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 understand the relationship between wavelength (λ), frequency (f), and wave velocity (v). The fundamental wave equation that relates these three quantities is:

\[ v = f \cdot \lambda \]

Where:
- \( v \) is the wave velocity,
- \( f \) is the frequency,
- \( \lambda \) is the wavelength.

Now, let's analyze the situation step by step.

### Step 1: Understand the Initial Conditions
Assume the initial wavelength is \( \lambda_1 \) and the initial frequency is \( f_1 \). The initial wave velocity can be expressed as:

\[ v_1 = f_1 \cdot \lambda_1 \]

### Step 2: Doubling the Wavelength
According to the problem, the wavelength is doubled. Therefore, the new wavelength \( \lambda_2 \) can be expressed as:

\[ \lambda_2 = 2 \cdot \lambda_1 \]

### Step 3: Analyze the Wave Velocity
If the medium remains unchanged, the wave velocity \( v \) is determined by the properties of the medium. For a stretched string, the wave velocity is constant and depends on the tension and mass per unit length of the string. Thus, we can say:

\[ v_2 = v_1 \]

### Step 4: Relate Frequency and Wavelength
Using the wave equation for the new conditions, we can express the new frequency \( f_2 \):

\[ v_2 = f_2 \cdot \lambda_2 \]

Since we established that \( v_2 = v_1 \), we can substitute:

\[ v_1 = f_2 \cdot (2 \cdot \lambda_1) \]

### Step 5: Solve for the New Frequency
We can rearrange this equation to find the new frequency \( f_2 \):

\[ f_2 = \frac{v_1}{2 \cdot \lambda_1} \]

Since we know that \( v_1 = f_1 \cdot \lambda_1 \), we can substitute \( v_1 \):

\[ f_2 = \frac{f_1 \cdot \lambda_1}{2 \cdot \lambda_1} \]

This simplifies to:

\[ f_2 = \frac{f_1}{2} \]

### Conclusion
In conclusion, when the wavelength of the wave is doubled:
- The wave velocity remains constant (\( v_2 = v_1 \)).
- The frequency is halved (\( f_2 = \frac{f_1}{2} \)).