Class 11 Physics Chapter 15 Waves
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NCERT Notes For Class 11 Physics Chapter 15 Waves
Class 11 Physics Chapter 15 Waves
WAVES
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, xrays 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 xaxis 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 xaxis at different values of timet.
 A wave travelling in the negative direction of xaxis 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 = x_{1} and at x = x_{1} + 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 μ^{a} T^{b}, where c is a dimensionless constant.
 Taking dimensions on both sides [M^{0}L^{1}T^{1}] = [M^{1}L^{1}]^{a}[M L T^{–2}]^{b} =[Ma+bLa+bT2b]
 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
 Bulk modulus , B and
 Density of the medium, ρ
 Using dimensional analysis we may write
v = C B^{a} ρ^{ b}
 Taking dimensions [M^{0}L^{1}T^{1}] = [ML^{1}T^{2}]^{a} [M L^{3}] ^{b} =[M^{a+b} L^{a3} 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 y_{1}(x, t) and y_{2}(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 xaxis.
 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 xaxis be
 And the wave travelling in the negative direction of xaxis
 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 timedisplacement 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 V_{L}
 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 V_{S} = +ve , V_{L}= 0
 Thus
b) Source moves away from the listener
 Now V_{S} = – ve , V_{L}= 0
Source stationary, listener moving
a) Listener moves towards the source
 Now V_{L} = – ve , V_{S}= 0
b) Listener moves away from the source
 Now V_{L} = + ve , V_{S}= 0
Both source and listener moving
a) Source and listener move towards each other
 Now V_{S} = +ve, V_{L} = ve
b) Source and listener move away from each other

 Now V_{S} = ve, V_{L} = +ve
c) Source moves towards the listener and listener moves away

 Now V_{S} = +ve, V_{L} = +ve
 Source moves away from the listener and listener moves towards the source
 Now V_{S} = ve , V_{L} = 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.