The problem of translating a frequency domain impedance boundary condition to time domain involves the Fourier transform of the impedance function. This requires extending the definition of the impedance not only to all real frequencies but to the whole complex frequency plane. Not any extension, however, is physically possible. The problem should remain causal, the variables real, and the wall passive. This leads to necessary conditions for an impedance function.Various methods of extending the impedance that are available in the literature are discussed. A most promising one is the so-called z-transform by Ozyoruk & Long, which is nothing but an impedance that is functionally dependent on a suitable complex exponent e −iωκ . By choosing κ a multiple of the time step of the numerical algorithm, this approach fits very well with the underlying numerics, because the impedance becomes in time domain a delta-comb function and gives thus an exact relation on the grid points.An impedance function is proposed which is based on the Helmholtz resonator model, called Extended Helmholtz Resonator Model. This has the advantage that relatively easily the mentioned necessary conditions can be satisfied in advance. At a given frequency, the impedance is made exactly equal to a given design value. Rules of thumb are derived to produce an impedance which varies only moderately in frequency near design conditions. An explicit solution is given of a pulse reflecting in time domain at a Helmholtz resonator impedance wall that provides some insight into the reflection problem in time domain and at the same time may act as an analytical test case for numerical implementations, like is presented at this conference by the companion paper AIAA-2006-2569 by N. Chevaugeon, J.-F. Remacle and X. Gallez.The problem of the instability, inherent with the Ingard-Myers limit with mean flow, is discussed. It is argued that this instability is not consistent with the assumptions of the Ingard-Myers limit and may well be suppressed. wherev = (v · n),v is the acoustic velocity vector and n denotes the normal vector of the wall that points into the wall. Pressure is scaled on ρ 0 c 2 0 , velocity on c 0 and impedance on ρ 0 c 0 , while we use throughout the e +iωt * Associate professor,

For the relatively high frequencies relevant in a turbofan engine duct, the modes of a lined section may be classified in two categories: genuine acoustic 3D duct modes resulting from the finiteness of the duct geometry, and 2D surface waves that exist only near the wall surface in a way essentially independent of the rest of the duct. Per frequency and circumferential order there are at most four surface waves. They occur in two kinds: two acoustic surface waves that exist with and without mean flow, and two hydrodynamic surface waves that exist only with mean flow. The number and location of the surface waves depends on the wall impedance Z and mean flow Mach number. When Z is varied, an acoustic mode may change via small transition zones into a surface waves and vice versa.Compared to the acoustic modes, the surface waves behave-for example as a function of the wall impedance-rather differently as they have their own dynamics. They are therefore more difficult to find. A method is described to trace all modes by continuation in Z from the hard-wall values, by starting in an area of the complex Z-plane without surface waves.

An explicit, analytical, multiple-scales solution for modal sound transmission through slowly varying ducts with mean flow and acoustic lining is tested against a numerical finite-element solution solving the same potential flow equations. The test geometry taken is representative of a high-bypass turbofan aircraft engine, with typical Mach numbers of 0.5-0.7, circumferential mode numbers m of 10-40, dimensionless wavenumbers of 10-50, and both hard and acoustically treated inlet walls of impedance Z = 2 − i. Of special interest is the presence of the spinner, which incorporates a geometrical complexity which could previously only be handled by fully numerical solutions. The results for predicted power attenuation loss show in general a very good agreement. The results for iso-pressure contour plots compare quite well in the cases where scattering into many higher radial modes can occur easily (high frequency, low angular mode), and again a very good agreement in the other cases.

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