# NCERT Notes For Class 11 Physics Chapter 15 Waves

## Class 11 Physics Chapter 15 Waves

NCERT Notes For Class 11 Physics Chapter 15 Waves, (Physics) exam are Students are taught thru NCERT books in some of state board and CBSE Schools. As the chapter involves an end, there is an exercise provided to assist students prepare for evaluation. Students need to clear up those exercises very well because the questions withinside the very last asked from those.

Sometimes, students get stuck withinside the exercises and are not able to clear up all of the questions. To assist students solve all of the questions and maintain their studies with out a doubt, we have provided step by step NCERT Notes for the students for all classes. These answers will similarly help students in scoring better marks with the assist of properly illustrated Notes as a way to similarly assist the students and answering the questions right.

## WAVE MOTION

• The motion of a disturbance from one point to another by the vibrations of the particles of the medium about their mean position is known as wave motion.
• It is a mode of transfer of energy from one point to another.
• The waves are mainly of three types: (a) mechanical waves, (b) electromagnetic waves and (c) matter waves.

### Mechanical waves

• Exist only within a material medium, such as water, air, and rock
• examples : – water waves, sound waves, seismic waves, etc
• two types : –

1) transverse waves

2) longitudinal waves

### Electromagnetic waves

• The electromagnetic waves do not require any medium for their propagation.
• All electromagnetic waves travel through vacuum at the same speed c, given by c = 299, 792,458 m s–1
• Examples of electromagnetic waves are visible and ultraviolet light, radio waves, microwaves, x-rays etc.

## Matterwaves

• Matter waves are associated with moving electrons, protons, neutrons and other fundamental particles, and even atoms and molecules
• Matter waves associated with electrons are employed in electron microscopes

### TRANSVERSE WAVES

• In transverse waves, the constituents of the medium oscillate perpendicular to the direction of wave propagation.
• A point of maximum positive displacement in a wave is called crest, and a point of maximum negative displacement is called trough. • Transverse waves can be propagated only through solids and strings, and not in fluids.

### LONGITUDINAL WAVES

• In longitudinal waves the constituents of the medium oscillate along the direction of wave propagation. • Longitudinal sound waves propagates as compressions(high pressure region) and rarefactions(low pressure regions)
• longitudinal waves can propagate in all elastic media (solids and fluids)
• transverse and longitudinal waves travel with different speeds in the same medium.

### The waves on the surface of water

• The waves on the surface of water are of two kinds: capillary waves and gravity waves.
• Capillary waves are ripples of short wavelength.
• The restoring force that produces capillary waves is the surface tension of water.
• Gravity waves have wavelengths typically ranging from several metres to several hundred metres.
• The restoring force that produces gravity waves is the pull of gravity, which tends to keep the water surface at its lowest level.
• The waves in an ocean are a combination of both longitudinal and transverse waves.

### Travelling or progressive wave

• A wave which travels from one point of the medium to another is called a travelling wave. ## DISPLACEMENT RELATION IN A PROGRESSIVE WAVE

• At any time t, the displacement of a wave travelling in positive x-axis is given by • Where , a- amplitude , k- angular wave number or propagation constant , ω- angular frequency , φ- initial phase angle and (kx- ωt+ φ) – phase Plots for a wave travelling in the positive direction of an x-axis at different values of timet.  • A wave travelling in the negative direction of x-axis can be represented by ### Amplitude

• The amplitude a of a wave is the magnitude of the maximum displacement of the elements from their equilibrium positions as the wave passes through them.
• It is a positive quantity, even if the displacement is negative.

#### Phase

• It describes the state of motion as the wave sweeps through a string element at a particular position x
• The constant φ is called the initial phase angle.

The value of φ is determined by the initial (t = 0) displacement and velocity of the element (say, at x = 0).

### Wavelength (λ)

• It is the minimum distance between two consecutive troughs or crests or two consecutive points in the same phase of wave motion. Propagation constant or the angular wave number (k)

• For t = 0 and φ = 0 • By definition, the displacement y is same at both ends of this wavelength, that is at x = x1 and at x = x1 + λ.
• Thus • This condition can be satisfied only when, • where n = 1, 2, 3… Since λ is defined as the least distance between points with the same phase, n =1 and therefore • k is called the propagation constant or the angular wave number ; its SI unit is radian per metre or rad m–1

### Period

• The period of oscillation T of a wave is the time any string element takes to move through one complete oscillation. #### Angular Frequency

• The angular frequency of the wave is given by • Its SI unit is rad s-1

#### Frequency

• It is the number of oscillations per unit time made by a string element as the wave passes through it
• The frequency v of a wave is defined as 1/T and is related to the angular frequency ω by • It is usually measured in hertz

#### Displacement relation of a longitudinal wave

• In a longitudinal wave, the displacement of an element of the medium is parallel to the direction of propagation of the wave.
• The displacement function for a longitudinal wave is written as, • where s(x, t) is the displacement of an element of the medium in the direction of propagation of the wave at position x and time t.

