If the frequency of a wave is doubled, its wavelength

Options

A. becomes doubled
B. remains same
C. becomes half of the original
D. none of these

Answer

To solve the problem of how the wavelength of a wave changes when its frequency is doubled, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Speed, Frequency, and Wavelength**:
The speed of a wave (v) is given by the equation:
\[ v = f \cdot \lambda \]
where \( f \) is the frequency and \( \lambda \) is the wavelength.

2. **Assume Constant Speed**:
For this problem, we will assume that the speed of the wave remains constant. This means that if we change the frequency, the wavelength must change accordingly to maintain the equation.

3. **Initial Conditions**:
Let the initial frequency be \( f \) and the initial wavelength be \( \lambda \). Therefore, we can express the speed of the wave as:
\[ v = f \cdot \lambda \]

4. **Doubling the Frequency**:
If the frequency is doubled, the new frequency becomes:
\[ f' = 2f \]
We need to find the new wavelength \( \lambda' \) when the frequency is doubled.

5. **Using the Wave Speed Equation**:
Since the speed of the wave remains constant, we can write:
\[ v = f' \cdot \lambda' \]
Substituting \( f' = 2f \) into the equation gives:
\[ v = 2f \cdot \lambda' \]

6. **Setting the Two Speed Equations Equal**:
Since both expressions equal the wave speed \( v \), we can set them equal to each other:
\[ f \cdot \lambda = 2f \cdot \lambda' \]

7. **Canceling \( f \)**:
Assuming \( f \) is not zero, we can divide both sides by \( f \):
\[ \lambda = 2 \cdot \lambda' \]

8. **Solving for the New Wavelength**:
Rearranging the equation gives:
\[ \lambda' = \frac{\lambda}{2} \]
This means that the new wavelength \( \lambda' \) is half of the original wavelength \( \lambda \).

### Conclusion:
If the frequency of a wave is doubled, its wavelength is halved. Thus, the answer is:
\[ \lambda' = \frac{\lambda}{2} \]