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 solve the problem regarding the intensity of sound produced in beats from two sound waves of equal intensity, we can follow these steps:

### Step 1: Define the Intensities of the Sound Waves
Let the intensity of the two sound waves be \( I_1 = I \) and \( I_2 = I \), where \( I \) is the given intensity.

### Step 2: Write the Equations for the Sound Waves
Assume the two sound waves can be expressed as:
- \( y_1 = A \sin(2\pi f_1 t) \)
- \( y_2 = A \sin(2\pi f_2 t) \)

Here, \( A \) is the amplitude of the waves, and \( f_1 \) and \( f_2 \) are their respective frequencies.

### Step 3: Understand the Concept of Beats
When two sound waves of slightly different frequencies interfere, they produce a phenomenon called beats. The beat frequency is given by the absolute difference of the two frequencies:
\[ f_{beat} = |f_1 - f_2| \]

### Step 4: Resultant Amplitude of the Combined Waves
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 sines, we can rewrite this as:
\[ y = 2A \cos\left(\pi (f_1 - f_2) t\right) \sin\left(\pi (f_1 + f_2) t\right) \]

### Step 5: Determine the Maximum and Minimum Amplitude
The amplitude of the resultant wave varies between:
- Maximum amplitude: \( 2A \) (when \( \cos\left(\pi (f_1 - f_2) t\right) = 1 \))
- Minimum amplitude: \( 0 \) (when \( \cos\left(\pi (f_1 - f_2) t\right) = 0 \))

### Step 6: Relate Amplitude to Intensity
The intensity \( I \) of a wave is proportional to the square of its amplitude:
\[ I \propto A^2 \]

Thus, the maximum intensity \( I_{max} \) when the amplitude is maximum \( 2A \) is:
\[ I_{max} = k(2A)^2 = 4kA^2 \]
Where \( k \) is a proportionality constant.

The intensity of the original waves is:
\[ I = kA^2 \]

### Step 7: Determine the Range of Resultant Intensity
From the above relations:
- Minimum intensity \( I_{min} = 0 \)
- Maximum intensity \( I_{max} = 4I \)

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

### Conclusion
The intensity of sound produced in beats will be between \( 0 \) and \( 4I \). Therefore, the possible values for the intensity \( I_s \) can be \( I, 2I, 3I, \) or \( 4I \), depending on the phase relationship of the two waves.