## THE SPEED OF A TRAVELLING WAVE

• The speed of a wave is related to its wavelength and frequency by the relation  • The speed is determined by the properties of the medium.

### Speed of a Transverse Wave on Stretched String

• The speed of transverse waves on a string is determined by two factors,

(i) the linear mass density or mass per unit length, μ, and (ii) (ii) the tension T.

• The linear mass density, μ, of a string is the mass m of the string divided by its length l. therefore its dimension is [ML–1].
• The tension T has the dimension of force [M L T–2].
• Let the speed v = C μa Tb, where c is a dimensionless constant.
• Taking dimensions on both sides [M0L1T-1] = [M1L-1]a[M L T–2]b =[Ma+bL-a+bT-2b]
• Equating the dimensions on both sides we get a+b = 0 , therefore a=-b, -a+b = 1, therefore 2b=1 or b= ½ and a= – ½
• Thus

v = C μ– ½ T ½ ,

or • It can be shown that C=1, therefore the speed of transverse waves on a stretched string is • The speed of a wave along a stretched ideal string depends only on the tension and the linear mass density of the string and does not depend on the frequency of the wave.

### Speed of a Longitudinal Wave – Speed of Sound

• In a longitudinal wave the constituents of the medium oscillate forward and backward in the direction of propagation of the wave.
• The sound waves travel in the form of compressions and rarefactions of small volume elements of air.
• The speed of sound waves depends on
1. Bulk modulus , B and
2. Density of the medium, ρ
• Using dimensional analysis we may write

v = C Ba ρ b

• Taking dimensions [M0L1T-1] = [ML-1T-2]a [M L-3] b =[Ma+b L-a-3 bT-2a]
• Equating the dimensions on both sides we get

a+b = 0 , therefore a=-b, -2a=-1, a=1/2 , therefore b=-1/2

• Therefore • where C is a dimensionless constant and can be shown to be unity.
• Thus, the speed of longitudinal waves in a medium is given by, • The speed of propagation of a longitudinal wave in a fluid therefore depends only on the bulk modulus and the density of the medium.
• The bulk modulus is given by • Here ΔV/V is the fractional change in volume produced by a change in pressure ΔP.

Speed of sound wave in a material of a bar

• The speed of a longitudinal wave in the bar is given by, • where Y is the Young’s modulus of the material of the bar.

### Speed of sound in different media ### Newton’s Formula

• In the case of an ideal gas, the relation between pressure P and volume V is given by • Therefore, for an isothermal change it follows that • Thus B=P
• Therefore, the speed of a longitudinal wave in an ideal gas is given by, • This relation was first given by Newton and is known as Newton’s formula. Laplace correction According to Newton’s formula for the speed of sound in a medium, we get for the speed of sound in air at STP, • This is about 15% smaller as compared to the experimental value of 331 m s–1
• Laplace pointed out that the pressure variations in the propagation of sound waves are adiabatic and not isothermal.
• For adiabatic processes the ideal gas satisfies the relation,   • Thus for an ideal gas the adiabatic bulk modulus is given by, • where γ is the ratio of two specific heats, Cp/Cv.
• The speed of sound is, therefore, given by, • This modification of Newton’s formula is referred to as the Laplace correction.
• For air γ = 7/5,therefore the speed of sound in air at STP, we get a value 331.3 m s–1, which agrees with the measured speed.

## THE PRINCIPLE OF SUPERPOSITION OF WAVES

• The principle of super position of waves states that the net displacement at a given time of a number of waves is the algebraic sum of the displacements due to each wave. • Let y1(x, t) and y2(x, t) be the displacements that any element of the string would experience if each wave travelled alone.
• The displacement y (x,t) of an element of the string when the waves overlap is then given by, • Let a wave travelling along a stretched string be given by, • And another wave, shifted from the first by a phase φ, • Both the waves have the same angular frequency, same angular wave number k (same wavelength) and the same amplitude a.
• Applying the superposition principle • Using the trigonometric relation  • Thus, the resultant wave is also a sinusoidal wave, travelling in the positive direction of x-axis.
• The resultant wave differs from the constituent waves in two respects:

i) its phase angle is (½)φ and

II) its amplitude is the quantity given by • If φ = 0,the amplitude of the resultant wave is 2a, which is the largest possible value of A(φ).
• If φ = π, the two waves are completely out of phase, the amplitude of the resultant reduces to zero.

