The second overtone of an open pipe A and a closed pipe B have the same frequencies at a given temperature. Both pipes contain air. The ratio of fundamental frequency of A to the fundamental frequency

Question

The second overtone of an open pipe A and a closed pipe B have the same frequencies at a given temperature. Both pipes contain air. The ratio of fundamental frequency of A to the fundamental frequency of B is:

TestHub · Accepted Answer

The second overtone of an open pipe is given by:

$\frac{3 v}{2 ℓ _{1}}$

The second overtone of a closed pipe is given by:

$\frac{5 v}{4 ℓ _{2}}$

Since these frequencies are equal:

$\frac{3 v}{2 ℓ _{1}} = \frac{5 v}{4 ℓ _{2}}$

This implies:

$\frac{ℓ _{1}}{ℓ _{2}} = \frac{4 \times 3}{2 \times 5} = \frac{6}{5}$

Now, the ratio of fundamental frequencies is:

$\frac{f _{1}}{f _{2}} = \frac{\frac{v}{2 ℓ _{1}}}{\frac{v}{4 ℓ _{2}}} = \frac{2 ℓ _{2}}{ℓ _{1}}$

Substituting the ratio of lengths:

$= \frac{2 \times 5}{6} = \frac{10}{6} = \frac{5}{3}$

The answer is $5 : 3$ .

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