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
- 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.
- 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
- 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.
- 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
- 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.
- 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
- 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.
- 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).
- 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.
- 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
- The period of oscillation T of a wave is the time any string element takes to move through one complete oscillation.
- The angular frequency of the wave is given by
- Its SI unit is rad s-1
- 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
v = C μ– ½ T ½ ,
- 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 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
- 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
- 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
- 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
♦ 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.
- 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.
- 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
Source moving and listener stationary
a) Source moves towards the listener
- Now VS = +ve , VL= 0
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
- 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.