MHT CET · Physics · Waves and Sound

If two waves of the same frequency and amplitude respectively on superposition produce a resultant disturbance of the same amplitude, the waves differ in phase by

A π
B Zero
C π/3
D 2π/3

Verified Solution

Answer & Solution

Correct Answer

(D) 2π/3

Step-by-step Solution

Detailed explanation

Let the equation of two waves are

y_{1} = a \sin (ω t - k x) \dots (i)

And

y_{2} = a \sin (ω t - k x + ϕ) \dots (i i)

When they superpose, the resultant wave is

y = y_{1} + y_{2}

= a [\sin (ω t - k x) + \sin (ω t - k x + ϕ)]

= a [2 \sin (ω t - k x + \frac{ϕ}{2}) \cos (- \frac{ϕ}{2})]

= 2 a \sin (ω t - k x + \frac{ϕ}{2}) \cos \frac{ϕ}{2}

= (2 a \cos \frac{ϕ}{2}) \sin (ω t - k x + \frac{ϕ}{2}) \dots (i i i)

Comparing Eq. (iii) with (i) or (ii), we get

a = 2 a \cos \frac{ϕ}{2} ⟹ \cos \frac{ϕ}{2} = \frac{1}{2}

⟹ \cos \frac{ϕ}{2} = \cos \frac{π}{3}

∴ \cos \frac{ϕ}{2} = \cos \frac{π}{3}

∴ \frac{ϕ}{2} = \frac{π}{3}

o r ϕ = \frac{2 π}{3}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Explore more questions on app

From MHT CET

Explore more questions on app