The velocity of sound in a gas is 4 times that in air at the same temperature. When a tuning, fork is sounded in air a wave of frequency 480 and wavelength `lamda _1` is produced. The same fork is sounded in the gas and if `lamda_2` is the wavelength of the wave, then `lamda_2/lamda_1`

Options

A. 1
B. 2
C. 4
D. `1/2`

Explanation

To solve the problem, we need to find the ratio of the wavelengths of sound in gas and air when a tuning fork is sounded in both media. Here’s a step-by-step solution:

### Step 1: Understand the relationship between velocity, frequency, and wavelength
The fundamental relationship between the speed of sound (v), frequency (f), and wavelength (λ) is given by the formula:
\[ v = f \times \lambda \]

### Step 2: Define the variables for air
Let:
- \( v_a \) = velocity of sound in air
- \( \lambda_1 \) = wavelength of sound in air
- \( f \) = frequency of the tuning fork = 480 Hz

Using the formula for air:
\[ v_a = f \times \lambda_1 \]
Substituting the frequency:
\[ v_a = 480 \times \lambda_1 \]
(Equation 1)

### Step 3: Define the variables for gas
According to the problem, the velocity of sound in gas \( v_g \) is 4 times that in air:
\[ v_g = 4 v_a \]

### Step 4: Write the equation for the gas
Using the same relationship for the gas:
\[ v_g = f \times \lambda_2 \]
Since the frequency remains the same when the tuning fork is sounded in gas:
\[ v_g = 480 \times \lambda_2 \]
(Equation 2)

### Step 5: Substitute \( v_g \) in terms of \( v_a \) into Equation 2
From the relationship established in Step 3:
\[ 4 v_a = 480 \times \lambda_2 \]

### Step 6: Substitute \( v_a \) from Equation 1 into the equation
Substituting \( v_a \) from Equation 1 into the equation:
\[ 4 (480 \times \lambda_1) = 480 \times \lambda_2 \]

### Step 7: Simplify the equation
Dividing both sides by 480:
\[ 4 \lambda_1 = \lambda_2 \]

### Step 8: Find the ratio \( \frac{\lambda_2}{\lambda_1} \)
To find the ratio of the wavelengths:
\[ \frac{\lambda_2}{\lambda_1} = 4 \]

### Conclusion
Thus, the ratio of the wavelengths \( \frac{\lambda_2}{\lambda_1} \) is equal to 4.

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