Question

A composite sound wave is represented by \( y = A \cos \omega t \cdot \cos \omega' t \). The observed beat frequency is:

• A. \( \frac{\omega - \omega'}{2\pi} \)
• B. \( \frac{\omega - \omega'}{\pi} \)
• C. \( \frac{\omega}{2\pi} \)
• D. \( \frac{\omega'}{\pi} \)

Hint:

When two waves with frequencies \( \omega \) and \( \omega' \) combine, the beat frequency is given by \( \frac{\omega - \omega'}{2\pi} \).

Answer 1

The Correct Option is A

The equation for the composite sound wave is given by: \[ y = A \cos(\omega t) \cdot \cos(\omega' t) \] Using the trigonometric identity for the product of cosines: \[ \cos A \cos B = \frac{1}{2} \left( \cos(A - B) + \cos(A + B) \right) \] we can rewrite the equation as: \[ y = \frac{A}{2} \left( \cos[(\omega - \omega') t] + \cos[(\omega + \omega') t] \right) \] The first term represents the beat frequency, which is the frequency of the oscillation of the amplitude. The frequency of the beats is given by: \[ f_{\text{beat}} = \frac{\omega - \omega'}{2\pi} \] Thus, the observed beat frequency is \( \frac{\omega - \omega'}{2\pi} \). Therefore, the correct answer is (1) \( \frac{\omega - \omega'}{2\pi} \).