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 solve the problem, we need to determine the wavelength of sound waves in the second medium when they enter from the first medium. Here’s a step-by-step solution:

### Step 1: Understand the relationship between speed, frequency, and wavelength
The speed of a wave (v) is related to its frequency (f) and wavelength (λ) 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: Identify the parameters in the first medium
In the first medium:
- Speed of sound waves, \( v_1 = v \)
- Wavelength, \( \lambda_1 = \lambda \)

Using the relationship:
\[ v = f \cdot \lambda \]
We can express the frequency as:
\[ f = \frac{v}{\lambda} \]

### Step 3: Identify the parameters in the second medium
In the second medium:
- Speed of sound waves, \( v_2 = 2v \)
- Wavelength, \( \lambda_2 \) (this is what we need to find)

Using the same relationship for the second medium:
\[ v_2 = f \cdot \lambda_2 \]
Substituting \( v_2 \):
\[ 2v = f \cdot \lambda_2 \]

### Step 4: Relate the frequencies in both media
Since the frequency of the wave does not change when it passes from one medium to another, we have:
\[ f = \frac{v}{\lambda} \]
Substituting this into the equation for the second medium:
\[ 2v = \left(\frac{v}{\lambda}\right) \cdot \lambda_2 \]

### Step 5: Solve for the new wavelength \( \lambda_2 \)
Now, we can rearrange the equation to solve for \( \lambda_2 \):
\[ 2v = \frac{v}{\lambda} \cdot \lambda_2 \]
Multiplying both sides by \( \lambda \):
\[ 2v \lambda = v \cdot \lambda_2 \]
Now, divide both sides by \( v \) (assuming \( v \neq 0 \)):
\[ 2\lambda = \lambda_2 \]

### Conclusion
Thus, the wavelength of sound waves in the second medium is:
\[ \lambda_2 = 2\lambda \]

### Final Answer
The wavelength of sound waves in the second medium is \( 2\lambda \).

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