Let a disturbance y be propagated as a plane wave along the x-axis. The wave profile at the instants `t=t_(1) and t=t_(2)` are represent respectively as: `y_(1)=f(x_(1)-vt_(1)) and `y_(2)=f(x_(2)-vt_(2)). The wave is propagating withoute change of shape.

Options

A. the velocity of the wave is `v`
B. the velocity of the wave is `v=(x_(2)+x_(1)//(t_(2)+t_(1))`
C. the particle verlocity is `v_(p)=-vf(x-vt)`
D. the phase velocity of the wave is `v`

Explanation

To solve the problem, we need to analyze the wave profiles given at two different times and determine various properties of the wave. Let's go through the solution step by step.

### Step-by-Step Solution

1. **Understanding the Wave Profile**:
The wave profiles are given as:
- \( y_1 = f(x_1 - vt_1) \)
- \( y_2 = f(x_2 - vt_2) \)
Here, \( f \) is a function representing the shape of the wave, \( v \) is the wave speed, \( t_1 \) and \( t_2 \) are two different times, and \( x_1 \) and \( x_2 \) are the corresponding positions at those times.

**Hint**: Identify the variables involved in the wave equation and their meanings.

2. **Wave Propagation**:
Since the wave is propagating without change of shape, the function \( f \) remains the same at different times. This means that the argument of the function \( f \) must be constant for corresponding points on the wave.

**Hint**: Recognize that the shape of the wave does not change as it moves.

3. **Velocity of the Wave**:
The wave velocity \( v \) can be calculated using the coefficients from the wave equation. The general form of a wave traveling in the positive x-direction is:
\[ y = f(x - vt) \]
Here, the coefficient of \( t \) is \( v \) and the coefficient of \( x \) is \( 1 \). Thus, the wave velocity \( V \) is given by:
\[ V = \frac{\text{Coefficient of } t}{\text{Coefficient of } x} = \frac{v}{1} = v \]

**Hint**: Use the relationship between the coefficients of \( t \) and \( x \) to find the wave velocity.

4. **Particle Velocity**:
The particle velocity \( V_p \) at any point in the wave can be derived from the wave function. The particle velocity is given by:
\[ V_p = -V \frac{\partial y}{\partial x} \]
Since \( y = f(x - vt) \), we can differentiate:
\[ \frac{\partial y}{\partial x} = f'(x - vt) \]
Thus, the particle velocity becomes:
\[ V_p = -v f'(x - vt) \]

**Hint**: Differentiate the wave function with respect to \( x \) to find the particle velocity.

5. **Phase Velocity**:
The phase velocity of the wave is defined as the speed at which a particular phase of the wave (e.g., a crest) travels through space. For a wave of the form \( y = f(x - vt) \), the phase velocity is equal to the wave velocity:
\[ V_{\text{phase}} = v \]

**Hint**: Remember that the phase velocity is the same as the wave velocity for a wave traveling in a uniform medium.

### Summary of Correct Options
Based on the analysis:
- **Option 1**: The velocity of the wave is \( V \) (Correct).
- **Option 2**: The velocity of the wave is \( \frac{x_2 + x_1}{t_2 + t_1} \) (Incorrect).
- **Option 3**: The particle velocity is \( V_p = -V f'(x - vt) \) (Correct).
- **Option 4**: The phase velocity of the wave is \( V \) (Correct).

Thus, the correct options are 1, 3, and 4.