Notes-Part-1-Class-12-Physics-Chapter-6-Superposition of Waves-MSBSHSE

Difference between mechanical waves and electromagnetic waves :

Properties of progressive waves:

1) Each particle in a medium executes the same type of vibration. Particles vibrate about their mean positions performing simple harmonic motion.

2) All vibrating particles of the medium have the same amplitude, period and frequency.

3) The phase, (i.e., state of vibration of a particle), changes from one particle to another.

4) No particle remains permanently at rest. Each particle comes to rest momentarily while at the extreme positions of vibration.

5) The particles attain maximum velocity when they pass through their mean positions.

6) During the propagation of wave, energy is transferred along the wave. There is no transfer of matter.

7) The wave propagates through the medium with a certain velocity. This velocity depends upon properties of the medium.

8) Progressive waves are of two types : transverse and longitudinal.

• In a transverse mechanical wave, the individual particles of the medium vibrate perpendicular to the direction of propagation of the wave. The progressively changing phase of the successive particles results in the formation of alternate crests and troughs that are periodic in space and time. In an electromagnetic wave, the electric and magnetic fields oscillate in mutually perpendicular directions, perpendicular to the direction of propagation.
• In a longitudinal mechanical wave, the individual particles of the medium vibrate along the line of propagation of the wave. The progressively changing phase of the successive particles results in the formation of typical alternate regions of compressions and rarefactions that are periodic in space and time. Periodic compressions and rarefactions result in periodic pressure and density variations in the medium. There are no longitudinal em wave.

9) A transverse wave can propagate only through solids, but not through liquids and gases while a longitudinal wave can propagate through any material medium.

Physical quantities related to a progressive wave :

• Wave speed : The distance covered by a progressive wave per unit time is called the wave speed.
• Frequency: The number of Waves that pass per unit time across a given point of the medium is called the frequency of the wave. It is equal to the number of vibrations per unit time made by a particle of the medium.
• Wavelength : Wavelength is the distance between consecutive particles of the medium which are moving in exactly the same way at the same time and have the same displacement from their equilibrium positions.  Such particles are said to be in the same phase (the same state of vibration).
• Amplitude : The magnitude of the maximum displacement of a particle of the medium from its equilibrium position is called the amplitude of the wave.
• Period : The time taken for a complete wave (one wavelength long) to pass a given point in the medium is called the period of the wave. It is equal to the periodic time of the vibrational motion of a particle of the medium.
• Wave number : The number of waves present per unit distance is called the wave number.

Equation of a simple harmonic progressive wave in different forms :

A simple progressive wave travelling along the positive x-direction is given by

y=A sin (ωt−kx)  ….(1)

where A is the amplitude of the wave, k is the wave number and ω is the angular frequency.

Wave number, k = \(\frac{2π}{λ}\)

∴ y = A sin (ωt − \(\frac{2π}{λ}x\))   …..(2)

Angular frequency, ω = 2πn, Eq. (2) can be written as

∴ y = A sin (2πnt − \(\frac{2π}{λ}x\))

y = A sin 2π(nt − \(\frac{x}{λ}\))  …….(3)

y = A sin 2πn(t − \(\frac{x}{nλ}\))

But nλ = v, the velocity of the wave.

y = A sin 2πn(t − \(\frac{x}{v}\))  ……….(4)

Also, writing n = v/λ in Eq. (3), we get,

y = A sin \(\frac{2π}{λ}(vt − x)\)  …… (5)

Frequency of vibrations, n= 1/T Eq. (2) can be written as

y = A sin \(2π(\frac{t}{T}-\frac{x}{λ})\)   …… (6)

Equations (1), (2), (3), (4), (5) and (6) are the different forms of the equation of a simple harmonic progressive wave.

Reflection of transverse waves :

Reflection of transverse waves from a denser medium : Suppose that a crest of a transverse wave travels along a string and is incident on the surface of a denser medium such as a rigid wall at point B, as shown in Fig. As the crest cannot travel further, it is reflected.

