Question #42877

A uniform plane wave of initial intensity $I_0 = 2500 \text{ W m}^{-2}$ propagates in the $+x$ direction through a viscous fluid. The power dissipated per unit volume by the wave, $p(x) = -\frac{dI}{dx}$, locally heats the fluid. In steady state, the local temperature rise above the ambient is given by $\Delta T(x) = \frac{p(x)}{h}$, where $h = 50 \text{ W m}^{-3} \text{ K}^{-1}$ is the volumetric cooling coefficient. The attenuation coefficient of the fluid increases with temperature as $\alpha(x) = \alpha_0 + \gamma \Delta T(x)$, where $\alpha_0 = 0.1 \text{ m}^{-1}$ and $\gamma = 0.01 \text{ m}^{-1} \text{ K}^{-1}$. What is the ratio of the power dissipated per unit volume at $x=0$ to the power dissipated per unit volume at a distance $x=L$ where the wave intensity has decreased to $I_L = 1000 \text{ W m}^{-2}$?