A harmonic wave is travelling along +ve x-axis, on a stretched string. If wavelength of the wave gets doubled, then

Options

A. frequency of wave may change
B. wave speed may change
C. both frequency and speed of wave may change
D. only frequency will change

Explanation

To solve the problem, we need to analyze how the wavelength of a harmonic wave affects its frequency and speed. Let's break it down step by step.

### Step 1: Understand the relationship between wavelength, frequency, and wave speed
The relationship between the wavelength (λ), frequency (f), and wave speed (v) is given by the equation:
\[ v = f \cdot \lambda \]
Where:
- \( v \) is the wave speed,
- \( f \) is the frequency,
- \( \lambda \) is the wavelength.

### Step 2: Analyze the effect of doubling the wavelength
If the wavelength is doubled, we can express this as:
\[ \lambda' = 2\lambda \]
Where \( \lambda' \) is the new wavelength.

### Step 3: Determine the implications on frequency and wave speed
Using the wave speed equation, we can express the new frequency \( f' \) when the wavelength is doubled:
\[ v = f' \cdot \lambda' \]
Substituting \( \lambda' \):
\[ v = f' \cdot (2\lambda) \]
This can be rearranged to find the new frequency:
\[ f' = \frac{v}{2\lambda} \]
Since \( v = f \cdot \lambda \), we can substitute \( v \) into the equation:
\[ f' = \frac{f \cdot \lambda}{2\lambda} = \frac{f}{2} \]
Thus, when the wavelength is doubled, the frequency is halved.

### Step 4: Consider the wave speed
The wave speed \( v \) depends on the medium through which the wave is traveling. If the medium remains unchanged, the wave speed will remain constant. Therefore, doubling the wavelength does not affect the wave speed if the medium is unchanged.

### Step 5: Summarize the findings
1. The wavelength has doubled (λ' = 2λ).
2. The frequency has halved (f' = f/2).
3. The wave speed remains unchanged (v is constant if the medium is unchanged).

### Conclusion
Based on the analysis:
- The frequency of the wave may change (it is halved).
- The wave speed may change (it remains constant if the medium is unchanged).
- Both frequency and speed of the wave may change (only frequency changes).
- Only frequency will change (this is correct).

### Final Answer
The correct options are:
- Frequency of the wave may change (correct).
- Wave speed may change (not correct if the medium is unchanged).
- Both frequency and speed of the wave may change (not correct).
- Only frequency will change (correct).