Solved Examples: Calculating Sound Intensity with Distance

📚 Quick Study Guide

• 🔊 Sound intensity ($I$) is the power ($P$) of a sound wave per unit area ($A$). The formula is: $I = \frac{P}{A}$.
• 📏 The area ($A$) is often the surface area of a sphere at a certain distance ($r$) from the sound source: $A = 4\pi r^2$.
• 📉 Therefore, sound intensity can also be expressed as: $I = \frac{P}{4\pi r^2}$. This means sound intensity decreases with the square of the distance.
• 👂 The threshold of hearing ($I_0$) is the minimum sound intensity detectable by the human ear, approximately $1.0 \times 10^{-12} \, W/m^2$.
• ➕ To compare sound intensities at two different distances ($r_1$ and $r_2$) from the same source: $\frac{I_1}{I_2} = \frac{r_2^2}{r_1^2}$
• 💡 When the distance doubles, the sound intensity reduces to one-fourth of its original value (inverse square law).

🧪 Practice Quiz

1. What is the formula for sound intensity ($I$) in terms of power ($P$) and area ($A$)?
   1. $I = P \times A$
   2. $I = \frac{A}{P}$
   3. $I = \frac{P}{A}$
   4. $I = P + A$
2. If the power of a sound source is 100 W, what is the intensity at a distance of 2 meters, assuming the sound radiates uniformly in all directions?
   1. $1.99 \, W/m^2$
   2. $0.99 \, W/m^2$
   3. $2.99 \, W/m^2$
   4. $3.99 \, W/m^2$
3. If the distance from a sound source doubles, what happens to the sound intensity?
   1. It doubles.
   2. It quadruples.
   3. It is reduced to one-half.
   4. It is reduced to one-fourth.
4. A sound has an intensity of $8.0 \times 10^{-8} \, W/m^2$ at a distance of 5 meters. What is the intensity at a distance of 10 meters?
   1. $4.0 \times 10^{-8} \, W/m^2$
   2. $2.0 \times 10^{-8} \, W/m^2$
   3. $1.0 \times 10^{-8} \, W/m^2$
   4. $6.0 \times 10^{-8} \, W/m^2$
5. Which of the following factors does NOT affect sound intensity?
   1. Frequency of the sound wave
   2. Distance from the sound source
   3. Power of the sound source
   4. Area over which the sound is spread
6. What is the relationship between sound intensity ($I_1$) at distance $r_1$ and sound intensity ($I_2$) at distance $r_2$ from the same source?
   1. $\frac{I_1}{I_2} = \frac{r_1^2}{r_2^2}$
   2. $\frac{I_1}{I_2} = \frac{r_1}{r_2}$
   3. $\frac{I_1}{I_2} = \frac{r_2^2}{r_1^2}$
   4. $\frac{I_1}{I_2} = \frac{r_2}{r_1}$
7. If the intensity of a sound is $5 \times 10^{-6} \, W/m^2$ at a certain location, and the power of the source is doubled, what is the new intensity at the same location?
   1. $2.5 \times 10^{-6} \, W/m^2$
   2. $5 \times 10^{-6} \, W/m^2$
   3. $10 \times 10^{-6} \, W/m^2$
   4. $20 \times 10^{-6} \, W/m^2$