ˆ
peiωtdω . (2.16)
A quantity gained by the inverse Fourier transform is complex-valued. Without loss of generality, the real part of this quantities can be considered as the physical quantity. The 3-D wave equation (2.11) without mean flow is transformed to
∂2pˆ
∂xi∂xi
+k2pˆ= 0 (2.17)
and is named Helmholtz equation. The parameter k is called wavenumber and is defined as k ≡ ω/c0. In a 1-D setup, the propagation of a f-wave in a duct with length l can be written in terms of the wavenumber as: fout = exp(−ilk)fin, where fin/out denote the f-wave entering and leaving the duct, respectively.
2.3 Impedance and Reflection Coefficient
Solving the Helmholtz equation (2.17) on a given domain requires setting boundary condi-tions (BCs) on the entire boundary. For an ideal opening to the environment, no pressure fluctuations are assumed, i. e., a Dirichlet BC ˆp= 0 is set on such a boundary. At a hard wall, no wall-normal velocity fluctuations can take place, which means that the gradi-ent of the pressure vanishes in wall-normal direction ni. This means the Neumann BC (∂p/∂xˆ i)ni = 0 is set. In addition to these ideal cases, situations occur where the pressure and velocity fluctuations are coupled in a certain manner. This is modeled as a Robin BC with the so-called (surface) impedance Z. This quantity is defined in the frequency domain as the ratio of the pressure fluctuations ˆpto the wall-normal velocity fluctuation ˆ
uini:
Z(ω) = p(ω)ˆ ˆ ui(ω)ni
. (2.18)
Hence, the impedance describes the frequency response function for the pressure when the boundary is excited by a velocity fluctuation. Impedance values should be regarded as an “effective, averaged boundary condition the incident sound experiences rather than as a detailed quantity associated with a point measurement” [10]. The real and imag-inary part of the complex valued impedance are denoted as the resistance and the re-actance, respectively. It can be shown that an acoustically passive boundary exhibits a non-negative resistance Re(Z)≥0 and that the boundary absorbs sound for Re(Z)>0, see Rienstra [11]. The two special cases of an ideal opening and a hard wall correspond to impedances Zopen = 0 and |Zwall| = ∞, receptively. The impedance perceived by an acoustic wave traveling within a fluid is called specific impedance and is given byρ0c0. To allow for a more intuitive interpretation, the impedance is often normalized with respect to that specific value. The normalized impedance z =Z/(ρ0c0) is denoted by lower case z in the following. By a 1-D acoustic analysis [12], the impedance of a loss-free quarter-wave
2.3 Impedance and Reflection Coefficient tube with effective lengthlecan be calculated. An f-wave is entering the tube, propagates the distance le along the tube, is reflected at the hard wall end (with reflection coefficient R = 1, see below Eq. (2.21)), and finally travels as g-wave along the tube in opposite direction. This yields the normalized impedance
z(ω) = 1 ρ0c0
ˆ p ˆ
u = f +g
f −g = 1 + exp(−i 2lek)
1−exp(−i 2lek) =−i cot ω le
c0
, (2.19)
where k ≡ω/c0 denotes the wavenumber, see Eq. (2.17).
The relation between the acoustic velocity and the pressure on a boundary Z(ω)ˆu(ω) = ˆ
p(ω) can also be described in the time domain. The multiplication in the frequency domain results in a convolution (marked by the symbol ∗) of u(t) with the so-called impulse response of the impedance Zt(t):
p(t) = F−1{Z} ∗u (t) =
Z ∞
−∞
Zt(τ)u(t−τ) dτ . (2.20) The impulse response Zt is given by the inverse Fourier transformed impedance Zt(t) = F−1{Z}(t) = 1/(2π)R∞
−∞Z(ω) exp(iωt) dω. A physical system formulated in a ‘cause and effect manner’ can be assumed to be causal, which means that its response cannot depend on future inputs. In formula, this means that Zt(t) = 0 for all t <0.
