Complete & Easy Class 12 Physics Notes on Motion and Waves | Clear Concepts, Progressive Wave Equation, Derivations & Solved Numericals

Table of Contents

Wave

A wave is the pattern of disturbance that travels through the medium from one point to another point without actual movement of particles. It is also the continuous transfer of momentum and energy from one point to another. It is the disturbance that travels, but the particles oscillate about their mean position.

Types of waves

Mechanical wave (Elastic Wave)
The wave that requires a medium for its propagation is called a mechanical wave. For eg: sound wave, water wave, waves in strings, etc. For the propagation of a mechanical wave, the medium should possess three properties, i.e, inertia, elasticity, and minimum friction.
Non-mechanical wave
The wave that does not require a medium for its propagation is called a non-mechanical wave. For eg: electromagnetic wave (light wave, X-ray, infrared ray).
Matter wave
The waves associated with the microscopic particles (electron, proton, neutron, etc.) when they are in motion are called matter waves.

Types of mechanical waves

Based on the direction of vibration of particles, it is classified into two types:

Transverse Wave
The wave in which the particles vibrate about their mean position perpendicularly to the direction of propagation of the wave is called a Transverse wave. For eg: waves in a string, light waves, waves in the surface of water.
It travels in the form of a crest and trough. It travels through solid and liquid. It acts in two dimensions.
Longitudinal Wave
It is the wave in which a particle vibrates about its mean position along the direction of propagation of the wave. For eg: sound wave, wave at the bottom of the sea, wave in spring, hold horizontal, etc.
It travels in the form of compression and rarefaction. It travels in all media, solid, liquid, and gas. It travels in one dimension.

Basic terminologies of waves

Crest (C): The maximum displacement of the particle above the equilibrium position.
Trough (T): It is the maximum displacement of the particle below the equilibrium position.

Compression
It is the region of medium in which particles are very close together. It is also the region of maximum pressure and density.
Rarefaction
It is the region of medium in which particles are far away from each other. It is also the region of minimum pressure and density.
Amplitude
The maximum displacement of particles from the mean position is called amplitude.

Wavelength (λ):
The distance travelled by a wave in one complete cycle is called the wavelength. The distance between two consecutive crests or troughs is also called the wavelength. Its unit isthe meter (m).
Frequency (f):
The number of oscillations in one second is called frequency (f). Its unit is Hertz (Hz).
Time period (T):
The time taken to complete one cycle or one oscillation is called the time period.

Wave velocity (v):

Phase
The angular displacement of a wave at any instant of time is called the phase. It is measured in terms of angle. The phase indicates the position and state of motion of particles at any instant of time.

Wave Equation for Simple Harmonic Motion (S.H.M)
The wave equation for S.H.M is:
y = a sin(ωt)
Here, ‘ωt’ represents the phase, ‘a’ is the amplitude, and ‘y’ is the displacement of the particle.

Phase Difference and Path Difference
The angular displacement between two points is called the phase difference, while the linear distance between two points is known as the path difference.

For a path difference equal to λ, the phase difference is 2π. When the path difference is 1 unit, the phase difference remains 2π. For a general path difference of x, the phase difference is given by the formula:

Phase difference (Ø) = (2π × x) / λ

The SI unit of phase is Hertz (Hz).

Progressive Wave

A form of crest or compression is called a progressive wave. In a progressive wave, all the particles vibrate simply harmonically with the same amplitude and frequency. No particles are permanently addressed, meaning there are no nodes and antinodes.

Equation of a Progressive Wave

Let us suppose the wave is travelling from left to right along the positive x-axis. Consider a particle at the origin $O$ , which is vibrating simple harmonically about its mean position. Then, the displacement equation of the particle at any instant of time $t$ is:

y=asin(ωt)…………………(i)

Here, $a$ is the amplitude and is the phase of the wave.

