Wave Motion Note | Class 12 Physics | NEB | NEPALI EDUCATE

WAVE MOTION

A wave is a disturbance from an equilibrium condition that travel with finite velocity from one region to another region of space. Normally the wave is in motion so it is called wave motion.

Types of Wave:

      i. Mechanical Waves or Elastic Waves:  The waves which require a material medium for their propagation are called mechanical waves.

Example: Water waves, sound waves, wave on spring etc. for the propagation of mechanical wave, the medium should have three properties; elasticity, inertia and low damping.

    ii.Non-mechanical Waves or Electromagnetic Waves: The waves which do not require a material medium for their propagation are called electromagnetic waves.

Example:  Radio waves, visible and ultraviolet light, microwaves, x-rays etc. 

   iii.Matter Waves: The waves which are associated with the motion of the particles of atomic or subatomic size such as electrons, protons, etc. are called matter waves.

Types of mechanical wave:

On the basis of mode of vibration of the particles of the medium, mechanical waves are divided into two types:                 

(a) Transverse wave and

(b) Longitudinal wave 

a. Transverse wave 

The wave motion, in which the particles of the medium vibrate about their mean position at right angle to the direction of propagation of the wave, is called the transverse wave. These waves travel in the form of crest and troughs. 

Example: Light wave, vibration of string, rods etc. 

b. Longitudinal wave

A wave motion in which the particles of the medium vibrate about their mean position along the direction of the propagation of the wave is known as longitudinal wave. These wave travel in the form of compression and rarefaction.

Example:  Sound wave in air, Wave on springs along length etc.

Some important terms 

Displacement (y): The displacement of a particle at any time’t’ is the distance it has moved from the mean position at that time. Displacement can be expressed as the function of time as

y = A sinωt  …..(i)     

Where A is the amplitude and is defined as the maximum displacement of the particle on either side of the mean position. 

Particle velocity (u): The particle velocity is that with which the particles of the medium vibrate about their mean position. Taking derivative of equation (i), w.r.t. time t, we get:

\[v = \frac{{dy}}{{dt}} = A\omega {\rm{ }}cos{\rm{ }}\omega t\]

\Particle velocity (v) = Aω cosωt               

Wavelength (λ): It is defined as the distance between adjacent identical parts of a wave (that may be crest or trough) or the linear distance travelled by the vibrating particle in one complete cycle.  It is measured in meter (m) in S.I. system.

Wave velocity (v): The wave velocity is the linear distance travelled per unit time by the wave, which can be written as:

\[v = \frac{\lambda }{t}\]    

v = λ. f

Time period (T): It is the time required for the point to complete one full cycle of its motion. It is denoted by T.

Wave frequency (f): The number of waves produced per second is called frequency of the wave motion. The frequency of a wave is related to the time period as 1

                                \[f = \frac{1}{T}\] (measured in Hz)

Phase, Phase Difference and Path Difference

Any two points in the cycle of waveform are said to be in same phase if they are in the same side, equally displaced and oscillating in the same direction. In Fig points (a & i), (b & j) are in same phase and these points are separated by an angle 2π, which is called phase difference.

A linear distance between two point a and i is λ, which is called path difference. 

Relation between path difference and phase difference

let us take any two points c and e, which are located at a distance x₁ and x₂ at which their phases w.r.t origin are φ₁ and φ₂; and time to reach the wave from origin to these points is t₁ and t₂ respectively, when path difference λ, phase difference= 2π, path difference 1, phase difference= 2π/λ, path difference x₁, phase difference= φ₁ = (2π/λ) x₁ 

Similarly, path difference x₂, phase difference= φ₂ = (2π/λ) x₂

Hence phase difference between two points = Δφ = φ₂ – φ₁ = $\frac{{{\rm{2π }}}}{{{\rm{ λ }}}}$(x₂ –x₁) = $\frac{{{\rm{2π }}}}{{{\rm{ λ }}}}$Δx

Where, Δx = x₂ –x₁ is the difference in distance between two points c and e is called their path difference.

 \Phase difference (Δφ ) =$\frac{{{\rm{2π }}}}{{{\rm{ λ }}}}$   × path difference (Δx). 

