Question #43053

A horizontal wind tunnel of length $L$ is designed such that the flow velocity of an ideal gas along its central axis $x$ (from $x=0$ to $x=L$) is given by $v(x) = v_0 \sqrt{1 + \frac{x}{L}}$. The gas has molar mass $M$ and is maintained at a constant temperature $T$. At $x=0$, the gas density is $\rho_0$. Assume the flow is steady and the density variations are small enough that the fluid can be treated as approximately incompressible when applying Bernoulli's equation. The refractive index of the gas depends on its local density $\rho$ according to the Gladstone-Dale relation $n(x) = 1 + K\rho(x)$, where $K$ is a positive constant. A laser beam of vacuum wavelength $\lambda$ travels along the central axis from $x=0$ to $x=L$. What is the magnitude of the phase shift $\Delta \phi$ of this laser beam relative to a reference beam traveling the same distance $L$ through a uniform stationary gas of constant density $\rho_0$? (Here $R$ is the universal gas constant).