Since point B is fixed, its displacement is always zero. Therefore, the crest must be reflected in such a way, that the displacement at B due to the reflected wave is exactly equal in magnitude and opposite in direction to that due to the incident wave. Therefore, the crest is reflected as a trough. Hence, there is a phase change of 180° or p radians when transverse waves are reflected from a denser medium.

Reflection of transverse waves from rarer medium : In this case, the particles of the medium are free to vibrate. Hence, (i) there is no change of phase (ii) a crest is reflected as a crest and a trough is reflected as a trough.

Superposition of Two Wave Pulses of Equal Amplitude and Same Phase Moving towards Each Other :

Consider two wave pulses of the same amplitude and phase moving towards each other, as shown in Fig.

When the two pulses cross each other, stages (b) to (d), the resultant displacement is equal to the vector sum of the displacements due to the individual pulses (shown by dashed lines). Therefore, the resultant displacement at that point becomes maximum. This phenomenon is called constructive interference. After crossing each other, both the pulses continue to propagate with their initial amplitude.

Superposition of Two Wave Pulses of Equal Amplitude and Opposite Phases Moving towards Each Other :

Consider two wave pulses of the same amplitude but opposite phase moving towards each other, as shown in Fig.

These wave pulses superimpose at t = 2 s and the resultant displacement (full line) is zero, due to individual displacements (dashed lines) differing in phase exactly by 180°. This is destructive interference.

Displacement due to one wave pulse is cancelled by the displacement due to the other wave pulse when they cross each other (Fig. (c)). After crossing each other, both the wave pulses continue and maintain their individual shapes.

An equation for the resultant wave produced due to superposition of two waves :

Consider two waves of the same frequency, different amplitudes A₁ and A₂ and differing in phase by φ. Let these two waves interfere at x = 0.

The displacement of each wave at x = 0 are

y₁ = A₁ sin ωt

y₂ = A₂ sin (ωt + φ)

According to the principle of superposition of waves, the resultant displacement at that point is

y = y₁ + y₂

= A₁ sin ωt + A₂ sin (ωt + φ)

Using the trigonometrical identity,

sin (C+D) = sin C cos D + cos C sin D,

y = A₁ sin ωt+A₂ ωt cos φ + A₂ cos ωt sin φ

y = (A₁+A₂ cos φ) sin ωt + A₂ sin φ cos ωt  ……(1)

Let (A₁+A₂ cos φ) = A cos θ   …..(2)

And A₂ sin φ = A sin θ  ……(3)

Substituting Eqs. (2) and (3) in Eq. (1), we get the equation of the resultant wave as

y = A cos θ  sin ωt + A sin θ cos ωt 

= A sin (ωt + θ) ...(4)

It has the same frequency as that of the interfering waves. The amplitude A of the resultant wave is given by squaring and adding Eqs. (2) and (3)

A² cos²θ  + A² sin²θ = (A₁+ A₂ cos φ)² + A₂² sin²φ  

A²(cos²θ  + sin²θ) = A₁²+ 2A₁A₂ cos φ + A₂² cos²φ + A₂² sin²φ

∴ A² = A₁²+ 2A₁A₂ cos φ + A₂²(cos²φ + sin²φ)

∴ A = \(\sqrt{A_1^2+2A_1A_2 cos φ + A_2^2}\)  …….(5)

Expression for the amplitude of the resultant wave when two waves are (1) in phase (2) out of phase :

Case (1) :When the two interfering waves are in phase, φ = 0. Then, the amplitude of the resultant wave is

A = \(\sqrt{A_1^2+2A_1A_2 cos 0 + A_2^2}\)

  ∴ A = \(\sqrt{A_1^2+2A_1A_2 + A_2^2}\)  = \(\sqrt{(A_1 + A_2)^2}\)

∴ A = A₁+ A₂   ……(6)

Thus, the amplitude of the resultant wave is maximum when the two interfering waves are in phase.

