Superposition of waves

Introduction

1. Wave Motion:

a) The sound waves can also be called as elastic waves and medium is needed for their propagation. Their velocity is different for different media.

b) The velocity of sound waves depends on the property of inertia and coefficient of elasticity of that medium.

c) In this type of motion the energy is transferred from one point to other by the propagation of disturbance in the medium. The motion of disturbance is called the wave motion.

d) In simple harmonic wave the particles of the medium executes simple harmonic motion. Each particle of the medium executes simple harmonic motion at all the times, though the phase of its vibration depends on its position and time.

On the basis of vibration of the particle waves are classified in two types:

(a) Transverse waves.

(b) Longitudinal waves.

(a) Transverse waves:

The particles of medium vibrate in a direction perpendicular to the direction of propagation of wave. Their polarization is possible. Example: Vibration of string, the surface waves produced on the surface of solid and liquid.

Fig (1)

(b) Longitudinal waves:

Vibrations of the particles of the medium are in the direction of wave propagation wave proceeds in form of compression and rarefaction. At places of compression the pressure and density are maximum, while at places of rarefaction those are minimum.

Example: Sound waves, waves is gases, waves is solids.

Fig (2)

2) Some important points related to Wave Motion:

1) Amplitude: The maximum displacement of a vibrating particle of the medium from the mean position.

2) Time period (T): The time taken by the particle to complete one vibration.

3) Frequency: The number of vibration in one second by the particle n = 1/T.

Angular Frequency

4) Wavelength : The distance between two consecutive particles vibrating in the same phase.

Or

The distance between two consecutive crests or trough.

Or

The distance between two consecutive compressions or rarefaction.

Here v is the velocity of wave.

is the wave propagation vector which is written as:

, being the unit vector along the direction of wave propagation then,

Fig (3)

3) Equations for the progressive Simple Harmonic Waves:

1) The equation of a wave moving along the direction of positive X-axis is

Where ‘a’ is the amplitude, is the phase difference and ‘y’ is the displacement of a particle.

2) The equation of a progressive wave moving along the direction of negative X-axis is

3) The general equation of wave moving along X-axis is:

4) Other equation for a progressive wave moving along the direction of positive X-axis is:

Is propagation constant or wave vector

Points to remember: Wave equation can be shown by any of the following relations:

All these waves are also called as ‘sinusoidal’ waves.

4) Necessary Condition for two particles to be in same phase or in opposite phase:

5) Intensity of waves, Energy Density:

1) Intensity (I): Amount of energy flowing per second from unit area of the medium in a direction perpendicular to the propagation of wave.

2) For a given medium ( and v are constants)

3) Unit of Intensity: Joule/m² second or watt/m².

4) Energy density (E): Energy per unit volume of the medium

5) For second waves the intensity obeys the inverse square law. If the distance between the source and the point of observation is ‘x’ then

6) Superposition of Waves:

1) If two or more waves propagate in a medium simultaneously, then the resultant displacement of the particle of the medium is given by the vector sum of displacements produced by the individual wave.

2) If the displacement produced by different waves are respectively

Then the resultant displacement of particle is given by,

3) Resultant displacement depends on

a) Amplitude

b) Phase difference

c) Frequencies

d) Direction of propagation of waves

4) Different phenomena occurring as a result of superposition of waves are

a) Interference

b) Beats

c) Stationary waves

d) Lissajous figures.

7) Interference:

1) As a result of superposition of two waves of same frequency traveling in the same direction simultaneously, the phenomenon of intensity of resultant wave becoming maximum and minimum is called as interference.

2) Equation of resultant wave formed by superposition of two waves:

Amplitude of the resultant wave

3) Intensity of the resultant wave

I₁ and I₂ are separate intensities of the superposing waves.

4) Law of conservation of energy is obeyed in interference also, only the energy of the medium is redistributed.

8) Constructive and Destructive Interference:

1) When superposing waves are in the same phase then

The amplitude of the resultant wave

In this condition the interference is called ‘constructive interference’

2) For if is zero or even multiple of ; , where n = 0, 1, 2, ---------------- or path difference of waves.

3) Then in above condition, the intensity of resultant wave is maximum.

