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Superposition of waves
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Superposition of Waves & Interference

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:

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.

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