Question

If two traveling waves, given by
𝑦₁=𝐴₀ sin(α𝑘𝑥 -𝜔𝑡) and 𝑦₂=𝐴₀ sin(α𝑘𝑥 − β𝜔𝑡) 
are superposed, which of the following statements is correct?

• A. For α= β=1, the resultant wave is a standing wave
• B. For α= β=-1, the resultant wave is a standing wave
• C. For α= β=2, the carrier frequency of the resultant wave is \(\frac{3}{2}\)𝜔
• D. For α= β=2, the carrier frequency of the resultant wave is 3𝜔

Answer 1

The Correct Option is C

When the two waves \( y_1 \) and \( y_2 \) are superposed, the resultant wave can be expressed as a combination of a carrier wave and a modulation wave. The carrier frequency is determined by the average of the angular frequencies, while the modulation frequency is determined by their difference.

Given:

\[ y_1 = A_0 \sin(kx - \omega t), \quad y_2 = A_0 \sin(\alpha kx - \beta \omega t) \]

Resultant Wave:

The resultant wave is given by the superposition of \( y_1 \) and \( y_2 \):

\[ y_{\text{resultant}} = 2A_0 \cos\left(\frac{\beta \omega - \omega}{2} t\right) \sin\left(\frac{\beta \omega + \omega}{2} t\right) \]

• The carrier frequency is determined by the term \( \frac{\beta \omega + \omega}{2} \).
• The modulation frequency is determined by the term \( \frac{\beta \omega - \omega}{2} \).

Substitute \( \alpha = \beta = 2 \):

For \( \alpha = \beta = 2 \), the carrier frequency becomes:

\[ \text{Carrier Frequency} = \frac{\omega + 2\omega}{2} = \frac{3}{2} \omega \]

Conclusion:

Thus, the correct statement is that the carrier frequency of the resultant wave is \( \frac{3}{2} \omega \).