4) If the interfering waves are in opposite phase then , and path difference

In this condition the amplitude of the resultant wave

Interference of this kind is called the ‘destructive interference’. In this condition the resultant intensity is also minimum.

5) Ratio of maximum intensity and minimum intensity of a wave is

6) To observe clear interference:

a) Phase difference of the waves must be fixed.

b) Amplitude of the waves must be equal.

9) Beats:

1) When two progressive sound waves of nearly equal frequencies superpose while traveling in the same direction, then the intensity of the resultant sound increases and decreases with time.

2) The increase and decrease (waxing and waning) of intensity of sound occurs with a regular interval. This regular waxing and waning of sound is called Beats. One decrease and one increase together make one BEAT.

3) Number of beats produced in one second is called to be beat frequency.

4) If the frequencies of the waves are m and n (m>n), then beat frequency = m-n. This difference must not be greater than 10Hz, otherwise beats are not heard.

5) Beat period = 1/ N = 1/ (m-n).

6) If the waves producing beats are

Then the equation of the resultant wave is

And the amplitude of the wave is;

10) APPLICATION OF BEATS:

1) In the tuning of radio receiver.

2) In manufacturing the electronic oscillators of less and stable frequencies.

3) In detecting poisonous gases produced in mines.

4) In the tuning of musical instruments.

5) In determining unknown frequency of tuning fork: Let the known frequency of the tuning fork be ‘m’, unknown frequency be ‘n’ and the number of beats produced be ‘N’ while playing these two together.

(a) If by putting some wax on the arm of unknown frequency tuning fork:

If the beat frequency decreases then n = m+N.

If the beat frequency increases then n = m-N.

(b) If by putting wax on the arm of known frequency tuning fork:

If the beat frequency decreases, then n = m-N.

If the beat frequency increases, then n = m+N.

(c) If the arms of unknown frequency tuning fork are filled then:

If the beat frequency increase n = m+N.

If the beat frequency decrease n = m-N.

11) Reflection of sound waves when wave posses from one medium to the other:

(a) Reflection from a rigid wall or a denser medium:

1) Phase change of .

2) Change of in the path difference

3) No change in the nature of wave.

4) Compression is reflected as compression and rarefaction as a rarefaction.

5) Node is always formed at rigid surface

6) Direction of wave is changed.

(b) Reflection from rare medium:

1) No change in the phase.

2) No change in the path difference

3) No change in the nature of wave.

4) Compression is reflected as rarefaction while rarefaction as compression.

5) Antinode is formed at the surface of a rare medium.

6) Direction of the wave is changed.

12) stationary waves:

1) This is the wave produced by the superposition of two identical waves traveling along the same straight line but in opposite direction.

2) Energy is not transferred by these waves in the medium. It is only redistributed.

3) If be the propagation wave along +ve x-axis and be the progressive wave along –ve x-axis, then for the resultant stationary wave.

this depends on position x.

5) Antinodes: At these points the amplitude of the vibrating particles is maximum

and the change in pressure and the density is minimum.

6) For the positions of antinodes:

7) Nodes: At these points the amplitude of particles is minimum and the change in pressure and density is maximum.

13) for stationary waves:

1) Waves produce nodes and antinodes at regular points in limited medium.

2) The distance between to consecutive nodes/ antinodes is .

3) Distance between a node and an adjacent antinode is .

4) Particles situated between two nodes execute simple harmonic motion whose amplitude is different but frequencies are same.

5) Amplitude depends on the position of the particle; maximum amplitude is obtained at antinodes and zero amplitude at the nodes.

6) The particles situated between two consecutive nodes vibrate in the phase with different amplitudes while the particles situated on either side of a node vibrate in the opposite phase.

14) vibration in a stretched string:

1) Transverse progressive wave is produced in a stretched string.

2) Node is always formed at fixed end of string.

3) Velocity of the waves produced in a stretched string is

Where T = tension and m = mass per unit length of string.

Given density of material of string is d; radius is r, then;

4) For a string fixed at both the ends, nodes are formed at the ends.

5) If the length of a string is L and p loops are formed in it, then the frequency of string

(a) When p=1, then . In this condition string vibrates in one loop. is called the fundamental frequency. This is also called first harmonic.