Progressive Wave

A wave which travels continuously in a medium in the same direction without any change in its amplitude and frequency is called a progressive wave or a travelling wave. The particles of the medium execute SHM about their mean position along or perpendicular to the direction of wave propagation according to its longitudinal or transverse nature. 

Equation of a progressive wave

Let us consider a progressive wave travelling from left to right along the positive x-direction starting from O. the equation of motion of a particle at point O at any instant of time‘t’ is given by,  

y = a sinѠt

Where 'a' is the amplitude of wave or particle and Ѡ is the angular velocity of the particles. Let P is a point at a distance x from O and their phase difference is f,

\[\Phi  = \frac{{2\pi x}}{\lambda }\]

Since the disturbance reaches in later time to the particles to right of O, they start to vibrate after some time w.r.t. the particle at O and the phase lag goes on increasing in this direction. Hence the displacement of a particle at P is given by,

y = a sin (Ѡt – f) 

= \[a\sin \left( {\omega t - \frac{{2\pi x}}{\lambda }} \right)\]

Here the quantity $\frac{{{\rm{2π }}}}{{{\rm{ λ }}}}$ is a constant for a given wave in a medium is known as propagation constant or wave number and denoted by k. Hence above equation can be written as,

\[\begin{array}{l}y{\rm{ }} = {\rm{ }}a{\rm{ }}sin{\rm{ }}\left( {\omega t{\rm{ }}--{\rm{ }}kx} \right) \ldots  \ldots  \ldots ..{\rm{ }}\left( i \right)\\ = a\sin \left( {\frac{{2\pi vt}}{\lambda } - \frac{{2\pi x}}{\lambda }} \right),\;{\rm{where}},{\rm{ }}v{\rm{ }}is{\rm{ }}the{\rm{ }}velocity{\rm{ }}of{\rm{ }}the{\rm{ }}wave.\\\therefore y = a{\rm{ sin  }}\frac{{2\pi }}{\lambda }(vt - x)........(ii)\\{\mathop{\rm Sin}\nolimits} ce\;\omega  = \frac{{2\pi }}{T},\;{\rm{From Equation (i) y  =  a sin(}}\frac{{2\pi }}{T}{\rm{t - }}\frac{{2\pi }}{T}{\rm{x)}}\\\therefore y = a{\rm{ sin }}2\pi (\frac{t}{T} - \frac{x}{\lambda }).....(iii)\end{array}\]

Equations i, ii and iii represent a plane progressive wave equation.  Similarly, a progressive wave travelling in opposite direction i.e. from right to left

The equation becomes, \[y{\rm{ }} = {\rm{ }}a{\rm{ }}sin{\rm{ }}\left( {\omega t{\rm{ }}--{\rm{ }}kx} \right) = a{\rm{ sin  }}\frac{{2\pi }}{\lambda }(vt - x) = a{\rm{ sin }}2\pi (\frac{t}{T} - \frac{x}{\lambda })\]

Principle of Superposition of Waves

It states that,“when a large number of waves travel through a medium simultaneously, the resultant displacement of any particle of the medium at any given time is equal to the vector sum of the displacements due to the individual waves.” 

If y₁, y₂, y₃, …, y_n are the displacements due to waves acting separately, then according to the principle of superposition the principle of superposition the resultant displacement y, when all the waves act together is given by:

y = y₁ + y₂ + y₃ + … + y_n, this is the principle of superposition of wave.

Stationary or Standing wave

When two identical progressive waves having same amplitude,  frequency and wavelength travelling in same medium with same speed but in opposite direction superimposed with each other, by their superposition ,a new wave is formed which is called standing wave.  

