**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, 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 = 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+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

- 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^{-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 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 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 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

- Now V

#### c) Source moves towards the listener and listener moves away

- Now V
_{S}= +ve, V_{L}= +ve

- Now V

- Source moves away from the listener and listener moves towards the source
- Now V
_{S}= -ve , V_{L}= -ve

- Now V

## 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.