# Four different independent waves are represented by,

i. ${y_1} = {a_1}\sin \omega t$

ii. ${y_2} = {a_2}\sin 2\omega t$

iii. ${y_3} = {a_3}\cos \omega t$

iv. ${y_4} = {a_4}\sin \left( {\omega t + \dfrac{\pi }{3}} \right)$

With which two waves interference is possible.

A. I and III

B. I and IV

C. III and IV

D. Not possible with any combination.

Answer

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**Hint:**Forbidden gap or band gap is same, for pure silicon at room temperature the forbidden gap or band gap is 1.1, that means the gap between the valance band and conduction band is called forbidden gap. This band implies without energy in the forbidden gap.

**Complete step-by-step solution:**

Given four independent waves,

$

{y_1} = {a_1}\sin \omega t \\

{y_2} = {a_2}\sin 2\omega t \\

{y_3} = {a_3}\cos \omega t \\

{y_4} = {a_4}\sin \left( {\omega t + \dfrac{\pi }{3}} \right) \\

$

In ${y_4}$they have mentioned $\dfrac{\pi }{3}$ that means phase difference

From the above mentioned data we can eight combinations of chances from those eight combinations, we have to prove the two waves interference,

For that we are using the condition for interference,

Condition for interference has two conditions; that condition is coherent and monochromatic condition,

In coherent condition there are three types of conditions which could be satisfied,

Those are 1.constant phase difference

2. Same frequency

3. Same waveform

From the given data there is the same waveform that means it has the same sinusoidal waveforms so that the same waveform is proved. Therefore the third condition in coherent is satisfied.

Now we can choose constant phase difference, from the given data the constant phase difference is also proved, therefore the first condition is also satisfied.

Now we can prove that the same frequency is not satisfied because any frequency is not equal to the remaining frequency, thus the second condition is dissatisfied.

Thus the coherent condition is not proved, so the correct option is D (not possible with any combination).

**Note:**In further we can choose the monochromatic condition, but in that condition also we cannot prove the two waves interfere. Hence the correct option from the given data is d. The sources are independent, the interference will not occur, so the given data is not possible with any combination

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