A closed organ pipe of length ' L_c ' and an open organ pipe of length ' L₀ ' contain different gases of densities ' ₁ ' and ' ₂ ' respectively. The compressibility of the gases is the same in both the pipes. The gases are vibrating in their first overtone with the same frequency. What is the length of open organ pipe?

Step-by-step solution

For open organ pipe: Fundamental frequency n = V 2 ~L _ o First overtone n ₁=2 n = V L _ 。 For closed organ pipe Fundamental frequency n^ = V^ 4 L_c First overtone n₁ ^ =3 n^ = 3 ~V ^ 4 ~L _ c aligned & n ₁= n ₁^ & V L _ o = 3 ~V ^ 4 ~L _ c & L _ o L _ c = 4 3 V V ^ & V = k ₂ , ~V ^ = k ₁ aligned where k is the adiabatic bulk modulus, which is reciprocal of compressibility. L₀ L_c = 4 3 ₁ ₂