In travelling waves, the relation between particle velocity `V_(p)`, wave velocity V wave shape slope m is given by

Options

A. `V_(p)=-vxxm`
B. `V_(p)=Vxxm`
C. `V=-V_(p)xxm`
D. `V=V_(p)xxm`

Answer

To solve the problem regarding the relationship between particle velocity \( V_p \), wave velocity \( V \), and the slope of the wave shape \( m \), we can follow these steps:

### Step 1: Understand the Wave Equation
The general equation of a traveling wave can be expressed as:
\[ y(x, t) = a \sin(kx - \omega t) \]
where:
- \( a \) is the amplitude,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency,
- \( x \) is the position,
- \( t \) is the time.

### Step 2: Define Wave Velocity
The wave velocity \( V \) can be defined in terms of the angular frequency \( \omega \) and the wave number \( k \):
\[ V = \frac{\omega}{k} \]

### Step 3: Differentiate to Find Particle Velocity
To find the particle velocity \( V_p \), we differentiate the wave equation with respect to time \( t \):
\[ V_p = \frac{\partial y}{\partial t} = a \omega \cos(kx - \omega t) \]

### Step 4: Differentiate to Find Slope
Next, we differentiate the wave equation with respect to position \( x \) to find the slope:
\[ \frac{\partial y}{\partial x} = a k \cos(kx - \omega t) \]

### Step 5: Relate Particle Velocity, Wave Velocity, and Slope
Now we can relate \( V_p \), \( V \), and the slope \( m \):
1. From the definitions, we have:
- \( V_p = a \omega \cos(kx - \omega t) \)
- \( V = \frac{\omega}{k} \)
- \( m = \frac{\partial y}{\partial x} = a k \cos(kx - \omega t) \)

2. Rearranging the definitions gives us:
\[ V_p = -V \cdot m \]
where \( m = \frac{\partial y}{\partial x} \).

### Conclusion
Thus, the relationship between particle velocity, wave velocity, and the slope of the wave shape can be expressed as:
\[ V_p = -V \cdot m \]