A closed organ pipe filled with a gas produces a fundamental resonance frequency of 200 Hz when the temperature of the gas is -17^ C . If the temperature of the gas is increased to 127^ C , what will be the frequency of the first overtone of the pipe? (Ignore the thermal expansion of the pipe)

Step-by-step solution

The fundamental frequency of a closed organ pipe is given by f = v 4L . The speed of sound in a gas is v = RT M , which implies v T . Therefore, the fundamental frequency f T . Given initial temperature T₁ = -17^ C = 256 K and final temperature T₂ = 127^ C = 400 K . Let f₁ be the fundamental frequency at T₁ and f₂ be the fundamental frequency at T₂ . f₂ f₁ = T₂ T₁ = 400 256 = 20 16 = 5 4 f₂ = 200 5 4 = 250 Hz For a closed organ pipe, the first overtone corresponds to the third harmonic, which has a frequency of 3f₂