Two sound waves of equal intensity l, generates beats. The intensity of sound `l_(s)` produced in beats will be

Options

A. l
B. 4.l
C. 2.l
D. `0ltl_(s)lt4l`

Answer

To find the intensity of sound \( I_s \) produced in beats by two sound waves of equal intensity \( I \), we can follow these steps:

### Step 1: Understand the relationship between intensity and amplitude
The intensity \( I \) of a wave is directly proportional to the square of its amplitude \( A \):
\[ I \propto A^2 \]
If the intensity of both waves is \( I \), then their amplitudes \( A_1 \) and \( A_2 \) are equal:
\[ A_1 = A_2 = A \]

### Step 2: Write the equations for the two waves
Assume the equations of the two waves are:
\[ y_1 = A \sin(2\pi f_1 t) \]
\[ y_2 = A \sin(2\pi f_2 t) \]

### Step 3: Determine the resultant wave
When two waves interfere, the resultant wave can be expressed as:
\[ y = y_1 + y_2 = A \sin(2\pi f_1 t) + A \sin(2\pi f_2 t \]
Using the trigonometric identity for the sum of sine functions, we can express this as:
\[ y = 2A \cos\left(\frac{(f_1 - f_2) \cdot 2\pi t}{2}\right) \sin\left(\frac{(f_1 + f_2) \cdot 2\pi t}{2}\right) \]

### Step 4: Analyze the amplitude of the resultant wave
The amplitude of the resultant wave is:
\[ A' = 2A \cos\left(\frac{(f_1 - f_2) \cdot 2\pi t}{2}\right) \]
This amplitude oscillates between \( 0 \) and \( 2A \).

### Step 5: Calculate the intensity of the resultant wave
The intensity \( I_s \) of the resultant wave is proportional to the square of the amplitude:
\[ I_s \propto (A')^2 = (2A \cos\theta)^2 = 4A^2 \cos^2\theta \]
Where \( \theta = \frac{(f_1 - f_2) \cdot 2\pi t}{2} \).

### Step 6: Determine the range of intensity
- The maximum intensity occurs when \( \cos^2\theta = 1 \):
\[ I_{max} = 4A^2 \propto 4I \]
- The minimum intensity occurs when \( \cos^2\theta = 0 \):
\[ I_{min} = 0 \]

Thus, the intensity of sound \( I_s \) produced in beats will range from \( 0 \) to \( 4I \):
\[ 0 \leq I_s \leq 4I \]

### Conclusion
The intensity of sound \( I_s \) produced in beats can take values from \( 0 \) to \( 4I \), inclusive.

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