The present paper examines, theoretically and experimentally, the sound field in the vicinity of a non-locally medium due to an airborne source. The non-locally reacting medium is characterized by a porous layer of finite thickness which is placed on a perfectly reflecting plane. According to an asymptotic analysis, the total sound field within the rigid porous medium consists of two components. Each of these two components can be represented by an integral expression. They can then be evaluated by a standard saddle path method to obtain a uniform asymptotic solution. Numerical validation with wave-based numerical schemes demonstrates the accuracy and computational efficiency of the derived asymptotic formula. Additional validation is provided through indoor experimental data obtained by using a layer of glass beads for modeling the rigid porous medium. When the receiver is situated within the porous layer near the perfectly reflecting plane, experimental measurements and theoretical predictions suggest that the interaction of the refracted wave with the perfectly reflecting plane has a significant impact on the total sound field. Experimental data and numerical simulations also indicate that it is rather difficult to distinguish the results between a thin rigid porous layer and a semi-infinite rigid porous medium for an airborne receiver.
A continuous source motion model has been developed to represent a cruising aircraft traveling at high altitudes. The numerical implementation is based on the Lorentz transform (LT) and a Fast Field Program (FFP) formulation, which is used to compute the sound fields due to a monopole point source traversing in a horizontally stratified atmosphere parallel to a ground surface. To reduce the computational expenses, a one-dimensional LT-FFP is performed to obtain the pressure time-history for overhead flight conditions. The continuous source motion model is compared against a heuristic model which applies a Doppler shift to the stationary source sound field. A linear sound speed profile was selected to simplify the ray model implementation. A parametric study involving the source Mach number and source emission frequency has been performed in a variety of environmental conditions. The differences in the predicted maximum sound pressure levels between the two models can be as large as 9 dB under certain conditions. Numerical simulations indicate that low source emission frequencies (e.g., 50–300 Hz) combined with high source Mach numbers tend to result in larger discrepancies. The numerical simulations suggest the importance of including the effects of convective source amplification, especially, for turboprop aircraft noise propagation.
A continuous source motion model is developed for computing the sound fields of a uniformly moving monopole source in a horizontally stratified atmosphere. The numerical model is based on the Lorentz transformation and the one-dimensional fast field formulation. The global matrix method and a bounded Green’s function solution expressed in the wavenumber domain is used in the numerical implementation. The solution is then expressed in the Lorentz frame. A transformation is performed to map the spatial Lorentz frame quantities into a reception time history in the stationary frame. Numerical results for an elevated source in a downward refracting atmosphere demonstrates the importance of a continuous source motion model especially at high Mach numbers in the presence of sound speed gradients where the pseudo-stationary source approximation becomes questionable. Coupled with the convective effects of continuous source motion and a frequency dependent ground impedance model, the resulting sound fields are significantly different from those predicted due to a pseudo-stationary source. Fluctuations on the order of 20 dB are predicted in the interference pattern in the far-field. A shift of the interference onset time to lower values is predicted when the source Mach number and/or sound speed gradient is increased.
This study examines the sound field within a hard-backed rigid porous medium due to an airborne source. The total sound field can be approximated by two terms: A transmitted wave component arriving at the receiver directly through the porous interface, and a second transmitted wave component reflected from the rigid backing plane before reaching the receiver. These two components can be expressed in an integral form that is amenable to further analyses. A standard saddle path method is applied to evaluate the integral analytically, leading to a uniform asymptotic solution that allows the prediction of the sound field within the rigid porous medium. The validity of the asymptotic formula is verified by comparison with the numerical results computed by a more accurate wave-based numerical scheme. The asymptotic solution is shown to provide a convenient means of rapid and accurate computations of sound field within the rigid porous medium. The accuracy of the numerical solutions is further confirmed by comparison with indoor experimental results. The measurement data and theoretical predictions suggest that when the receiver is located near the bottom of the hard-backed layer, the reflection of the refracted wave gives rise to a significant contribution to the total sound field.
The Lorentz transform, which can be applied to a uniformly moving source, effectively converts the time-domain wave equation to an equivalent problem in frequency-domain due to a stationary source. Standard fast field program (FFP) implementations apply the property of conjugate symmetry in the wavenumber domain for improved efficiency. Recent literature [D. Dragna et al., AIAA 52, 1928–1939 (2014)] suggests the use of a Dopplerized frequency-dependent impedance model to account for the effects of source motion. Consequently, the FFP kernel function is no longer identical for the positive and negative wavenumbers. Additional complications are introduced by the necessity to compute the positive and negative horizontal separation distances in the Lorentz frame to obtain a complete time history of sound pressures for the source approaching and receding from the receiver. Further development of the FFP algorithm in the Lorentz frame is explored such that a frequency-dependent impedance model is developed. Both moving line and point monopole sources are considered in the current investigation. Results are validated against direct numerical integration schemes and with the published time-domain solutions. Applications for an aircraft operating in cruise condition is also examined in the present study. [Work Sponsored by the Federal Aviation Administration.]
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