The lengths of the two pipes open at both ends are L and $(L + L_{1})$ . If they are sounded together, the beat frequency will be
( $v$ = velocity of sound in air)

Question

The lengths of the two pipes open at both ends are L and ( L + L 1 ​ ) . If they are sounded together, the beat frequency will be ( v = velocity of sound in air)

TestHub · Accepted Answer

The fundamental frequency of a pipe open at both ends of length $L$ is given by $f_{1} = \frac{v}{2 L}$ .

The fundamental frequency of a pipe open at both ends of length $L + L_{1}$ is given by $f_{2} = \frac{v}{2 ( L + L _{1} )}$ .

The beat frequency is the difference between the two frequencies:

$f_{B} = f_{1} - f_{2}$

$f_{B} = \frac{v}{2 L} - \frac{v}{2 ( L + L _{1} )}$

$f_{B} = \frac{v}{2} (\frac{1}{L} - \frac{1}{L + L _{1}})$

$f_{B} = \frac{v}{2} (\frac{L + L _{1} - L}{L ( L + L _{1} )})$

$f_{B} = \frac{v L _{1}}{2 L ( L + L _{1} )}$

Answer: $\frac{v L _{1}}{2 L ( L + L _{1} )}$

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