Wave motion is transfer of energy in the form of disturbance from one place to another place due to periodic motion of particle. Wave motion can be classified into two types according to medium:
Electromagnetic wave
Mechanical wave
Mechanical Wave and Electromagnetic Wave
i) Mechanical Wave
The wave which requires material medium for their propagation are called mechanical wave. e.g. sound wave, wave in string etc.
ii) Electromagnetic Wave
The wave which don’t require any material medium for their propagation are called electromagnetic wave. For example, light wave, radio wave etc.
On the basis of direction of vibration of particle, waves are of two types:
i) Transverse Wave
If the vibration of particle of the medium are at right angles to direction of propagation of wave, then such wave is called transverse wave. They are transmitted in the form of crest and trough.
Fig: Transverse wave
ii) Longitudinal Wave
If the vibration of particle of the medium are in the direction of propagation of a wave, such wave is called longitudinal wave. In longitudinal wave, particles are vibrated in the form of compression and rarefaction. e.g. sound wave.
Longitudinal wave: compression and rarefaction
Some Terms Related to Wave Motion
Wave diagram used for displacement, amplitude, wavelength and phase
a) Displacement
A distance of vibrating particle from mean position is called displacement. It is denoted by ‘y’.
b) Amplitude
Maximum displacement of vibrating particle from its mean position is called amplitude. It is denoted by ‘a’.
c) Wavelength
The distance between any two successive maxima or minima is called wavelength.
Or,
The distance travelled by wave in one complete cycle is called wavelength. It is denoted by ‘λ’.
Frequency
The number of complete oscillation made by wave in one second is called frequency. It is denoted by ‘f’ and given by:
f = 1/T
f = ω/2π
Time Period
Time taken by vibrating particle to complete one revolution is called time period. It is denoted by ‘T’ and given by:
T = 1/f
T = 2π/ω
Phase or Phase Angle
Phase angle along a wave
The state of vibrating particle at a given time is called phase. It is expressed in term of angle.
For distance λ, phase = 2π
For distance 1, phase = 2π/λ
For distance x, phase = (2π/λ)x
φ = 2πx/λ
Progressive Wave
The wave which is travelling in forward direction with constant amplitude and frequency is called progressive wave.
Fig: Progressive wave
Suppose a wave travelling in a direction from left to right with velocity ‘v’ in which all particle vibrate with simple harmonic motion. The displacement of vibrating particle from origin O is given by:
y = a sinωt — (i)
where ‘a’ is amplitude and ‘ω’ its angular velocity.
Also suppose another vibrating particle is at point ‘P’ which is at distance ‘x’ from origin as shown in figure. Let φ be the phase angle of a particle at point ‘P’. Then displacement of particle at point ‘P’ is given by:
y = a sin(ωt − φ) — (ii)
We know that for distance λ, phase angle is 2π and for distance x, phase angle is (2π/λ)x.
y = a sin(ωt − 2πx/λ)
= a sin(2πft − 2πx/λ)
= a sin[(2π/λ)(vt − x)]
y = a sin[(2π/λ)(vt − x)] — (iii)
Equation (iii) is required equation for progressive wave travelling from left to right.
If the progressive wave is travelling from right to left then its equation is:
y = a sin[(2π/λ)(vt + x)]
Equation of progressive wave also written as:
y = a sin(ωt − kx)
where, k = 2π/λ
k is called wave number or propagation constant.
Superposition Theorem
When number of waves having displacement y₁, y₂, y₃ … yₙ propagates in a medium, then resultant displacement of a wave at any instant is equal to the vector sum of displacement due to individual wave. This phenomenon is called superposition theorem.
⃗y = ⃗y₁ + ⃗y₂ + ⃗y₃ + … + ⃗yₙ
Stationary / Standing Wave
When two progressive waves of same amplitude and frequency travel with same speed in a medium in opposite direction, they will superpose each other and stationary wave is formed.
Fig: Formation of stationary wave
Let us consider two waves of same amplitude ‘a’ and same frequency ‘f’ having wavelength ‘λ’ travelling opposite direction with their displacement:
y₁ = a sin(ωt − kx) — (i)
y₂ = a sin(ωt + kx) — (ii)
Applying superposition theorem:
y = y₁ + y₂
= a sin(ωt − kx) + a sin(ωt + kx)
= a{sin(ωt − kx) + sin(ωt + kx)}
= 2a sinωt · coskx
y = 2a coskx · sinωt
y = A sinωt — (iii)
where A = 2a coskx is amplitude of stationary wave.
Equation (iii) is equation of stationary wave.
Condition of Nodes and Antinodes
1) For Nodes
For nodes amplitude is zero:
A = 0
coskx = 0
coskx = cos[(2n + 1)π/2]
kx = (2n + 1)π/2, where n = 0, 1, 2, 3 …
(2π/λ)x = (2n + 1)π/2
x = (2n + 1)λ/4
x = λ/4, 3λ/4, 5λ/4 … nodes are formed.
2) For Antinodes
coskx = 1
coskx = cosnπ
kx = nπ
(2π/λ)x = nπ
x = nλ/2
At x = 0, λ/2, λ, 3λ/2 … antinodes are formed.
Solved Numerical
Q.1
A wave has the equation (x in metres and t in seconds): y = 0.02 sin(30t − 4x). Find: (i) its frequency, speed and wavelength, (ii) the equation of wave with double amplitude but travelling in the opposite direction.
Given:
y = 0.02 sin(30t − 4x) — (i)
From equation of progressive wave:
y = a sin(ωt − kx) — (ii)
Comparing equations (i) and (ii):
a = 0.02
ω = 30
k = 4
(i) Frequency
ω = 2πf
2 × 3.14 × f = 30
f = 30/6.28
f = 4.77 Hz
Wavelength
k = 2π/λ
4 = 2π/λ
λ = 2 × 3.14 / 4
λ = 1.57 m
Speed
v = fλ
= 4.77 × 1.57
v = 7.5 m/s
(ii) Double amplitude and opposite direction
Amplitude a′ = 2a
a′ = 0.04
For opposite direction: y = 2a sin(ωt + kx)
y = 0.04 sin(30t + 4x)
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