To compute the intensity level (IL) in decibels (dB), we use the formula:

$IL = 10 \cdot \log_{10}\left(\frac{I}{I_0}\right)$

Where:

• $I$ is the intensity of the sound in $\text{W/m}^2$.
• $I_0$ is the reference intensity, typically $1 \times 10^{-12} \, \text{W/m}^2$, which corresponds to the threshold of hearing.

Given:

• $I = 10^{-13} \, \text{W/m}^2$
• $I_0 = 1 \times 10^{-12} \, \text{W/m}^2$

Now, let's compute the intensity level.

$IL = 10 \cdot \log_{10}\left(\frac{10^{-13}}{10^{-12}}\right) = 10 \cdot \log_{10}\left(10^{-1}\right)$

$IL = 10 \cdot (-1) = -10 \, \text{dB}$

The intensity level is -10 dB.

Would you like further details or clarification on any part of the calculation? Here are some related questions:

1. How is sound intensity measured in practice?
2. What is the reference level for sound intensity in decibels?
3. What happens when the intensity is lower than the reference $I_0$?
4. Can you explain the relationship between intensity and loudness perception?
5. How does distance affect the intensity level of a sound?

Tip: The decibel scale is logarithmic, meaning each increase of 10 dB represents a tenfold increase in sound intensity.