Two sound waves of wavelengths 2m and 2.02 m produce 20 beats in a gas in 5 seconds. What will be the speed of sound in this gaseous medium ?

Answer

To solve the problem, we need to find the speed of sound in a gaseous medium given two sound waves with different wavelengths and the number of beats produced. Here’s a step-by-step solution:

### Step 1: Understand the Concept of Beats
When two sound waves of slightly different frequencies interfere, they produce a phenomenon known as beats. The beat frequency (f_beat) is given by the absolute difference between the two frequencies (f1 and f2):
\[ f_{beat} = |f_1 - f_2| \]

### Step 2: Calculate the Beat Frequency
We are given that 20 beats occur in 5 seconds. Therefore, the beat frequency can be calculated as:
\[ f_{beat} = \frac{\text{Number of beats}}{\text{Time}} = \frac{20}{5} = 4 \text{ Hz} \]

### Step 3: Relate Frequency to Wavelength
The frequency of a wave is related to its speed (v) and wavelength (λ) by the formula:
\[ f = \frac{v}{\lambda} \]
For the two sound waves with wavelengths λ1 = 2 m and λ2 = 2.02 m, we can express their frequencies as:
\[ f_1 = \frac{v}{\lambda_1} = \frac{v}{2} \]
\[ f_2 = \frac{v}{\lambda_2} = \frac{v}{2.02} \]

### Step 4: Set Up the Equation for Beat Frequency
Using the definition of beat frequency:
\[ f_{beat} = |f_1 - f_2| = \left| \frac{v}{2} - \frac{v}{2.02} \right| \]
This can be simplified to:
\[ f_{beat} = \left| \frac{v(2.02 - 2)}{2 \times 2.02} \right| \]
\[ f_{beat} = \frac{v(0.02)}{2 \times 2.02} \]

### Step 5: Substitute the Beat Frequency
We know that \( f_{beat} = 4 \text{ Hz} \):
\[ 4 = \frac{v(0.02)}{4.04} \]

### Step 6: Solve for the Speed of Sound (v)
Rearranging the equation to solve for v:
\[ v = 4 \times 4.04 / 0.02 \]
\[ v = \frac{16.16}{0.02} \]
\[ v = 808 \text{ m/s} \]

### Final Answer
The speed of sound in the gaseous medium is **808 m/s**.

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