## REFLECTION OF WAVES

• When a pulse or a travelling wave encounters a rigid boundary it gets reflected.
• If the boundary is not completely rigid or is an interface between two different elastic media, a part of the wave is reflected and a part is transmitted into the second medium.
• The incident and refracted waves obey Snell’s law of refraction, and the incident and reflected waves obey the laws of reflection.
• A travelling wave, at a rigid boundary or a closed end, is reflected with a phase reversal.
• A travelling wave ,at an open boundary is reflected without any phase change.
• Let the incident wave be represented by • then, for reflection at a rigid boundary the reflected wave is represented by, • For reflection at an open boundary, the reflected wave is represented by ### Standing Waves and Normal Modes

• The waveform or the disturbance does not move to either side is known as stationary wave or standing wave.
• Let the wave travelling in the positive direction of x-axis be • And the wave travelling in the negative direction of x-axis • The principle of superposition gives, for the combined wave • The amplitude is zero for values of kx that give sin kx = 0 . Those values are given by • Substituting k = 2π/λ in this equation, we get ### NODES

• The positions of zero amplitude in a standing wave are called nodes.
• A distance of λ/2 or half a wavelength separates two consecutive nodes.
• The amplitude has a maximum value of 2a, which occurs for the values of kx that give ⎢sin k x ⎢= 1.
• The values are • Substituting k = 2π/λ in this equation, we get ### ANTINODES

♦ The positions of maximum amplitude are called antinodes.

♦ The antinodes are separated by λ/2 and are located half way between pairs of nodes.

### STANDING WAVES ON A STRETCHED STRING

• For a stretched string of length L, fixed at both ends, the two ends of the string have to be nodes.
• If one of the ends is chosen as position x = 0, then the other end is x = L. In order that this end is a node; the length L must satisfy the condition The standing waves on a string of length L have restricted wavelength given by • The frequencies corresponding to these wavelengths is given by • where v is the speed of travelling waves on the string.
• The set of frequencies possible in a standing wave are called the natural frequencies or modes of oscillation of the system.
• The frequency corresponding to n=1 is • The oscillation mode with this lowest frequency (n=1) is called the fundamental mode or the first harmonic.
• The second harmonic is the oscillation mode with n = 2. The third harmonic corresponds to n = 3 and so on.

The frequencies associated with these modes are often labelled as v1, v2, v3 and so on.

• The collection of all possible modes is called the harmonic series and n is called the harmonic number.  Modes of vibration of a pipe closed at one end

• In a closed pipe standing waves are formed such that a node at the closed end and antinode at open end.
• Now if the length of the air column is L, then the open end, x = L, is an antinode and therefore,

L = +(n )

• Where n=0,1,2,3….
• The modes, which satisfy the condition • The corresponding frequencies of various modes of such an air column are given by, • The fundamental frequency is v/4L and the higher frequencies are odd harmonics of the fundamental frequency, i.e. 3 v/4L,5 v/4L,…  ### Pipe open at both ends

• In the case of a pipe open at both ends, there will be antinodes at both ends, and all harmonics will be generated. ## BEATS

• The phenomenon of wavering of sound intensity when two waves of nearly same frequencies and amplitudes travelling in the same direction, are superimposed on each other is called beats.
• The beat frequency, is given by The time-displacement graphs of two waves of frequency 11 Hz and 9 Hz • Musicians use the beat phenomenon in tuning their instruments.
• If an instrument is sounded against a standard frequency and tuned until the beat disappears, then the instrument is in tune with that standard.

## DOPPLER EFFECT

• The apparent change in the pitch or the frequency of sound produced by a source due to relative motion of the source, listener or the medium is called Doppler effect. • It was proposed by Christian Doppler and tested experimentally by Buys Ballot
• All types of waves shows Doppler effect.
• S- source
• f – frequency of sound from source
• V – velocity of sound
• λ- wavelength

### When Source and Listener at Rest

• When the source and the listener are at rest , the frequency of sound heard by the listener ### When source and listener moving in the direction of sound

• The relative velocity of sound wave with respect to source is V – Vs
• Vs – velocity of source
• Thus, apparent wavelength is • The relative velocity of sound with respect to listener is V = −V VL
• The apparent frequency of sound heard by the listener is • Thus ## Special cases

Source moving and listener stationary

### a) Source moves towards the listener

• Now VS = +ve , VL= 0
• Thus ### b) Source moves away from the listener

• Now VS = – ve , VL= 0 Source stationary, listener moving

### a) Listener moves towards the source

• Now VL = – ve , VS= 0 ### b) Listener moves away from the source

• Now VL = + ve , VS= 0 ### Both source and listener moving

#### a) Source and listener move towards each other

• Now VS = +ve, VL = -ve #### b) Source and listener move away from each other

• Now VS = -ve, VL = +ve #### c) Source moves towards the listener and listener moves away

• Now VS = +ve, VL = +ve 1. Source moves away from the listener and listener moves towards the source
• Now VS = -ve , VL = -ve ## Effect of motion of the medium

• When the wind blows the air medium will moves with a velocity w

When wind moves towards the listener the velocity of sound is V+ w

• Thus, the apparent frequency • If the wind is blowing from listener to the source , velocity of sound is V – w ### Uses of Doppler Effect

• To estimate the speed of submarine , aero plane, automobile , etc
• To track artificial satellite
• To estimate velocity and rotation of star
• Doctors use it to study heart beats and blood flow in different part of the body. Here they use ulltrasonic waves, and in common practice, it is called sonography.
• In the case of heart, the picture generated is called echocardiogram.