Sound waves of wavelength `lambda` travelling in a medium with a speed of `upsilon m//s` enter into another medium where its speed is `2upsilon m//s.` Wavelength of sound waves iin the second medium is

Options

A. `lambda`
B. `(lambda)/(2)`
C. `2lambda`
D. `4lambda`

Explanation

To find the wavelength of sound waves in the second medium, we can follow these steps:

### Step 1: Understand the relationship between speed, frequency, and wavelength
The relationship between the speed of a wave (v), its frequency (f), and its wavelength (λ) is given by the equation:
\[ v = f \cdot \lambda \]
This means that the speed of the wave is equal to the product of its frequency and wavelength.

### Step 2: Define the variables for the first medium
Let:
- \( \lambda_0 \) = wavelength in the first medium
- \( v_0 \) = speed in the first medium = \( \upsilon \)
- \( f \) = frequency in the first medium

From the relationship, we can express the frequency in the first medium as:
\[ f = \frac{v_0}{\lambda_0} = \frac{\upsilon}{\lambda} \]

### Step 3: Define the variables for the second medium
In the second medium, we have:
- \( v_1 \) = speed in the second medium = \( 2\upsilon \)
- \( \lambda_1 \) = wavelength in the second medium

The frequency in the second medium remains the same as in the first medium, so:
\[ f = \frac{v_1}{\lambda_1} \]

### Step 4: Set the frequencies equal
Since the frequency remains constant when the wave enters the second medium, we can set the two expressions for frequency equal to each other:
\[ \frac{\upsilon}{\lambda} = \frac{2\upsilon}{\lambda_1} \]

### Step 5: Solve for the wavelength in the second medium
Now, we can rearrange the equation to solve for \( \lambda_1 \):
\[ \lambda_1 = \frac{2\upsilon \cdot \lambda}{\upsilon} \]
\[ \lambda_1 = 2\lambda \]

### Conclusion
The wavelength of sound waves in the second medium is:
\[ \lambda_1 = 2\lambda \]