Wave Motion

Wave

A wave is a continuous transfer of energy and momentum in thee form of disturbance travelling with a certain velocity from one point to another point into a medium.
Energy is transferred due to repeated simple harmonic motion of particles.

Types of waves

1. Mechanical Wave
   Wave that requires a medium to propagate is called the mechanical wave. For example :- Sound wave.
2. Non-Mechanical Wave
   Wave that does not require a medium to propagate is called non mechanical wave. For example :- Electro magnetic wave.

Some basic terms related to wave

1. Displacement
   The linear distance travelled by a particle from its mean position is called displacement. It is denoted by symbol 'y'.
   The displacement of a particle executing simple harmonic motion is given by : y= a Sin ωt
2. Amplitude
   The maximum value of displacement of a particle from its mean position is called amplitude. It is denoted by symbol 'a' or 'r'.
3. Time period
   The total time taken by a wave to complete one cycle is called its time period. It is denoted by 'T'.
4. Frequency
   The total number of revolution or oscillations made per second is called it's frequency. It is denoted by the symbol 'f'.
   It is equal to the reciprocal of time period i.e. \(f= \frac{1}{T}\) .
5. Wave length
   The linear distance travelled by a wave during one cycle is called wavelength. It is denoted by the symbol 'λ'.
6. Wave velocity
   Velocity with which a wave propagates into a medium is called wave velocity. It is generally denoted by symbol 'v'. The velocity of wave is different depending upon the medium.
7. Particle velocity
   The velocity with which particles of a medium vibrates when wave propagates through that medium is called particle velocity. It is denoted by symbol 'u'.
   In other words , the rate of change of displacement of the particle is called particle velocity.
   Since , \(y= a \sin \omega t\)
   then , \(u = \frac{dy}{dt} = \frac{d( a \sin \omega t)}{dt} \)
   \(or, u = a \omega \cos \omega t\)
   \(For u=u(max) , \cos \omega t=1\)
   \(u_{(max)} =a \omega\)

Relation between wave velocity, frequency and wave length of a wave

We have,
\(Wave\ velocity(v) = \frac{distance\ travelled\ by\ a\ wave\ during\ one\ cycle(\lambda)}{time\ taken\ to\ complete\ one\ cycle(T)}\)
\(or , v= \frac{\lambda}{T}\)
\(or, v= \lambda f\)

Phase angle

The angular displacement of a particle from it's mean position denotes it's phase.
The displacement of a particle is,
y= a Sin ωt
Here, ωt=phase angle

Relation between phase difference and path difference

For a path difference λ , phase difference = 2 π
For a path difference 1, phase difference = \(\frac{2 \pi}{\lambda}\)
For a path difference x, phase difference = \(\frac{2 \pi}{\lambda} . x\)

If φ be the phase difference between particle 'A' and 'P' where 'x' is the path difference then,

\(\phi = \frac{2 \pi}{\lambda} . x\)
\(\phi = k. x\)
where, \(k= \frac{2 \pi}{\lambda}\) is called propagation constant.

Wave motion

The repeated simple harmonic motion of particles by which energy get transferred in between particles in the form of disturbance is called wave motion.
Depending upon the direction of vibration of particles, there are two kinds of wave motion.

1. Transverse Wave Motion
   The wave motion where particles vibrates in a direction perpendicular to the direction of propagation of wave is called transverse wave motion.
   Wave having such kinds of motion are called transverse wave.
   Eg :- light wave, electro-magnetic wave etc.
2. Longitudinal Wave Motion
   The wave motion where particles vibrates in a direction parallel to the direction of propagation of wave is called longitudinal wave motion.
   Wave having such kinds of motion are called longitudinal wave.
   Eg:- sound wave , wave in pipe etc.
   
   A longitudinal wave propagates in the form of compression and rarefraction.
   During compression, vibrating particles come closer so that the volume of a medium decreases with increase in pressure as well as density whereas during rarefraction vibrating particles goes far away so that the volume of a medium increases with decrease in pressure as well as density .
   Here longitudinal wave are also called pressure wave.

Progressive wave

The wave that transfer energy from one point to another point into a medium is called progressive wave. It is also called travelling wave.

Let us consider a wave having wavelength 'λ' is travelling from A to B with velocity 'v' as shown in the figure. As soon as wave propagates particles of the mediun vibrates about its mean position. The displacement of particle at A is given by
y= a Sin ωt...... (i)
where a=amplitude ω=angular velocity
Again consider a point 'P" at a distance 'x' from point A. Particles at point P vibrates lately than particles at A. So, the displacement of particles at P is given by
y= a Sin (ωt-φ) ...... (ii)
where φ=phase angle of particle at P
Since, \(\phi= \frac {2 \pi}{\lambda} x =kx \)
Equation ii becomes ,
y= a Sin (ωt-kx).........(iii)
Since, \(\omega= 2 \pi f = \frac{2 \pi}{T}\)

Euation (iii) becomes,
\(y= a \sin(2 \pi \frac{t}{T}- \frac{2 \pi}{\lambda}x)\)
\(y= a \sin 2 \pi (\frac{t}{T}-\frac{x}{\lambda})\)............(iv)

Also, \(\omega = 2 \pi f = 2 \pi \frac{v}{\lambda}[v= f. \lambda]\)

Equation (iv) becomes,
\(y= a \sin (\frac{2 \pi}{\lambda} v t - \frac{2 \pi}{\lambda} x)\)
\(y= a \sin \frac{2 \pi}{\lambda}(v t - x)\)............(v)

Hence, eqn (iii), (iv) and (v) are the required progressive wave equations.

