The beat frequency (4 beat per second = 4 Hz ) corresponds to the difference between the frequencies of the two waves:
[tex]4 Hz = f_1 - f_2[/tex] (1)
The frequency of the first wave can be written as
[tex]f_1= \frac{v}{\lambda_1} [/tex]
where v is the speed of sound in the gas and [tex]\lambda_1 = 0.8 m[/tex] is the wavelength of the first wave, while the frequency of the second wave is
[tex]f_2 = \frac{v}{\lambda_2} [/tex]
where [tex]\lambda_2=0.81 m[/tex] is the wavelength of the second wave. If we substitute into the first equation (1), we find
[tex]4 Hz = \frac{v}{\lambda_1}- \frac{v}{\lambda_2} = v (\frac{1}{\lambda_2}- \frac{1}{\lambda_2} )[/tex]
and if we re-arrange it, we can find the velocity of the sound in the gas:
[tex]v= \frac{4 Hz}{ \frac{1}{\lambda_1}- \frac{1}{\lambda_2} }= \frac{4 Hz}{ \frac{1}{0.8 m} - \frac{1}{0.81 m} }=259.2 m/s [/tex]