(b) If p =2, then . In this condition is called the second harmonic or the first overtone.

(c) If p=3, then is called third harmonic or the second overtone.

(d) In strings all harmonics are produced and their ratio is ;

(e) Some harmonics are shown in the figure

Fig (4)

15) laws of vibration of a stretched string:

1) Law of length: where T and m are constants.

2) Law of tension: where L and m are constants.

3) Law of mass: where L and T are constants.

4) Law of radius: where d, L and T are constants.

5) Law of density: where r, T and L are constants.

On the basis of above Laws the formulae of frequency of vibration of string are:

M being the mass hanged on string.

16) melde’s experiment:

1) In a vibrating string of fixed length, the product of number of loops of loops in a vibrating string and square root of tension is a constant or

2)

3) In longitudinal vibration system the frequency of tuning fork is given by;

= 2 ´ (vibration frequency of string)

4) In this experiment vibrations of string are always transverse, but in longitudinal vibration system; the vibrations of arms of the tuning fork are along the direction of string. This experiment is also based on the stationary (transverse) waves.

17) vibration of air columns in pipes:

The pipe which contains air and in which sound vibrations are produced is called organ pipe.

1) CLOSED PIPE:

(a) One end of the pipe of this kind is closed and the other end is open.

(b) Node is formed at closed end and antinodes at the open end. In this pipe, number of antinodes and nodes are the same.

(c) Closed end of pipe reflects the compression as compression and rarefaction as rarefaction open end of the pipe reflects compression as rarefaction while rarefaction as compression.

(d) For a pipe of length L, the frequency of fundamental node of stationary waves produced. Fundamental frequency , is same as first harmonic while other are multiples of this frequency

wavelength . Frequency of third harmonic or the first overtone

wavelength

Frequency of the second overtone or the fifth harmonic

, wavelength

hese are shown in the following figure:

Fig (5)

(a) Fundamental node or first harmonic .

(b) First overtone or third harmonic .

(c) Second overtone or fifth harmonic .

Therefore the ratio of overtone is 3:5:7.

(e) It is clear that the only odd harmonics are produced in this pipe:

2) Vibration of an open pipe:

(a) These pipes are open at both ends where antinodes are formed. At these ends compression is reflected as rarefaction while rarefaction as compression. Here the number of antinodes is more than that of nodes.

(b) For a pipe of length L, the frequency of fundamental node and wavelength

First overtone or second harmonic frequency , wavelength

Second overtone or third harmonic frequency wavelength

These are shown in the following figure:

Fig (6)

(a) Fundamental node or first harmonic

(b) First overtone or second harmonic

(c) Second overtone or third harmonic

Clearly in open pipe all the harmonics are produced. The ratio of the frequencies is:

In this condition the ratio of overtone is 2:3:4:5: …………

End correction: e = 0.6r, where ‘r’ is the radius of pipe. Therefore for a closed pipe the effective length of the air column = L+e = L + 0.6r.

For open pipe the effective length of the air column is L+2e = L+1.2r.

For a closed pipe with end correction

By comparison:

(For fundamental only)

2) In open pipe all the harmonic are obtained while in closed pipe only odd harmonics are obtained.

3) The sound produced in open organ tube is pleasing and that of the closed organ tube is less pleasing.

18) resonance tube:

Resonance: If the frequency of a tuning fork used is same as the frequency of vibration of air column in the tube, then the intensity of sound becomes maximum. This is said to be the condition of resonance.

It is an example of a closed pipe in which the length of the air column can be changed by adjusting the level of water.

Application:

To determine the velocity of sound and the frequency of tuning fork

Formula:

If the first and second resonance lengths are L₁ and L₂ then

19)

(a) Velocity of sound (Longitudinal wave) in elastic medium:

Where E = Coefficient of elasticity of medium, d = density

For a solid medium:

, where E=Y = Young’s coefficient of elasticity

In a liquid medium:

, where E=K= Coefficient of volume elasticity.

In gaseous medium:

M = molecular, P = pressure, T = temperature.

(b) In stretched vibrating wire or a string, the velocity of a transverse wave: