Question #42866

A cylindrical string of length $L$, cross-sectional area $A$, and intrinsic linear mass density $\mu_0$ is stretched with a constant tension $T$ between two rigid supports inside a closed chamber. The chamber is filled with an ideal monoatomic gas ($\gamma = \frac{5}{3}$). Due to the fluid's added mass effect, the effective linear mass density of the string during transverse vibrations is given by $\mu_{\text{eff}} = \mu_0 + \rho_g A$, where $\rho_g$ is the density of the gas. Initially, the gas has an energy density (internal energy per unit volume) of $u_0$ and a mass density of $\rho_0$. In this initial state, it is given that $\mu_0 = \rho_0 A$, and the string's fundamental frequency is $f_0$. The gas is then rapidly and adiabatically compressed until its energy density becomes $32 u_0$. Assuming $L$, $A$, and $T$ remain unchanged, what is the new fundamental frequency $f_1$ of the string?