A U-tube having unequal arm-lengths contains water. A tuning fork of frequency 440 Hz can set up the air in the shorter arm in its fundamental mode of vibration, while the same tuning fork can set up the air in the longer arm in its first overtone vibration. Determine the lengths of the air columns. Disregard any end effect and assume that the speed of sound in air is 330 m s ⁻¹ .

Step-by-step solution

The air columns in the U-tube act as closed organ pipes since they are closed by water at the bottom and open at the top. The wavelength of the sound wave is given by: = v f = 330 440 = 0.75 m = 75 cm For the shorter arm vibrating in its fundamental mode, the length of the air column L₁ is: L₁ = 4 = 75 4 = 18.75 cm 18.8 cm For the longer arm vibrating in its first overtone, the length of the air column L₂ is: L₂ = 3 4 = 3 75 4 = 56.25 cm 56.3 cm The lengths of the air columns are 18.8 cm and 56.3 cm .