The goal of this work is to analyze
the acoustic field propagating at long distances from an aircraft, flying above
ground with given impedance and topography. In particular, due to the
importance of the subject in environmental engineering, we seek the acoustic
field near the ground, and its effect on human hearing. In this case, solving
the acoustic problem using a standard Helmholtz solver is impractical, for
various reasons that will be explained. One important issue is the irregular
boundary caused by ground topography. A procedure often used in under-water
acoustics, which is borrowed here for use for the large-scale aero-acoustic
problem, is to approximate the Helmholtz equation by a parabolic equation
justified asymptotically, and solve the latter numerically. This is known as
the Parabolic Approximation (PA). The advantage of doing this is that owing to
the parabolic character of the differential equation, the numerical solution
can be advanced in the range direction by simple range-stepping, very similarly
to the way in which time-stepping is preformed in time-dependant problems. The accuracy of the narrow-angle and wide-angle PA is
investigated in the present context both theoretically and numerically. A
solution scheme based on the PA is devised using the Finite Element method in
the height direction and implicit range-stepping. The computational scheme is verified
using some simple test problems. Numerical results for the
Sound Pressure Level (SPL) envelopes in some cases of interests (various impedance
values, various topographies) are presented.