Another way to acoustically characterize a boundary is the ratio of the reflected acoustics (g) to the normally incident (f), see Fig. 2.1 (θ = 0). This is done by the reflection coefficient, which is defined as
R(ω) = g(ω)ˆ
fˆ(ω) . (2.21)
Its connection to the impedance is given by R= z−1
z+ 1 = Z−ρ0c0 Z+ρ0c0
, or z = Z ρ0c0
= 1 +R
1−R . (2.22)
In the formula above, it can be observed that there are no reflections for a normalized impedance of unity, z = 1. A system with a normalized resistance below and above unity is referred to as normally damped andover-damped, respectively. When the plane wave is incident upon boundary with an angle θ (see Fig. 2.1a), the reflection is given by
Rθ = zcos(θ)−1
zcos(θ) + 1 . (2.23)
The maximal absorption is achieved for z = 1/cos(θ) in that case.
Frequency response functions such asRandZ describe the system behavior in the Fourier domain and, thus, are valid if the system dynamics are neither decaying nor growing. In many aeroacoustic systems, this assumption applies. For instance, the sound propagation in the inlet duct of an aero-engine can be studied under this assumption, since the sound pressure does not vary on short time scales. However, this assumption is violated when the (linear) thermoacoustic stabilization of a system is studied. Here, transfer functions
in the Laplace domain have to be considered. Exemplarily shown for the pressure, the Laplace transformation L is defined as:
P(s) = L{p}(s) = Z ∞
0
p(t)e−stdt . (2.24)
Functions in the Laplace domain are denoted by an capital letter, e. g., P(s) for the pressure. The complex-valued Laplace variable s = σs + iω describes both the angular frequencyωand the growth rateσsof a signal. All frequency response functions defined in the Fourier domain can also be defined analogously in the Laplace domain. For instance, the impedance reads here as:
Zs(s) = P(s)
U(s) . (2.25)
The frequency response functions in the Fourier domain define the behavior only on the imaginary axis of the complex-valued Laplace domain. The Laplace domain representation contains additional information for the behavior of the system responding to an decaying or increasing input signal u′(t) = ˆuexp[(σs+ iω)t].
The transfer behavior of any linear, time-invariant (LTI) system can fully be modeled by a transfer function in the Laplace domain. In this context, linear means that the input and the output signal (which are functions in time) can be related to each other by a linear function. Time-invariant implies that the characteristics of the system do not change in time. If the frequency response is known in the Fourier domain, say Z(ω), it generally cannot be extruded into the Laplace domain as Zs(s) = Zs(σs+ iω) 6= Z(ω) for σs 6= 0, see Schmid et al. [13]. A frequency response described by a holomorphic2 or meromorphic3 function can be extended to a transfer function in the Laplace domain by analytic continuation, which is by construction valid for the underlying LTI system.
In experiments, the impedance cannot be obtained directly since the velocity in the neck is difficult to measure. When there is no mean flow present, the two- or multi-microphone method offers a good possibility to determine the impedance, see for instance Temiz et al. [14]. Here, a sample, such as a resonator, is located at the end of a tube, the so-called impedance tube, which is equipped with an array of microphones. A loudspeaker placed at the other end excites this configuration harmonically. From the pressure data of the microphones, the characteristicf- andg-waves (see Eq. (2.13)) can be reconstructed.
Their ratio gives the reflection coefficient R, cf. Eq. (2.21), from which the impedance can be deduced, see Eq. (2.22). However, for acoustic resonators, the transformation from the reflection coefficient Rto the resistance Re(z) is ill-conditioned for frequencies clearly distinct from the eigenfrequency, see F¨orner and Polifke [15, Sec. 3.3]. This means that even a small deviation in the reflection coefficient can lead to a huge deviation in the resistance.
In presence ofgrazing flow—a mean flow normal to the resonator opening, this impedance tube approach is difficult to realize. The in-situ method developed by Dean [16] can be applied instead under such conditions. Here, at least one microphone is mounted on the
2complex differentiable
3holomorphic except for a set of isolated points
2.4 Impedance and Reflection Coefficient Describing Functions