Now, let us consider another particle at point $P$ , at a distance from the origin. Let be the phase difference of the particle at point $P$ . Then, its displacement equation is given by:

y=asin(ωt−ϕ)……………………….(ii)

We know that for a path difference of the phase difference is For a path difference of 1 unit, the phase difference is also For a general path difference , the phase difference is:

ϕ=2πx/λ

Substituting into equation (ii), we get:

equation progressive's wave prograssive wave2

Numerical Problem:

1.A wave has the equation (in meters and seconds):

Find:
(i) its frequency, wavelength, and speed
(ii) the equation of a wave with double amplitude but travelling in the opposite direction

Solution:

Given Equation:

Compare it with the standard progressive wave equation:

From this, we get:

2. y = 10 sin 2π (t – 0.005x) where y and x are in cm and t is in sec. Calculate amplitude, frequency, wavelength, and v.

Here, y = 10 sin 2π(t – 0.005x)
= 10 sin (2πt – 2π × 0.005x)

Now, comparing with y = a sin (ωt – kx)

ω = 2π, k = 2π × 0.005, a = 10 cm
∴ Now, a = 10 cm = 10/100 = 0.1 m

ω = 2πf
2π = 2πf
∴ f = 1 Hz

Now, k = 2π × 0.005
or, 2π/λ = 2π × 0.005
λ or, 1/0.005 = λ
∴ λ = 200 cm = 2 m

Now, v = λf= 2 × 1 = 2 m/s

y₁ = a sin (ωt – kx)

y₂ = a cos (ωt – kx)
y₂ = a sin (π/2 + (ωt – kx))
= a sin (π/2 + ωt – kx)

Now, φ₂ – φ₁ = π/2 + ωt – kx – ωt + kx
= π/2

y₁ = a sin (ωt + π/6)

y₂ = a cos ωt
Now, y₂ = a sin ωt
y₂ = a sin (ωt + π/2)

Δφ = ωt + π/2 – ωt – π/6
= π/3

3.A wave has a frequency of 500 Hz and a velocity of 360 m/s. The distance between two particles having a phase difference of 60° is

Here, Δφ = 60° = π/3
v = λ f
or, 360 = λ × 500
∴ λ = 0.72 m

Now, Δφ = 2π × Δx / λ

or, π/3 = 2π × Δx / 0.72 or, Δx = 0.12 m
= 12 cm

4.A wave is represented by an equation y = 7 sin (7π t – 0.04 x + π/3) where x is in meters and t is in sec. The speed of the wave is;

Here, y = 7 sin (7π t – 0.04 x + π/3)
∴ y = a sin (ω t – k x + φ₀)

v = ?, a = 7, ω = 7π, k = 0.04, φ₀ = π/3

ω = 2π f
or, 7π = 2π f
∴ f = 3.5 Hertz

Now, k = 2π / λ
or, 0.04 = 2π / λ
∴ λ = 2π / 0.04 = 157 m

Now, v = λ f
or, v = 157 × 3.5
= 550
= 175π m/s

Differential form of the wave equation:

Principle of Superposition

The principle of Superposition states that “If two or more than two waves pass through a medium at any point, then the resultant displacement is equal to the vector sum of individual displacements at that point.”

Stationary Waves (Standing)

The resultant wave formed due to the superposition of two progressive waves travelling in opposite directions with the same amplitude, frequency, and speed is called a stationary wave. It is called stationary because the wave formed seems not to be moving, and hence, there is no net transfer of energy.

In a stationary wave, there are points where particles are permanently at rest, called nodes. There are some points with maximum displacement called antinodes.

Equation of a stationary wave:

Equation of a stationary waveConsider two progressive waves having the same amplitude and frequency travelling with the same speed in opposite directions. Displacement equations of the two progressive waves are,

y₁=asin(ωt−kx)— i
=asin(ωt+kx)— ii

According to the principle of superposition,

y=y₁+y₂=asin(ωt−kx)+asin(ωt+kx)
=a[2sin(ωt)cos(kx)]
=2acoskx⋅sinωt
Asinωt— iii

Where, $\cos kx$ is the amplitude of the wave. Eqn III represents the required mathematical equation of a stationary wave.

Special Case:
i) Condition for maximum amplitude (Antinode formation)

The amplitude will be maximum if:

coskx=±1[A=±2a]

ii) Condition for minimum amplitude (Node formation)

The amplitude will be minimum if,

Complied By: Er. Basant Kumar Yadav

Post Views: 714