Velocity of Sound Calculation from Beats and Wavelengths

This problem involves calculating the velocity of sound using the concept of beats, which arise from the interference of two sound waves with slightly different frequencies.

Key Concepts

• The relationship between velocity ($v$), frequency ($f$), and wavelength ($\lambda$) of a wave is given by the formula: $ v = f \lambda $.
• The frequency can be expressed as: $ f = \frac{v}{\lambda} $.
• The beat frequency ($f_b$) is the absolute difference between the frequencies of two sound waves: $ f_b = |f_1 - f_2| $.

Problem Analysis

We are given two sound waves with wavelengths:

• $ \lambda_1 = 1 \text{ m} $
• $ \lambda_2 = 1.01 \text{ m} $

These waves produce 10 beats per second, so the beat frequency is:

• $ f_b = 10 \text{ Hz} $

We need to find the velocity of sound ($v$).

Step-by-Step Solution

1. Express Frequencies: Using the formula $ f = \frac{v}{\lambda} $, the frequencies of the two waves are:
   • $ f_1 = \frac{v}{\lambda_1} = \frac{v}{1} $
   • $ f_2 = \frac{v}{\lambda_2} = \frac{v}{1.01} $
2. Apply Beat Frequency Formula: Since $ \lambda_1 < \lambda_2 $, the frequency $ f_1 $ will be greater than $ f_2 $. Therefore, the beat frequency is: $ f_b = f_1 - f_2 $ $ 10 = \frac{v}{1} - \frac{v}{1.01} $
3. Solve for Velocity ($v$): Factor out $v$ from the equation: $ 10 = v \left( 1 - \frac{1}{1.01} \right) $ Calculate the term in the parenthesis: $ 1 - \frac{1}{1.01} = \frac{1.01 - 1}{1.01} = \frac{0.01}{1.01} $ Substitute back into the equation: $ 10 = v \left( \frac{0.01}{1.01} \right) $ Rearrange to solve for $v$: $ v = 10 \times \frac{1.01}{0.01} $ $ v = 10 \times 101 $ $ v = 1010 \text{ m/s} $

Conclusion

The velocity of sound in the gas is $1010 \text{ m/s}$.