Mathematical Treatment of Stationary wave 

Let us consider two identical progressive waves of amplitude a, frequency f and having wavelength  are travelling in same medium in opposite direction simultaneously. Then these two wave equation are given by, y₁ = a sin (Ѡt + kx)        and y₂ = a sin (Ѡt – kx)

Since they are travelling simultaneously, the new wave is formed by their superposition and resultant displacement of the particles of medium after their superimposition is given by, 

\[\begin{array}{l}y{\rm{ }} = {\rm{ }}{y_1} + {\rm{ }}{y_2}\\y{\rm{ }} = {\rm{ }}a{\rm{ }}\left[ {sin{\rm{ }}\left( {\omega t{\rm{ }} + {\rm{ }}kx} \right){\rm{ }} + {\rm{ }}sin\left( {\omega t{\rm{ }}--{\rm{ }}kx} \right)} \right]\\y = {\rm{ }}2a{\rm{ }}sin\left[ {\frac{{\omega t{\rm{ }} + {\rm{ }}kx + \omega t{\rm{  -  }}kx}}{2}} \right]\cos \left[ {\frac{{\omega t{\rm{ }} + {\rm{ }}kx - \omega t{\rm{  +  }}kx}}{2}} \right]\\y = 2a{\rm{ }}cos{\rm{ }}kx{\rm{ }}sin{\rm{ }}\omega t{\rm{ }}\end{array}\]

Since initial waves are sinusoidal wave, their resultant also should be sinusoidal wave.

Here 2acos kx = A should be the amplitude of the resultant wave

Thus, y=2acos kx Sin Ѡt= ASin Ѡt

This is the equation of stationary wave. At a point, k =$\frac{{{\rm{2π }}}}{{{\rm{ λ }}}}$ is constant whereas amplitude of the resultant wave, A = 2a cos kx changes with x, i.e. amplitude of vibration of the different points is not constant.

Position of Nodes: 

At nodes amplitudes of the resultant wave, i.e. A=2a cos kx should be zero and this is possible only when cos kx =0.

Hence at the distances x = $\frac{{{\rm{ λ }}}}{{{\rm{ 4 }}}}$, $\frac{{{\rm{ 3λ }}}}{{{\rm{ 4 }}}}$,$\frac{{{\rm{ 5λ }}}}{{{\rm{ 4 }}}}$……. From the boundary, nodes are formed.

Also distance between any two consecutive nodes =$\frac{{{\rm{ 3λ }}}}{{{\rm{ 4 }}}}$-$\frac{{{\rm{ λ }}}}{{{\rm{ 4 }}}}$ = $\frac{{{\rm{ λ }}}}{{{\rm{ 2 }}}}$

Position of Antinodes:  

At antinodes, the amplitude of the resultant wave, i.e. A =2a cos kx should be maximum and this is possible only when cos kx = ± 1.

\[\begin{array}{l}or,{\rm{ Cos }}\frac{{2\pi x}}{\lambda } =  \pm 1\\or,\;{\rm{Cos }}\frac{{2\pi x}}{\lambda } =  \pm 1 = \cos n\pi ;{\rm{ }}\,{\rm{where n = 0,1,2,3}}.....\\or,{\rm{ }}\frac{{2\pi x}}{\lambda } = n\pi \\or,\,{\rm{ }}x = \frac{{n\lambda }}{2};\end{array}\]

Hence at the distance x=0, $\frac{{{\rm{ λ }}}}{{{\rm{ 2 }}}}$, λ, $\frac{{{\rm{ 3λ }}}}{{{\rm{ 2 }}}}$, ….. From the boundary, antinodes are formed.

Also distance between any two consecutive antinodes =$\frac{{{\rm{ λ }}}}{{{\rm{ 2 }}}}$-0=$\frac{{{\rm{ λ }}}}{{{\rm{ 2 }}}}$......(iii)

This concludes that the distance between two consecutive nodes or the distance between two consecutive antinodes is equal to =$\frac{{{\rm{ λ }}}}{{{\rm{ 2 }}}}$ and that distance between node and antinode is $\frac{{{\rm{ λ }}}}{{{\rm{ 4 }}}}$.

Wave properties

a) Reflection of waves

When the waves are incident on a boundary between two media, part of energy of incident waves returns back into the same medium and this phenomenon is called reflection of waves.

b) Refraction of waves

The bending of wave due to the change in its speed when it travels from one medium to another medium is called refraction of sound.

c) Diffraction of waves

The phenomenon of spreading of the waves when they pass through an aperture or round obstacle is known as diffraction

d) Interference of waves

When two waves of same frequency and in same phase travel in the same direction along a straight line simultaneously they superpose in such a way that the intensity of the resultant wave is maximum at certain points and minimum at certain other points. This phenomenon is called the interference.