Lecture Notes

Wave:

The continuous transfer of disturbance from one part of a medium to another through the successive vibrations of the medium in the mean position is called wave. Through the motion, the energy and momentum are carried out from one region to another. It is called the oscillations instead of wave if the energy is not transferred.

Some important terms:

Crest:

Crest is the maximum displacement of the particles of a medium above the equilibrium (rest) position.

Trough:

Trough is the maximum displacement of the particles of a medium below the equilibrium position.

Amplitude:

When a mechanical wave passes through the medium, the maximum displacement produced by the wave from its equilibrium position is called its amplitude.

Wavelength:

The distance covered by the wave over one complete cycle when it is traveled over any time is called wavelength. It is denoted by \lambda and measured in meter(m) unit. In another way, it is also defined as the distance between two successive crest and trough.

Frequency:

The number of oscillations made by the particle in one second is called its frequency. It is denoted by f and its unit is S-1 or Hertz (Hz).

  • Velocity and wavelength is changed but frequency remains always constant.
  • Frequency is considered as fundamental parameter of wave.

Period:

The time taken by vibrating particle to complete one oscillation is called its period. It is denoted by T and its unit is second (S). It is simply reciprocal of frequency. i.e.

T = \frac {1}{f}

 

Wave velocity:

The distance traveled by wave per second is called wave-velocity (v). i.e.

v = \frac{distance\ travelled\ by\ wave}{time\ taken}

At time T, wave travels ‘\lambda‘ distance, so, v = \frac{\lambda}{T}

= \lambda \times \frac{1}{T}  = \lambda x f

∴ v = \lambdaf.

Phase:

The state of motion of a particle at a given place and time is called its page. That is, where is particle? And what is the direction of the wave in that time? It is measured in terms of angle, called ‘phase angle’.

 Types of waves:

Transverse wave:

A wave in which the particles of the medium vibrate up and down at right angles to the direction in which wave is moving is called transverse wave. A transverse wave consist of crest and trough. Example, water waves formed on the surface of water, waves produced in a string when its end is jerked and light waves (EM waves).

Longitudinal wave:

The waves in which the particles of the medium vibrate parallel to the direction of the wave in which it is moving is called a longitudinal wave. It consists of compressions and rarefactions. Example sound waves and waves in a spring.

Differences between mechanical wave and electromagnetic wave:

Mechanical wave  Electromagnetic wave
1. It needs a material medium for its propagation. 1. It does not need a material medium for its propagation.
2.  Speed is less than electromagnetic wave.

Example sound waves.

2. Speed is more than mechanical wave.

Example light-wave, X-rays.

Differences between transverse wave motion and longitudinal wave motion:

Transverse wave motion  Longitudinal wave motion
1. The particles of medium vibrate perpendicularly to the direction of propagation of wave. 1. The particles of the medium vibrate to the direction of the propagation of the wave.
2.  It consists of crest and trough.

3. It can be observed in solid and at the liquid surface.

4. There is no variation in pressure.

2.  It consists of compressions and rarefactions. Example sound waves.

3. It can be observed in solid, liquid and gases.

4. There is variation in pressure.

Path differences:

In a wave, the positions or stages of vibrations of particles can be expressed in terms of distances from the origin or some reference point. If the position of two particles is given in terms of the distance between the particle, then, it is called as path difference.

 

Relation between phase difference and path difference:

Consider, a wave having wavelength \lambda and frequency f is travelling with velocity v as shown in the figure.

A point ‘A’ is at a distance x from the particle at the origin O, then, the path difference between A and O is x, for the path difference of \lambda, the phase difference is 2π. For the path difference of x, the phase difference is (\frac{2\pi}{\lambda},x).

Therefore, phase difference,

\Delta \phi = \frac{2\pi}{\lambda}. \Delta x .

Progressive wave:

The wave that travels from one region of medium to another is called the progressive wave. In other way, the wave that the wave profile travels in forward direction with constant amplitude and frequency is called progressive wave. The wave profiles move with a speed of the wave.

Derivation of progressive wave equation:

Consider a progressive wave moving with velocity v along the positive x direction as indicated in figure.

All the particles in the medium oscillates with time period T and amplitude A. Suppose, a particle P, at x = 0 starts moving along positive x direction. The displacement of the particle at x = 0 at any instant of time T is given by the equation:

y (0,t) = ASinωt

where, ω is the angular velocity of the particle = \frac{2\pi}{T}

Again, suppose, second particle Q is at a distance x from P. So, the wave takes x/v second to travel from P to Q. Thus, the displacement, y (x.t) of a particle at Q at any time T will be same as the displacement of the particle at P at an earlier time (t - \frac{x}{v}) . Then,

y(x,t) = y(0, t - \frac{x}{v})

= ASinω(t - \frac{x}{v}) = ASin (ωt-\frac{ω}{v}x)

y(x,t) = ASin(ωt-kx) …………… (i)

Where, k = \frac{\omega}{v}, is propagation constant or wave constant.

