fundamental frequency,

This mode of vibration of two nodes and two antinodes as shown in fig. b. The note so produced is called third harmonic.

It is the first overtone produced in a closed organ pipe

Overtone: These are the notes/sounds of frequency twice/thrice/four times ….the fundamental frequency

Third normal mode of vibration

Let , be the wavelength corresponding to n = 3

From (2).

The frequency of vibration in this mode is given by

The frequency of vibration in the 3rd normal mode is five times the fundamental frequency.

This mode of vibration of three nodes and three antinodes as shown c. The note or sound so produced is called fifth harmonic, this is the second overtone produced in the closed organ pipe.

Proceeding as above, the frequency of note produced in nth normal mode of vibration of closed organ pipe would be

This is harmonic or overtone.

38 STANDING WAVES IN OPEN ORGAN PIPES: ANALYTICAL TREATMENT

An open organ pipe is open at both ends. Therefore, antinode is formed at each end as shown in Fig… Proceeding as in the case of closed organ-pipe,

And setting S= max at and at L

We get….(1)

First normal mode of vibration

From (1).

The frequency of vibration in this mode is given by

This is the lowest frequency of vibration and is called the fundamental frequency. There are two antinodes and one node as shown. The note or sound of this frequency is called fundamental note or first harmonic,

(ii) second normal mode of vibration

For n = 2. From (1))

The frequency of vibration in this mode is given by

i.e., frequency of vibration in second normal mode is twice the fundamental frequency. there are two nodes and three antinodes as shown in Fig. The note so produced is called second harmonic or first overtone.

(iii) Third normal mode of vibration

For n= 3.

From (1 )

The frequency of vibration in this mode is given by

i.e., frequency of vibration in third normal mode of open organ pipe is thrice the fundamental frequency. There are three nodes and four antinodes as shown in fig. The note so produced is called third harmonic or second overtone.

In general, the frequency of vibration in nth normal mode of vibration in open organ pipe would be

The note so produced would be called nth harmonic or th overtone. It would contain n modes and antinodes.

Special cases:

Comparison of closed and open organ pipes shows that fundamental note in open organ pipe has double the frequency of the fundamental note in closed organ pipe.

Further, in an open organ pipe, all harmonics are present whereas in a Closed organ pipe, only alternate harmonics of frequencies. Etc. is present. The harmonics of frequencies are missing.

Hence the overall musical sound produced by an open organ pipe is richer than the musical sound produced by a closed organ pipe.

2. Harmonies are the notes/sounds of frequency equal to or an integral multiple of fundamental frequency (). Thus, first second third…. harmonics have frequencies respectively.

3. Overtones are the notes/sounds of frequency twice thrice four times the fundamental frequency Thus; first second third overtones have frequencies .respectively and so on.

Problem A pipe 20 cm long is closed at one end. Which harmonic mode of the pipe resonantly excited by a 430 Hz source ? Will the same source be in resonance with the pipe if both ends are open? Take speed of sound in air 340 m/s

Sol. Here,

Fundamental frequency of vibration in a closed organ pipe

Hence a source of freq 430 Hz may excite resonantly the fundamental mode of closed pipe.

If pipe is open at both ends, fundamental frequency

Same source of frequency 430 will not be in resonance with the pipe open at both ends.

39 BEATS

This phenomenon of alternate variation in the intensity of sound with time at a particular position, when two sound waves of nearly equal frequencies and amplitudes travelling in the same direction superimpose on each other is called beats.

The time interval between two successive beats (I.e. two successive maxima of sound) is called beat period. The number of beats produced per second is called beat frequency.

For example, when two sounds of very close frequencies, say 256 Hz and 260 Hz reach our ears simultaneously, we hear a sound of frequency 258 Hz, which is the average of two combining frequencies. In addition, the intensity of sound heard increases and decreases slowly.

The number of beats heard per second is equal to 4. This is the difference in frequencies of two incoming sounds.

40 FORMATIONS OF BEATS

(a) Graphical method

Suppose we have two tuning forks A and B. Let the frequency of fork A be 6 and frequency of fork B be 8. Let the waves of compression and rarefaction given by the forks A and B be represented by curves (a) and (b). crest represents a compression and a trough represents a rarefaction

Fig. (c) Shows superimposition of the two waves from forks A and B

And in Fig. (d), we have represented the resultant wave according to the principle of superposition.

Thus one beat is formed in 1/2 second between P and R. Similarly, another beat is formed in the next 1/2 second between R and T. Hence number of beats per second is equal to two, which is also the difference in frequencies of the two forks A and B.

(b) Analytical Method

Let us consider two waves of equal amplitude and slightly different frequencies , and , travelling in a medium in the same direction.

Let the time dependent variations of the displacements due to two sound waves at a particular location be

And

According to superposition principle, the resultant displacement s at the same time is

Using trigonometric relation

We get

= A

Here A= represents the amplitude of the resultant wave given by equation. (3).