Case (2) : When the two interfering waves are out of phase, φ = π. Then, the amplitude of the resultant wave is,

A = \(\sqrt{A_1^2+2A_1A_2 cos π + A_2^2}\)

∴ A = \(\sqrt{A_1^2+2A_1A_2 (-1) + A_2^2}\)  = \(\sqrt{(A_1 - A_2)^2}\)

∴ A = A₁ − A₂   ……(7)

Thus, the amplitude of the resultant wave is minimum when the two interfering waves are in opposite phase.

• Thus, the maximum amplitude is the sum of the two amplitudes when the phase difference between the two waves is zero and the minimum amplitude is the difference of the two amplitudes when the phase difference between the two waves is π.

Intensity of the resultant wave :

The intensities of the waves are proportional to the squares of their amplitudes.

Hence, when φ = 0

I_min ∝ (A_min)² where (A_min)²  = (A₁ − A₂)²

And when φ = π

I_max ∝ (A_max)² where (A_max)²  = (A₁ + A₂)²

Therefore intensity is maximum when the two waves interfere in phase while intensity is minimum when the two waves interfere out of phase.

Expression for the equation of a stationary wave on a stretched string :

When two progressive waves having the same amplitude, wavelength and speed propagate in opposite directions through the same region of a medium, their superposition under certain conditions creates a stationary interference pattern called a stationary wave.

Consider two simple harmonic progressive waves, of the same amplitude A, wavelength λ and frequency n = ω/2π, travelling on a string stretched along the x-axis in opposite directions. They may be represented by

y₁ = A sin (ωt − kx)   (along the + x-axis) and  ….(1)

y₂ = A sin (ωt + kx) (along the — x-axis)  …..(2)

where k = 2π/λ is the propagation constant.

By the superposition principle, the resultant displacement of the particle of the medium at the point at which the two Waves arrive simultaneously is the algebraic sum

y = y₁ + y₂ = A[sin (ωt − kx) + sin (ωt + kx)]

Using the trigonometrical identity,

Sin C + sin D = 2 sin\((\frac{C−D}{2})\)  cos\((\frac{C−D}{2})\)

y = 2A sin ωt cos (−kx)

= 2A sin ωt cos kx      ….[… cos (−kx) = cos (kx)]

= 2A cos kx sin ωt  ….(3)

 y = R sin ωt, ...(4)

where R = 2A cos kx.  ….(5)

Equation (4) is the equation of a stationary wave.

Nodes and antinodes : Nodes are the points of minimum displacement. This is possible if the amplitude is minimum (zero),

At nodes, R = 0.

∴ cos \(\frac{2πx}{λ}\) = 0      …(.. A ≠ 0 and k = \(\frac{2π}{λ}\)  )

\(\frac{2πx}{λ}=\frac{π}{2}=\frac{3π}{2}=\frac{5π}{2}\)

∴ x = \(\frac{λ}{4}=\frac{3λ}{4}=\frac{5λ}{4}\)….. (2p+1) \(\frac{λ}{4}\)  ....(1)

where p =0, 1, 2, …. Therefore, the distance between successive nodes is

[2(p+1) + 1]\(\frac{λ}{4}\)  − (2p+1)\(\frac{λ}{4}\)  = \(\frac{λ}{2}\)

The points at which the particles of the medium vibrate with the maximum amplitude are called the antinodes.

At antinodes, R = ± 2A

∴ cos \(\frac{2πx}{λ}\)= ± 1

 ∴ \(\frac{2πx}{λ}\)   = 0, π, 2π, 3π ……     

∴ x = 0, \(\frac{λ}{2}\), λ, \(\frac{3λ}{2}\) .... \(\frac{pλ}{2}\)  …. (p = 0,1, 2, 3…..)  …. (2)

Therefore, the distance between successive antinodes is

\(\frac{(p+1)λ}{2}\) − \(\frac{pλ}{2}\)  = \(\frac{λ}{2}\) 

∴ Distance between successive nodes = distance between successive antinodes = \(\frac{λ}{2}\).

