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The problem asks for the pressure and intensity of a harmonic plane wave in a fluid with a density of 1000 kg/m³. The particle velocity is given as:

\[
\vec{v}(x,t) = 2 \sin(2000t - x) \hat{k} \, \text{cm/s}.
\]

### Step-by-step Solution:

1. **Convert velocity to SI units:**
   The velocity is provided in cm/s, and we need to convert it to m/s for consistency in SI units:
   \[
   \vec{v}(x,t) = 2 \times 10^{-2} \sin(2000t - x) \, \hat{k} \, \text{m/s}.
   \]

2. **General form of a plane wave:**
   The particle velocity for a harmonic plane wave is generally given by:
   \[
   \vec{v}(x,t) = v_0 \sin(\omega t - kx),
   \]
   where \( v_0 \) is the amplitude, \( \omega \) is the angular frequency, and \( k \) is the wavenumber.

   From the given velocity expression, we identify:
   - Amplitude \( v_0 = 2 \times 10^{-2} \, \text{m/s} \),
   - Angular frequency \( \omega = 2000 \, \text{rad/s} \),
   - Wavenumber \( k = 1 \, \text{m}^{-1} \).

3. **Pressure amplitude:**
   The pressure amplitude \( p_0 \) is related to the velocity amplitude \( v_0 \) by the formula:
   \[
   p_0 = \rho v_0 c,
   \]
   where \( \rho \) is the density of the fluid and \( c \) is the speed of sound in the fluid. However, the speed of sound is not given in the problem, so we assume it to be a standard value for a fluid. Let's use \( c = 1500 \, \text{m/s} \) (which is typical for water).

   Substituting values:
   \[
   p_0 = (1000 \, \text{kg/m}^3)(2 \times 10^{-2} \, \text{m/s})(1500 \, \text{m/s}) = 30 \, \text{Pa}.
   \]

4. **Intensity of the wave:**
   The intensity \( I \) of the wave is related to the pressure amplitude \( p_0 \) by:
   \[
   I = \frac{p_0^2}{2 \rho c}.
   \]
   Substituting the values:
   \[
   I = \frac{30^2}{2(1000)(1500)} = \frac{900}{3000000} = 3 \times 10^{-4} \, \text{W/m}^2.
   \]

### Final Answers:
- **Pressure amplitude**: \( p_0 = 30 \, \text{Pa} \).
- **Intensity**: \( I = 3 \times 10^{-4} \, \text{W/m}^2 \).

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Do you need further clarification or details on any step?  
Here are some related questions you may want to explore:

1. How does the speed of sound in different fluids affect the pressure amplitude?
2. What role does the wavenumber play in wave propagation?
3. How would the results change if the fluid's density was different?
4. What is the relationship between intensity and amplitude for sound waves?
5. How does the angular frequency influence the behavior of harmonic waves?

**Tip**: Always ensure that units are consistent throughout your calculations to avoid errors.

Find the pressure and intensity of a harmonic plane wave in an unknown fluid with a density of 1000 kg/m³ when the particle velocity is given by: v(x,t) = 2sin(2000t - x) k̂ cm/s.

Learn how to calculate the pressure and intensity of a harmonic plane wave in a fluid with a density of 1000 kg/m³. This problem involves key wave dynamics concepts and formulas for pressure and intensity in fluid physics.

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