If the intensity of sound is doubled, by how many decibels does the sound level increase ?

Explanation

To solve the problem of how many decibels the sound level increases when the intensity of sound is doubled, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Decibel Formula**:
The sound level in decibels (β) is given by the formula:
\[ \beta = 10 \log \left( \frac{I}{I_0} \right) \]
where \( I \) is the intensity of the sound and \( I_0 \) is the reference intensity.

2. **Define the Initial and Final Intensities**:
Let the initial intensity be \( I_0 \). If the intensity is doubled, the new intensity \( I \) becomes:
\[ I = 2I_0 \]

3. **Substitute the New Intensity into the Formula**:
Substitute \( I = 2I_0 \) into the decibel formula:
\[ \beta = 10 \log \left( \frac{2I_0}{I_0} \right) \]

4. **Simplify the Expression**:
The \( I_0 \) terms cancel out:
\[ \beta = 10 \log (2) \]

5. **Calculate the Logarithm**:
The value of \( \log(2) \) is approximately 0.3010. Therefore:
\[ \beta = 10 \times 0.3010 = 3.010 \]

6. **Conclusion**:
The increase in sound level when the intensity is doubled is approximately:
\[ \beta \approx 3 \text{ decibels} \]

### Final Answer:
The sound level increases by approximately **3 decibels** when the intensity of sound is doubled.