From Eqs. (1) and (2), it can be seen that

(i) the nodes and the antinodes occur alternately and are equally spaced

(ii) the distance between a node and an adjacent antinode

=\(\frac{(2p+1)λ}{4}\) − \(\frac{pλ}{2}\)  =\(\frac{(2p+1)λ}{4}\) − \(\frac{2pλ}{4}\) = \(\frac{λ}{4}\) 

Properties of Stationary Waves:

• Stationary waves are produced due to superposition of two identical waves (either transverse or longitudinal waves) traveling through a medium along the same path in opposite directions
• The overall appearance of a standing wave is of alternate intensity maximum (displacement antinode) and minimum (displacement node).
• The distance between adjacent nodes (or antinodes) is λ/2.
• The distance between successive node and antinode is λ/ 4.
• There is no progressive change of phase from particle to particle. All the particles in one loop, between two adjacent nodes, vibrate in the same phase, while the particles in adjacent loops are in opposite phase.
• A stationary wave does not propagate in any direction and hence does not transport energy through the medium.
• In a region where a stationary wave is formed, the particles of the medium (except at the nodes) perform SHM of the same period, but the amplitudes of the vibrations vary periodically in space from particle to particle.

Since the nodes are points where the particles are always at rest, energy cannot be transmitted across a node. The energy of the particles within a loop remains localized, but alternates twice between kinetic and potential energy during each complete vibration. When all the particles are in the mean position, the energy is entirely kinetic. When they are in their extreme positions, the energy is entirely potential.

Distinguish between progressive waves and stationary waves :

Free vibrations ; A body capable of vibrations is said to perform free vibrations when it is disturbed from its equilibrium position and left to itself. i.e. In free vibration, the body at first is given an initial displacement and the force is then withdrawn. The body starts vibrating and continues the motion on its own. No external force acts on the body further to keep it in motion.

• In the absence of dissipative forces such as friction due to surrounding air and internal forces, the total energy and hence the amplitude of vibrations of the body remains constant. The frequencies of the free vibrations of a body are called its natural frequencies and depend on the body itself.
• In the absence of a maintaining force, in practice, the total energy and hence the amplitude decreases due to dissipative forces and the vibration is said to be damped. The frequency of damped vibrations is less than the natural frequency.

Forced vibrations : The vibrations of a body in response to an external periodic force are called forced vibrations.

• The external force supplies the necessary energy to make up for the dissipative losses. The frequency of the forced vibrations is equal to the frequency of the external periodic force.
• The amplitude of the forced vibrations depends upon the mass of the vibrating body, the amplitude of the external force, the difference between the natural frequency and the frequency of the periodic force, and the extent of damping.

Applications of resonance :

• A radio or TV receiver set is tuned to the frequency of the desired broadcast station by adjusting the resonant frequency of its electrical oscillator circuit.
• The speed of sound at room temperature can be determined by making the air column of a resonance tube resonate with a vibrating tuning fork of known frequency.
• The frequency of a tuning fork can be determined by making a sonometer wire, of known mass per unit length and under known tension, resonate with the vibrating fork.
• The amplitude of the oscillations of a child on a swing is increased by pushing with a frequency equal to the natural frequency of the swing.

Disadvantages of resonance :

• A column of soldiers marching in regular step on a narrow and structurally flexible bridge can set it into dangerously large amplitude oscillations. The bridge may even collapse at the resonance.
• Structural resonance of a suspension bridge induced by the winds can lead to its catastrophic collapse. Several early suspension bridges were destroyed by structural resonance induced by modest winds.
• Vibrations of a motor or engine can induce resonant vibrations in its supporting structures if their natural frequency is close to that of the vibrations of the engine. A common example is the rattling sound of a bus body when the engine is left idling.
• Vibrations in an aircraft are caused by the engine and the aerodynamic effects. The vibrations cause metal fatigue, especially in the fuselage, wings and tail, and eventually lead to metal fracture.