To solve the problem step by step, we will calculate the intensity of sound and the amplitude of oscillations for both the faintest and loudest sounds.
### Given Data:
- Faintest sound pressure amplitude, \( \Delta P_1 = 2 \times 10^{-5} \, \text{N/m}^2 \)
- Loudest sound pressure amplitude, \( \Delta P_2 = 28 \, \text{N/m}^2 \)
- Density of air, \( \rho = 1.29 \, \text{kg/m}^3 \)
- Velocity of sound, \( v = 345 \, \text{m/s} \)
- Frequency, \( f = 500 \, \text{Hz} \)
### Part (a): Calculate Intensity in \( \text{W/m}^2 \) and in \( \text{dB} \)
#### Step 1: Calculate the intensity for the faintest sound
The formula for intensity \( I \) is given by:
\[
I = \frac{1}{2} \frac{(\Delta P)^2}{\rho v}
\]
Substituting the values for the faintest sound:
\[
I_1 = \frac{1}{2} \frac{(2 \times 10^{-5})^2}{1.29 \times 345}
\]
Calculating \( I_1 \):
\[
I_1 = \frac{1}{2} \frac{4 \times 10^{-10}}{444.05} \approx 4.49 \times 10^{-13} \, \text{W/m}^2
\]
#### Step 2: Convert intensity to decibels
The formula to convert intensity to decibels is:
\[
L = 10 \log_{10} \left( \frac{I}{I_0} \right)
\]
Where \( I_0 = 10^{-12} \, \text{W/m}^2 \) is the reference intensity.
Substituting the values:
\[
L_1 = 10 \log_{10} \left( \frac{4.49 \times 10^{-13}}{10^{-12}} \right) \approx 10 \log_{10} (4.49) \approx -3.48 \, \text{dB}
\]
#### Step 3: Calculate the intensity for the loudest sound
Using the same formula for intensity:
\[
I_2 = \frac{1}{2} \frac{(28)^2}{1.29 \times 345}
\]
Calculating \( I_2 \):
\[
I_2 = \frac{1}{2} \frac{784}{444.05} \approx 0.88 \, \text{W/m}^2
\]
#### Step 4: Convert loudest intensity to decibels
\[
L_2 = 10 \log_{10} \left( \frac{0.88}{10^{-12}} \right) \approx 10 \log_{10} (8.8 \times 10^{11}) \approx 119.45 \, \text{dB}
\]
### Part (b): Calculate the amplitude of oscillations
#### Step 5: Calculate amplitude for the faintest sound
Using the formula for amplitude \( A \):
\[
A_1 = \frac{\Delta P_1}{\rho v (2 \pi f)}
\]
Substituting the values:
\[
A_1 = \frac{2 \times 10^{-5}}{1.29 \times 345 \times (2 \pi \times 500)}
\]
Calculating \( A_1 \):
\[
A_1 \approx \frac{2 \times 10^{-5}}{1.29 \times 345 \times 3141.59} \approx 1.43 \times 10^{-11} \, \text{m}
\]
#### Step 6: Calculate amplitude for the loudest sound
Using the same formula:
\[
A_2 = \frac{\Delta P_2}{\rho v (2 \pi f)}
\]
Substituting the values:
\[
A_2 = \frac{28}{1.29 \times 345 \times (2 \pi \times 500)}
\]
Calculating \( A_2 \):
\[
A_2 \approx \frac{28}{1.29 \times 345 \times 3141.59} \approx 2 \times 10^{-5} \, \text{m}
\]
### Final Results:
- **Faintest Sound:**
- Intensity: \( 4.49 \times 10^{-13} \, \text{W/m}^2 \)
- Decibels: \( -3.48 \, \text{dB} \)
- Amplitude: \( 1.43 \times 10^{-11} \, \text{m} \)
- **Loudest Sound:**
- Intensity: \( 0.88 \, \text{W/m}^2 \)
- Decibels: \( 119.45 \, \text{dB} \)
- Amplitude: \( 2 \times 10^{-5} \, \text{m} \)
