Malbéqui P.

Several complementary methods have been developed for predicting outdoor propagation. The ray-model can be used for the non-linear sonic boom propagation; the Fast Field Program is efficient when the meteorological conditions are independent of the range; more recently, the Euler’s equations have been applied to rotational flows. The so-called Parabolic Equation (PE) is well suited to long range sound propagation in the atmosphere, taking into account temperature and wind velocity gradients, and including the kinematic and the thermal turbulences, the soil absorption and the terrain topography. The PE based on the approximation of the wave equation computes quickly the forward-field with a step by step marching method. In particular, the wide-angle parabolic equation provides accurate results, including a 45 degrees propagation aperture angle. However, in the framework of aircraft certification, a wider angle of propagation can be required to predict sound pressure levels almost straight to the acoustical source. This paper shows that the Higher Order Parabolic Equation (HOPE) allows us to compute outdoor propagation at very wide angles, close to 90 degrees. The derivation of the HOPE based on the approximation of a square operator using a Padé expansion with a high order is presented. Its numerical implementation using a finite difference technique and the alternative directions technique is then investigated. Comparisons between the wave equation, the wide-angle PE and the HOPE show that the HOPE improves the results both at wide angles of propagation and at very long range. In particular, as expected, increasing the order of the Padé expansion of the HOPE opens up the aperture angle of propagation.