Question

Two waves \( y_1 = A_1 \sin (\omega t - \beta_1 x) \) and \( y_2 = A_2 \sin (\omega t - \beta_2 x) \) superimpose to form a resultant wave whose amplitude is

• A. \( A_1 + A_2 \)
• B. \( \sqrt{A_1^2 + A_2^2} \)
• C. \( A_1^2 + A_2^2 \)
• D. \( \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos (\beta_2 - \beta_1)} \)

Hint:

When two waves superimpose, the resultant amplitude depends on the phase difference between them.

Answer 1

The Correct Option is D

Step 1: Superposition of waves.
The resultant amplitude of two superimposed waves is given by the formula: \[ A = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos (\beta_2 - \beta_1)} \] where \( A_1 \) and \( A_2 \) are the amplitudes of the individual waves, and \( \beta_1 \) and \( \beta_2 \) are the wave numbers.

Step 2: Conclusion.
The correct amplitude of the resultant wave is given by the formula in option (4).