Optical distortions caused by turbulent airflow surrounding an aircraft, known as aero-optical phenomena, are a major impediment to applications of airborne laser systems. To better understand the spectral properties of aero-optical distortions, a general expression for the wavenumber spectrum of the refractive index is derived from the ideal-gas law and Gladstone-Dale relation. The derived index-of-refraction spectrum accounts for changes in air density due to both temperature and pressure fluctuations and is used to calculate the phase-distortion spectrum of an optical beam propagating through a weakly compressible, turbulent flow field. Numerical simulations of weakly compressible, temporally evolving shear layers are used to verify theoretical results and confirm that if the log slope of the one-dimensional density spectrum in the inertial subrange is -mρ, the optical phase distortion spectral slope is given by -(mρ + 1). The value of mρ is shown to be dependent on the ratio of shear-layer free-stream densities and bounded by the spectral slopes of temperature and pressure fluctuations.