Question #43928

An open cylindrical organ pipe of initial length $L$ and initial radius $r$ is made of a material with a linear coefficient of thermal expansion $\alpha_1$. Both open ends are fitted with bimetallic irises that control the effective radius of the openings, such that the radius expands with an equivalent linear coefficient of thermal expansion $\alpha_2$. The pipe is sounded at its fundamental frequency. If the temperature of the pipe and the irises is increased by $\Delta T$ (while the temperature of the air inside is independently maintained constant, ensuring the speed of sound $v$ remains constant), what is the fractional change in the fundamental frequency, $\frac{\Delta f}{f}$? (Assume $\alpha_1 \Delta T \ll 1$, $\alpha_2 \Delta T \ll 1$, and the end correction at each open end is $e = 0.6r$).