The relation between wavelength `(lamda)` and angular wave number (k) of a wave is

Options

A. `k=2pilamda`
B. `k=(lamda)/(2pi)`
C. `k=(1)/(lamda)`
D. `k=(2pi)/(lamda)`

Answer

To find the relation between wavelength (λ) and angular wave number (k) of a wave, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions:**
- The angular wave number (k) is defined as the number of radians per unit distance and is given by the formula:
\[ k = \frac{2\pi}{\lambda} \]
- The wavelength (λ) is the distance between successive crests (or troughs) of a wave.

2. **Relate Angular Frequency (ω) and Wave Speed (c):**
- The angular frequency (ω) is related to the wave speed (c) and the angular wave number (k) by the equation:
\[ c = \frac{\omega}{k} \]
- This equation states that the speed of the wave (c) is equal to the angular frequency (ω) divided by the angular wave number (k).

3. **Substitute Angular Frequency (ω):**
- The angular frequency (ω) can be expressed in terms of frequency (f):
\[ \omega = 2\pi f \]
- The frequency (f) can also be expressed in terms of wave speed (c) and wavelength (λ):
\[ f = \frac{c}{\lambda} \]
- Therefore, substituting the expression for frequency into the angular frequency gives:
\[ \omega = 2\pi \left(\frac{c}{\lambda}\right) \]

4. **Substitute ω in the Wave Speed Equation:**
- Now, substituting this expression for ω into the wave speed equation:
\[ c = \frac{2\pi \left(\frac{c}{\lambda}\right)}{k} \]
- This simplifies to:
\[ c = \frac{2\pi c}{\lambda k} \]

5. **Cancel c from Both Sides:**
- Assuming \( c \neq 0 \), we can cancel c from both sides:
\[ 1 = \frac{2\pi}{\lambda k} \]

6. **Rearranging the Equation:**
- Rearranging gives:
\[ k = \frac{2\pi}{\lambda} \]

7. **Final Relation:**
- Thus, the relation between the wavelength (λ) and the angular wave number (k) is:
\[ k = \frac{2\pi}{\lambda} \]

### Summary:
The relation between the wavelength (λ) and angular wave number (k) of a wave is given by:
\[ k = \frac{2\pi}{\lambda} \]