Question #43915

A rectangular organ pipe of length $L_0 = 73 \text{ cm}$, width $w = 40 \text{ cm}$, and height $h = 10\pi \text{ cm}$ is closed at one end and open at the other. The walls of the pipe have negligible thermal expansion. However, the top boundary of the open end is formed by a flat bimetallic strip of length $w = 40 \text{ cm}$ and thickness $t = 2 \text{ mm}$, clamped at its ends to the side walls. At temperature $T_0$, the strip is perfectly straight. When heated by $\Delta T = 100 \text{ K}$, the strip bows outward into a circular arc, increasing the cross-sectional area of the open end. The difference in the coefficients of linear expansion of the two metals is $\Delta \alpha = 4.25 \times 10^{-5} \text{ K}^{-1}$. Assume the speed of sound is $v = 340 \text{ m/s}$. Use the following approximations: (1) Radius of curvature $R = \frac{t}{\Delta \alpha \Delta T}$, (2) Sagitta $s = \frac{w^2}{8R}$, (3) Added area is a parabolic segment $\Delta A = \frac{2}{3}ws$, (4) Effective radius for end correction $r_{\text{eff}} = \sqrt{A/\pi}$ where $A$ is the total open area, (5) End correction $e = 0.6 r_{\text{eff}}$, and (6) $\Delta A \ll A_0$ allowing differential approximations. Find the magnitude of the shift in the fundamental resonant frequency.