A wave moves ar a constant speed along a stretched string.Mark the incorrect statement out of the following:

Options

A. particle speed is constant and equal to the wave speed
B. particle speed is independent of amplitude of the periodic mmotion of the source.
C. particle speed is independent of frequency of periodic motion of the source
D. particle speed is dependent on tension and linear mass density the string

Explanation

To solve the problem, we need to analyze the statements related to the wave moving along a stretched string and identify the incorrect one.

### Step-by-Step Solution:

1. **Understand the Wave Equation**:
The wave equation for a wave traveling along a stretched string can be expressed as:
\[ y = A \sin(\omega t - kx) \]
where:
- \( y \) is the displacement,
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number,
- \( t \) is time,
- \( x \) is the position along the string.

2. **Identify Particle Velocity**:
The particle velocity \( v_p \) can be derived by differentiating the wave equation with respect to time:
\[ v_p = \frac{\partial y}{\partial t} = A \omega \cos(\omega t - kx) \]
This shows that the particle velocity varies with time and position, indicating it is not constant.

3. **Determine Wave Velocity**:
The wave velocity \( v \) is given by:
\[ v = \frac{\omega}{k} \]
This is a constant value for a given wave, independent of time and position.

4. **Analyze the Statements**:
Now, we need to evaluate the provided statements:
- **Statement 1**: "Particle speed is constant and equal to the wave speed."
- **Analysis**: Incorrect, as particle speed varies with time and position.
- **Statement 2**: "Particle speed is independent of amplitude."
- **Analysis**: Incorrect, as particle speed is directly proportional to amplitude.
- **Statement 3**: "Particle speed is independent of frequency."
- **Analysis**: Incorrect, as particle speed depends on angular frequency \( \omega \).
- **Statement 4**: "Particle speed is dependent on the tension and the linear mass density of the string."
- **Analysis**: Correct, since wave speed depends on tension \( T \) and linear mass density \( \mu \) as \( v = \sqrt{\frac{T}{\mu}} \).

5. **Conclusion**:
All the first three statements are incorrect, but the question asks for one incorrect statement. The first statement, which claims that particle speed is constant and equal to wave speed, is the most fundamentally incorrect in terms of wave mechanics.

### Final Answer:
The incorrect statement is: "Particle speed is constant and equal to the wave speed."