The ratio of maximum particle velocity to wave velocity is [ where symbols have their usual meanings ]

Options

A. kA
B. `A omega `
C. `k omega `
D. `(omega ) /k`

Answer

To find the ratio of maximum particle velocity to wave velocity, we can follow these steps:

### Step 1: Write the wave equation
The general wave equation is given by:
\[ y(x, t) = A \sin(\omega t - kx) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

### Step 2: Differentiate the wave equation with respect to time
To find the particle velocity, we differentiate the displacement \( y \) with respect to time \( t \):
\[ v_p = \frac{\partial y}{\partial t} = A \frac{\partial}{\partial t} \sin(\omega t - kx) \]
Using the chain rule, this gives:
\[ v_p = A \omega \cos(\omega t - kx) \]

### Step 3: Find the maximum particle velocity
The maximum value of \( \cos(\omega t - kx) \) is 1. Therefore, the maximum particle velocity \( v_{p, \text{max}} \) is:
\[ v_{p, \text{max}} = A \omega \]

### Step 4: Determine the wave velocity
The wave velocity \( v \) can be expressed in terms of angular frequency \( \omega \) and wave number \( k \):
\[ v = \frac{\omega}{k} \]

### Step 5: Calculate the ratio of maximum particle velocity to wave velocity
Now, we can find the ratio of maximum particle velocity to wave velocity:
\[ \text{Ratio} = \frac{v_{p, \text{max}}}{v} = \frac{A \omega}{\frac{\omega}{k}} = A k \]

### Conclusion
Thus, the ratio of maximum particle velocity to wave velocity is:
\[ \text{Ratio} = A k \]