A wave of angular frequency `30` rad//sec propagates so that a certain phase of oscillationn moves along x-axis, y-axis , z-axis with speeds `1m//s`, `2m//s` and `2m//s` respectively. The propagation constant `K` is

Options

A. `30hati+15hatj+15hatk`
B. `10hati+10hatj+10hatk`
C. `30hati+30hatj+30hatk`
D. `6hati+6hatj+6hatk`

Explanation

To find the propagation constant \( K \) for the given wave, we can follow these steps:

### Step 1: Identify Given Values
- Angular frequency \( \omega = 30 \, \text{rad/s} \)
- Velocity along the x-axis \( v_x = 1 \, \text{m/s} \)
- Velocity along the y-axis \( v_y = 2 \, \text{m/s} \)
- Velocity along the z-axis \( v_z = 2 \, \text{m/s} \)

### Step 2: Calculate Propagation Constant for Each Axis
The propagation constant \( K \) can be calculated using the formula:
\[ K = \frac{\omega}{v} \]

#### For the x-axis:
\[ K_x = \frac{\omega}{v_x} = \frac{30 \, \text{rad/s}}{1 \, \text{m/s}} = 30 \, \text{rad/m} \]

#### For the y-axis:
\[ K_y = \frac{\omega}{v_y} = \frac{30 \, \text{rad/s}}{2 \, \text{m/s}} = 15 \, \text{rad/m} \]

#### For the z-axis:
\[ K_z = \frac{\omega}{v_z} = \frac{30 \, \text{rad/s}}{2 \, \text{m/s}} = 15 \, \text{rad/m} \]

### Step 3: Write the Propagation Constant Vector
The propagation constant vector \( \mathbf{K} \) can be represented as:
\[ \mathbf{K} = K_x \hat{i} + K_y \hat{j} + K_z \hat{k} \]

Substituting the values calculated:
\[ \mathbf{K} = 30 \hat{i} + 15 \hat{j} + 15 \hat{k} \]

### Final Answer
The propagation constant \( \mathbf{K} \) for the wave is:
\[ \mathbf{K} = 30 \hat{i} + 15 \hat{j} + 15 \hat{k} \, \text{(in rad/m)} \]

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