Three-dimensional infrasonic wave propagation above irregular boundaries

  • Codor Khodr

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)


This thesis deals with the propagation of infrasonic waves above irregular boundaries using a numerical approach based on the parabolic equation (PE) method. The first part of the thesis is dedicated to the development of a two-dimensional parabolic equation (2DPE) method, based on the well-known Beilis-Tappert coordinate shift map. While the original theory was limited to a narrow-angle formulation, it is here extended to wide-angle propagation by using higher-order Padé series to approximate the pseudo-differential square-root operator. The equation is numerically solved using the implicit Crank-Nicolson finite-difference scheme and validated against both analytical solutions (in the case of propagation above a flat impedance surface) and a COMSOL Finite-Element model (in the case of propagation above an irregular Gaussian surface). Simulations show a good agreement between the analytical models and the 2DPE. The wide-angle formulation allows for a more accurate representation of the pressure field at higher-altitude while maintaining the accuracy of the original first-order shift map.

A significant part of the thesis is dedicated to the derivation of a novel three-dimensional parabolic equation (3DPE) that takes into account both irregular boundaries and refraction from a layered atmosphere, with an effective sound speed profile. This development represents a first attempt at applying the PE method to three-dimensional surface scattering, in the context of atmospheric acoustics. Here, a coordinate transformation method is used on the Helmholtz equation to express the problem in a flattened domain, where simplified impedance boundary conditions can be easily enforced. Realistic propagation problems in infrasonic wave modeling typically involve large-scale simulations. As a result, the main focus of the proposed method is the derivation of an efficient numerical scheme, avoiding the inversion of a large sparse system. This has been achieved by a combination of a small-slope assumption and the use of an iterative gradient scheme, which involves the computation of tridiagonal matrices only. Another solution, based on the Alternate Direction Implicit (ADI) and the Split-Step Padé scheme, has also been proposed to extend the method to wide-angle propagation.

A parametric study has also been proposed to quantify the diffraction of infrasonic waves by a bivariate Gaussian surface of variable height. The frequency range considered spans from 0.5 Hz to 10 Hz, for a propagation range of 10 km and an obstacle size that varies between 0 and 300 m of height. The obstacle scale, which is simply the ratio of the Gaussian surface height to the wavelength (i.e h0/λ) is proven to be the governing parameter of the existence of diffractive effects in the shadow zone, downstream of the obstacle. A comparison with the 2DPE shows a difference in Sound Pressure Level (SPL) between 3 dB and 10 dB in the shadow zone. More simulations have been performed to investigate the coupled effects of irregular boundaries and atmospheric refraction, for linear, logarithmic and jet atmospheric sound speed profiles. It is shown that in the presence of terrain, transversal scattering effects become significant, even for smooth profiles and low frequencies. The parametric study has confirmed the existence of three-dimensional topographic effects in the higher end of the infrasound spectrum (f>1 Hz).

In order to check the novel 3DPE solver for a realistic propagation problem, a Wind-Turbine noise prediction is carried out in a three-dimensional environment. Numerical PE simulations are compared against the experimental pressure data, extracted from recordings made at the IS50 micro-barometer array, part of the International Monitoring System (IMS), located in the Ascension Island. Based on the results, it is highlighted that topography plays an important role in the correct prediction of the pressure field in a realistic environment, as three-dimensional modeling enabled us to correctly match experimental data at different receiver locations.
Date of Award12 May 2020
Original languageEnglish
Awarding Institution
  • University of Bristol
SupervisorMahdi Azarpeyvand (Supervisor) & Alison C Rust (Supervisor)


  • Infrasound, Physical Acoustics, Wave Scattering, Parabolic Equation, Outdoor Sound Propagation

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