For wave travelling along negative x-axis the above equations becomes,
y= a sin (ω t + k x)
\(y= a \sin 2 \pi (\frac{t}{T}+ \frac{x}{\lambda})\)
\(y= a \sin \frac{2 \pi}{\lambda} (vt + x)\)

Principle of Superposition of Waves

When 'n' no. of waves are travelling simultaneously into a medium then each particle of the medium is displaced by these waves.

Principle of Superposition of Waves state that " The total displacement of a particle due to 'n' no. of waves is equal to the sum of displacement given to meet by each wave."

Let y₁ , y₂ , ......., y_n be the displacement of a particle due to 'n' no of waves then the total displacement is ,
y = y₁ + y2 + ..........+ yn 

Stationary Wave

The resultant wave formed by the super position of two progressive wave having same amplitude and same frequency travelling with equal velocity in opposite direction is called stationary wave or standing wave. It is called so because there is no transmission of energy during wave propagation.

Stationary Wave Equation

Let us consider two progressive wave having same amplitude and same frequency travelling with same velocities in opposite direction. Their wave equation can be written as,

\(y_{1} = a \sin (\omega t + k x).........(i)\)
\(y_{2} = a \sin (\omega t - k x).........(ii)\)
where a = amplitude 

Using principle of superposition of wave,
\(y = y_{1} + y_{2}\)
\(or, y = a \sin (\omega t + k x) + a \sin (\omega t - k x)\)
\(or, y = a [\sin (\omega t + k x) + a \sin (\omega t - k x)]\)
\(or, y = 2 a \sin \frac{\omega t + k x + \omega t - k x }{2} \cos \frac{\omega t + k x - \omega t + k x }{2}\)
\(or, y = 2 a \sin \omega t . \cos k x\)
\(or, y = 2 a \cos kx . \sin \omega t\) ............(iii)

This is the stationary wave equation . Comparing eqn (iii) with \(y= A \sin \omega t\) we get,
A = 2 a Cos kx is the amplitude of stationary wave.

Formation of Nodes

The points where amplitude of the resultant wave becomes zero are called nodes. For formation of nodes,
We must have,

\(\cos k x = 0\)
\(or, \cos k x_n = \cos (2n+1) \frac{\pi}{2}\) where n= 0,1,2,3 .....
\(or, k x_n = (2n+1) \frac{\pi}{2}\)
\(or, \frac{2 \pi}{\lambda} . x_n = (2n+1) \frac{\pi}{2}\)
\(or, x_n = \frac{(2n+1) \lambda}{4}\)
where, x = position of nodes

Hence, nodes are formed at a distance \(\frac{\lambda}{4}\), \(\frac{3 \lambda}{4}\) , \(\frac{5 \lambda}{4}\)........... from mean position.

Distance between corresponding nodes,
\(= \frac{3 \lambda}{4} - \frac{\lambda}{4}\)
\(= \frac{2 \lambda}{4}\)
\(= \frac{\lambda}{2}\)

Formation of Anti nodes

The points where the amplitude of the resultant wave becomes maximum are called antinodes.

For the formation of anti nodes we must have,

\(Cos kx = 1\)
\(or, Cos kx_{n} = Cos n \pi\)
where n = 0,1,2,3,....
\(kx_{n} = n \pi\)
\(\frac{2 \pi}{\lambda} x_{n} = n \pi\)
\(x_{n} = \frac{n \lambda}{2}\)

Hence, antinodes is formed at origin and other at a distance of \(frac{\lambda}{2}, \(\lambda\) , \(\frac{3 \lambda}{2}\)..... from the mean position.

Difference between two corresponding anti nodes,
\(\lambda - \frac{\lambda}{2} = \frac{\lambda}{2}\)

Hence, during the formation of stationary wave nodes and anti nodes are equally seperated .

Natural Frequency

Every object in the universe has a tendency to oscillate. The frequency with which a body oscillates is called it's natural frequency.

Free oscillation

The oscillation of a body in the absence of external resistive force(air resistance) is called free oscillation. The amplitude and energy of a freely oscillating body is always constant. Example : motion of simple pendulum.

Damped oscillation

The oscillation of a body in the presence of external resistive force (air resistance) is called damped oscillation. During damped oscillation , amplitude and energy of the oscillating body decreases continuously. This condition is called damping condition.

Forced oscillation

When a body is under damping condition, an external force has to be applied to continue it's oscillation. The oscillation of a body in the presence of external periodic force is calld forced oscillation.

Resonance

When frequency of external source(f) is equal to the natural frequency (f_o) of a body then the body vibrates with maximum amplitude. This condition is called resonance and the source frequency f is called resonant frequency.

Special Thanks(Credit) : Hari Saran Regmi Sir