The propagation of sonic boom through kinematic turbulence is known to have an important impact on the noise perceived at the ground. In this work, a recent numerical method called FLHOWARD3D based on a one-way approach is used to simulate the propagation of classical and low-boom waveforms. Kinematic turbulence is synthesized following a von Kármán energy spectrum. Two- and three-dimensional (2D and 3D) simulations are compared to experimental measurements, and 2D simulations are found to be slightly less accurate than 3D ones but still consistent with experimental levels around 98% of the time. A stochastic study is carried out on the 2D simulation using the generalized polynomial chaos method with parameters of the von Kármán spectrum as uncertain parameters. Differences between the propagation of a classical N-wave and low booms are observed: the classical N-wave shows higher peak pressure and variations than low-boom signatures. The standard deviation for the peak pressure, the D-weighted sound exposure level (D-SEL), and the perceived level in dB (PLdB) metrics all show a linear increase with the distance, with a faster increase for the classical N-wave for the peak pressure and D-SEL and a similar increase between the different booms for PLdB. In general, it is found that low-boom waveforms show less sensitivity to turbulence.