Clearly, the resultant amplitude A changes with time.

(1 ) The amplitude A will be maximum,

When = max = + 1 = cos n where,

Hence the amplitude of resultant wave and therefore, resultant intensity of sound will be maximum at

Time interval between two successive maxima of sound

I.e., frequency of maxima =

(2) The amplitude A will be minimum,

When = minimum = 0

cos

Or ….. (4).

Hence the amplitude of resultant displacement and therefore, resultant intensity of sound will be minimum at times,

Time interval between two successive minima of sound

I.e. Frequency of minima

= ………………(5)

Comparison of values for maxima and minima intensity shows that maximum and minimum intensity of sound occurs alternately.

Combining (4) and (5),

we get, frequency of beats =

i.e.,

Applications of Beats:

For tuning musical instruments.
For producing colorful effects in music.
Beats are used in electronics.
For detection of marsh gas in mines.

41 DOPPLER’S EFFECT IN SOUND

According to Doppler Effect, whenever there is a relative motion between a source of sound, the listener and the intervening medium, the apparent frequency of sound heard by the listener is different from the actual frequency of sound emitted by the source.

When the distance between the source and listener is decreasing, the apparent frequency increases the reverse is also true. For example:

1. The frequency of a whistling engine heard by a person standing on the platform appears to increase, when the engine is approaching the platform, and it appears to decrease when the engine is moving away from the platform

Thus basically, Doppler Effect is the motion related change in frequency of sound.

 Note that Doppler Effect is essentially a wave phenomenon. It holds not only for sound waves, but also for electromagnetic waves.

Expression for Apparent Frequency

Let S be a source of sound and L, the listener. Both are initially at rest. Let be the actual frequency of sound emitted by the source and be the actual wavelength of the sound emitted.

If is velocity of sound in still air, then

Let us now consider a situation, in which medium, source and listener move in the same direction from S to L with velocities, respectively,

As a result of the motion of the medium, the sound waves in one second travel a distance

This is the resultant velocity of sound along SL.

Further, the distance moved by the source in one second along SL.

Relative velocity of sound with respect to the source =

So the apparent wavelength

………….(1)

These waves of wavelength travel towards the listener, who himself moves through a distance in one second.

Relative velocity of sound waves with respect to listener

Apparent frequency of sound waves heard by the listener is

………..(2)

Using (1) ……………..(3)

Sign Convention. All velocities along the direction S to L are taken as positive and all velocities along the direction L to S are taken negative.

As case the medium is moving in the opposite direction from listener to source, is negative.

Now the effective velocity of sound waves

Hence, observed frequency is given by

……………….(4)

In case medium is stationary, 0.

Therefore, from (3) & (4)

……….(5)

Special Cases.

If the source is moving towards the listener but the listener is at rest,

Then, is positive and

Therefore, from (5),

If the source is moving away from the listener, but the listener is at rest,

then is negative and , Therefore, from (5),

we have

If the source is at rest and listener is moving away from the source,

then and is positive,

Therefore, from (5),

If the source is at rest and listener is moving towards the source,

then and is negative,

Therefore, from (5)
If the source and listener are approaching each other.

then , is positive and is negative,

From (5),

If the source and listener are moving away from each other,

then , is negative and is positive,

Therefore, from (5), we have

If the source and listener are both in motion in the same direction and with same velocity, then (say).

From (5),

It means, there is no change in the frequency of sound heard by the listener

If the source and listener move at right. angles to the direction of wave propagation

The velocity of source and listener in direction of wave propagation

= cos 90 =0 and

From (5), or .

It means there is no change in the frequency of sound heard if there is a small displacement of the source and listener at right angle to the direction of wave propagation.

42 APPLICATIONS OF DOPPLER’S EFFECT

The change in frequency caused by a moving source/observer t/listener is called Doppler shift. The measurement of Doppler shift has been used.

(1) By police to check over speeding of vehicles,

(2) At airports to guide the aircraft,

(3) In the military to detect enemy aircrafts,

(4) By astrophysicists to measure the velocities of planets and stars,

(5) To study heart beats and blood flow in different parts of the

43 Characteristics of Musical Sounds:

Music:

The sound which has a pleasing sensation to the ears is called music.

e.g: The sound produced by sitar, violin and table are music.

Noise:

The sound which has none pleasing or jarring effect on ears is called noise.

e.g: The sound produced by explosion, from market etc.

44 Characteristics:

There are three characteristics of musical sounds

(i) Loudness (ii) Pitch (iii) Quality or Timbre

(i) Loudness:

The amount of energy crossing per unit area around a point in one second is called loudness.

(ii) Pitch:

It is a sensation which helps a listener to distinguish between a high and a grave note. It depends on frequency. The pitch of a lady’s sound is higher than that of a man.

(iii) Quality or Timbre:

Quality of sound helps us to distinguish between two sounds of same pitch and loudness. It is due to quality due to which we can recognize one’s friend without seeing him.