Question

An open pipe of length l1 and a closed pipe of length l2 both resonate with the same tuning fork in their fundamental mode of vibration. Calculate the value of l1 /l2 ?

Answer 1

This is what I get

Explanation:

We know that an open pipe is open at both ends and closed pipe is closed at one end. Also the fundamental for an open end pipe needs to be an antinode at both ends, since the air can move at both ends. While the fundamental for a closed pipe has a node at the closed end and an antinode at the open end.

For `l_1` being the length of the open pipe , then for its fundamental we have

`l_1 = λ_1/2` ......(1)

If `n_"1open"` is the fundamental frequency and `v` is the velocity of sound in air, then

`n_"1open" = v/λ_1` .....(2)

Using (1) we get

`n_"1open" = v/(2l_1)` …... (3)

For `l_2` being the length of the closed pipe then for its fundamental we have the relationship

`l_2=lambda_1/4` .......(4)

Using expression similar to (2) and (4) we get

`n_"1close"= v/(4l_2)` .........(5)

It is given that both resonate with the same tuning fork in their fundamental mode of vibration.

`=> n_"1open"=n_"1close"`

Dividing (3) by (5) we get

`1=( v/(2l_1))/(v/(4l_2))`
`=>l_1/l_2=2`