Question

Two organ pipes closed at one end have the same diameters but different lengths. Show that the end correction at each end is e = `(n_1l_1 - n_2l_2)/(n_2 - n_1)`, where the symbols have their usual meanings. Take `γ = 5/3`.

Answer 1

Suppose two organ pipes, closed at one end and of the same inner diameter d, have lengths l₁ and l₂.

Then, the effective lengths of the air columns are respectively

L₁ = l₁ + e = l₁ + 0.3 d and L₂ = l₂ + e = l₂ + 0.3 d

where e = 0.3 d is the end correction for the open end. The fundamental frequencies of the corresponding air columns are

`n_1 = v/(4L_1) = v/(4(l_1 + e)` and `n_2 = v/(4L_2) = v/(4(l_2 + e)`

where v is the speed of sound in air.

∴ v = 4n₁ (l₁ + e) = 4n₂ (l₂ + e)

∴ n₁l₁ + n₁e = n₂l₂ + n₂e

∴ n₁l₁ − n₂l₂ = (n₂ − n₁)e

∴ e = `(n_1l_1 - n_2l_2)/(n_2 - n_1)`

which is the required expression.