To find the amplitude of the traveling wave, we will use the given information and relevant equations step by step.
### Step 1: Write down the known values
We have the following values from the problem:
- Angular frequency, \( \omega = 1000 \, \text{rad/s} \)
- Density, \( d = 1 \, \text{kg/m}^3 \)
- Intensity, \( I = 10^{-10} \, \text{W/m}^2 \)
- Maximum pressure change, \( \Delta P_m = 2 \times 10^{-4} \, \text{N/m}^2 \)
### Step 2: Use the intensity formula
The intensity \( I \) of a longitudinal wave can be expressed as:
\[
I = \frac{1}{2} \rho v \omega^2 A^2
\]
where \( A \) is the amplitude, \( \rho \) is the density, and \( v \) is the velocity of the wave.
### Step 3: Express velocity in terms of pressure change
For a longitudinal wave, the maximum pressure change \( \Delta P_m \) can be related to the amplitude and angular frequency as follows:
\[
\Delta P_m = \rho v A \omega
\]
From this, we can express \( v A \):
\[
v A = \frac{\Delta P_m}{\rho \omega}
\]
### Step 4: Substitute known values
Substituting the known values into the equation for \( v A \):
\[
v A = \frac{2 \times 10^{-4}}{1 \times 1000} = 2 \times 10^{-7} \, \text{m}^2/\text{s}
\]
### Step 5: Use the intensity equation
Now, we can substitute \( v A \) back into the intensity equation. First, we need to express \( v \) in terms of \( A \):
From the intensity equation:
\[
I = \frac{1}{2} \rho v \omega^2 A^2
\]
We can express \( v \) as:
\[
v = \frac{I}{\frac{1}{2} \rho \omega^2 A^2}
\]
### Step 6: Substitute \( v A \) into the intensity equation
Using the relation \( v A = 2 \times 10^{-7} \):
\[
I = \frac{1}{2} \rho \left(\frac{2 \times 10^{-7}}{A}\right) \omega^2 A^2
\]
This simplifies to:
\[
I = \rho \omega^2 A \times 10^{-7}
\]
### Step 7: Solve for amplitude \( A \)
Now we can rearrange the equation to solve for \( A \):
\[
A = \frac{I}{\rho \omega^2 \times 10^{-7}}
\]
Substituting the values:
\[
A = \frac{10^{-10}}{1 \times (1000)^2 \times 10^{-7}} = \frac{10^{-10}}{10^6 \times 10^{-7}} = \frac{10^{-10}}{10^{-1}} = 10^{-9} \, \text{m}
\]
### Conclusion
Thus, the amplitude of the traveling wave is:
\[
\boxed{10^{-9} \, \text{m}}
\]
This corresponds to option (c).
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