Since, ω = 2πf and v = \lambdaf, we can write,

k =\frac{\omega}{v} = \frac{2\pi f}{\lambda f} = \frac{2\pi}{\lambda}

also, ω = 2πf = \frac{2\pi}{T}

then, the equation (i) can be expressed in terms of frequency as:

y (x,t) = ASin(2πft – kx). ……………………….. (ii)

Again, as f = \frac{1}{T} and substituting for k, the equation can also be written as:

y (x,t) = ASin(\frac{2\pi t}{T} - \frac{2\pi x}{\lambda})

y (x,t) = ASin2π (\frac{t}{T} - \frac{x}{\lambda}) ………………… (iii)

Equations (i), (ii) & (iii) are the required equation of progressive wave.

When the wave is travelling from right to left (i.e. in negative x – direction), the progressive wave equation takes the form:

y(x,t) = ASin(ωt+kx) …………… (iv)

Principle of Superposition:

When two or more waves travels simultaneously in a medium, they move independently without affecting the motion of one another. The resulting displacement of a particle at any instant of time is obtained by the vector sum of individual displacement of the superposing waves and this is called principle of superposition.

If y1, y2, y3,……………,yn are the displacement of n superposing waves at a point in a medium then, the resultant displacement at that point is given by:

\vec{y}=\vec{y_1}+\vec{y_2}+\vec{y_3}+............+\vec{y_n} .

Stationary wave:

When two progressive wave having same frequency and amplitude but moving in opposite direction superimposed upon each other, they give rise to a new type of wave called stationary wave. Since there is no flow of energy along the way as in case of progressive wave, so, stationary waves are also called standing waves.

Characteristics:

They may be transverse or longitudinal in nature. In stationary wave, there are certain points where the amplitude of vibration is maximum. Such points are called anti nodes. In mid -way between the antinodes, there are some points, where the amplitude of vibration is zero, such points are called nodes.

Differences between progressive wave and stationary wave:

Progressive wave  Stationary wave
1. The amplitude of oscillation is same at all position in medium.  1. The amplitude of oscillation are different at different place in medium.
 2. No particle is permanently at rest. 2. The particles  at nodes are permanently at rest.
3. The particles of the medium pass through their mean position one by one. 3. All particles of the medium pass simultaneously through their mean position.
4. Pressure vibration takes place at every point. 4. Pressure variation are maximum at nodes and nodes and antinodes.
5. There exist a regular phase difference between successive particles.  5. All the particles in between two successive nodes are in phase.

Equation of stationary wave:

Consider two progressive waves of the same amplitude and frequency are travelling with same speed in opposite direction. The equation of the wave travelling from left to right from positive x direction is:

y1 = aSin(ωt-kx) …………… (i)

Again, the equation of the wave travelling from right to left is:

y2 = aSin(ωt+kx)………….(ii)

where, a = amplitude of vibration.

ω = 2πf = angular velocity of the wave.

When these two wave superimpose upon each other, they form a stationary wave. According to principle of superposition, the resultant displacement at any point is:

y = y1 + y2

= aSin(ωt-kx) + aSin(ωt+kx)

= a[Sin(ωt-kx) + Sin(ωt+kx)]

= a.2Sinωt.Coskx = (2aCoskx). Sinωt

∴ y = ASinωt ………… (iii)

Where, A = 2aCoskx = amplitude of stationary wave, which depends on the distance x.

The amplitude will be maximum when ‘A’ becomes maximum.

When, Coskx = ±1

Or, kx = 0, π, 2π, 3π

Or, \frac{2\pi}{\lambda}x = n\pi  (n = 0, 1, 2, …………)

Or, x = \frac{n\lambda}{2}

Hence, the amplitude becomes maximum i.e. antinodes are occurred at x = 0, \frac{\lambda}{2}, \lambda, \frac{3\lambda}{2}, ………..

This means if one antinode is at x = 0, the next antinode will be at a distance of \frac{\lambda}{2} and second antinode will be at a distance of \lambda and so on.

The distance between any two consecutive antinode is: \frac{\lambda}{2}

The amplitude will be minimum when:

Coskx = 0

Or, kx = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, ............

Or, \frac{2\pi}{\lambda}x = (2n+1)\frac{\pi}{2}            (n = 0, 1, 2, ………….)

∴ x = (2n+1) \frac{\lambda}{4}

Hence, amplitude becomes minimum, i.e. nodes occurred at:

x = \frac{\lambda}{4}, \frac{3\lambda}{4}, \frac{5\lambda}{4}, .............

The distance between any two consecutive node is:

\frac{3\lambda}{4} - \frac{\lambda}{4} = \frac{\lambda}{2}

Reflection, Refraction and Diffraction of Sound waves:

Reflection:

The phenomena of returning back of sound waves after striking an obstacle is called reflection of sound. During reflection, angle of incidence (i) and angle of reflection (r) are equal. For e.g. Echo. The minimum distance between source and obstacle to observe echo is 17 m.

Refraction:

The phenomena of bending of sound waves when it travels from one medium to another is called refraction of sound. The velocity of wave changes during refraction. Sounds are easier to listen during nights then day time due to refraction.

Diffraction:

The phenomena of spreading of wave around the edge of an obstacle or aperture when it pass through it is called diffraction of sound. The sound is heard inside the room even the source is outside is an example of diffraction.