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NVAEasyJEE 2023Superposition Principle & Standing Waves

JEE Physics 2023 Question with Solution

The displacement equations of two interfering waves are given by y1=10sin⁡(ωt+π3) cm,y2=5[sin⁡(ωt)+3cos⁡(ωt)] cm.y_1 = 10 \sin(\omega t + \frac{\pi}{3}) \, cm, \quad y_2 = 5 [\sin(\omega t) + \sqrt{3} \cos(\omega t)] \, cm. The amplitude of the resultant wave is ..... cm.

Answer

Correct answer:20

Step-by-step solution

Standard Method

Given:

y1=10sin⁡(ωt+π3)y_1 = 10 \sin(\omega t + \frac{\pi}{3})

and

y2=5[sin⁡(ωt)+3cos⁡(ωt)]y_2 = 5[\sin(\omega t) + \sqrt{3}\cos(\omega t)]

Find: The amplitude of the resultant wave.

From the solution, the resultant displacement is the sum of y1y_1 and y2y_2. First rewrite y2y_2 as

y2=5sin⁡(ωt)+53cos⁡(ωt)y_2 = 5\sin(\omega t) + 5\sqrt{3}\cos(\omega t)

Then use the amplitude formula for two interfering waves:

Aresult=A12+A22+2A1A2cos⁡(ϕ1−ϕ2)A_{\text{result}} = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos(\phi_1 - \phi_2)}

where A1=10 cmA_1 = 10 \, \text{cm}, A2=5 cmA_2 = 5 \, \text{cm}, and the phase difference is π3\frac{\pi}{3}.

As stated in the solution, after calculation the amplitude is found to be 20 cm20 \, \text{cm}.

Therefore, the required numerical answer is 20.

Using the hint and given phase relation

Given: Two interfering waves are added by the principle of superposition.

Find: Resultant amplitude.

The hint says to use vector addition for waves with a phase difference. The provided explanation identifies the phase difference as π3\frac{\pi}{3} and concludes the resultant amplitude after calculation.

So the final result extracted from the solution is 20 cm20 \, \text{cm}.

Hence, the answer is 20.

Common mistakes

  • Treating y2=5[sin⁡(ωt)+3cos⁡(ωt)]y_2 = 5[\sin(\omega t) + \sqrt{3}\cos(\omega t)] as having amplitude 5 cm5 \, \text{cm} directly is incorrect, because the sine and cosine terms must first be combined properly. Use superposition or phasor addition before reading the amplitude.

  • Ignoring the phase difference between the two waves is wrong, because resultant amplitude depends on relative phase. Use the interference amplitude relation with the phase difference identified from the expressions.

  • Adding amplitudes algebraically without checking whether the waves are exactly in phase is incorrect. Only when the phase difference is zero can amplitudes be added directly; otherwise use vector addition.

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