Question

The resultant amplitude of superposition of two waves \( y_1 = a \cos(\omega t) \) and \( y_2 = a \cos(\omega t + \phi) \) is:

• A. \( 2a \cos(\omega t) \)
• B. \( a \cos(\omega t + \phi) \)
• C. \( 2a \cos \left( \frac{\phi}{2} \right) \)
• D. \( 2a \)

Hint:

When two waves superimpose, their amplitudes add up vectorially. The resultant amplitude can be calculated using the formula \( y = a \sqrt{2(1 + \cos(\phi))} \).

Answer 1

The Correct Option is C

The resultant amplitude when two waves superimpose is given by: \[ y = \sqrt{a^2 + a^2 + 2a^2 \cos(\phi)} = a \sqrt{2(1 + \cos(\phi))} \] Simplifying further: \[ y = 2a \cos \left( \frac{\phi}{2} \right) \] Thus, the resultant amplitude is \( 2a \cos \left( \frac{\phi}{2} \right) \).