Question

What is the meaning of interference? Derive the expression for the intensity of the resulting wave due to interference of the waves \( y_1 = a_1 \sin(\omega t) \) and \( y_2 = a_2 \sin(\omega t + \phi) \). If \( a_1 = 5 \, \text{cm} \) and \( a_2 = 3 \, \text{cm} \), then find the ratio of maximum and minimum intensities of the resulting wave.

Hint:

The intensity in interference phenomena depends on the phase difference between the waves. Maximum intensity occurs when the waves are in phase, and minimum intensity occurs when the waves are out of phase.

Answer 1

Step 1: Meaning of Interference.
Interference is the phenomenon that occurs when two or more waves overlap and combine to form a resultant wave. This can lead to constructive interference (when the waves add up) or destructive interference (when the waves cancel out).
Step 2: Deriving the Expression for Intensity.
The total displacement \( y \) due to the interference of two waves \( y_1 \) and \( y_2 \) is given by: \[ y = y_1 + y_2 = a_1 \sin(\omega t) + a_2 \sin(\omega t + \phi) \] Using the trigonometric identity: \[ \sin + \sin = 2 \sin\left( \frac{A + B}{2} \right) \cos\left( \frac{A - B}{2} \right) \] we get: \[ y = 2 \sin\left( \omega t + \frac{\phi}{2} \right) \cos\left( \frac{\phi}{2} \right) \] The intensity \( I \) of the wave is proportional to the square of the amplitude, so: \[ I \propto y^2 = 4 a_1 a_2 \cos^2\left( \frac{\phi}{2} \right) \]
Step 3: Maximum and Minimum Intensity.
- For maximum intensity, \( \cos^2\left( \frac{\phi}{2} \right) = 1 \), so the maximum intensity \( I_{\text{max}} \) is: \[ I_{\text{max}} \propto 4 a_1 a_2 \] - For minimum intensity, \( \cos^2\left( \frac{\phi}{2} \right) = 0 \), so the minimum intensity \( I_{\text{min}} \) is: \[ I_{\text{min}} \propto 0 \]
Step 4: Ratio of Maximum and Minimum Intensities.
The ratio of maximum to minimum intensities is: \[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{4 a_1 a_2}{0} = \infty \] Thus, the intensity can vary from maximum to zero, depending on the phase difference \( \phi \).