An impedance-based formulation of frequency-domain nonlinearity indicators in finite-amplitude sound propagation Open Access

A nonlinearity indicator is a measure by which the strength of nonlinearity in a wave process is quantified. Examples of nonlinearity indicators include the Gol'dberg number,^1–4 the average steepening factor,^5,6 the derivative skewness,⁷ the bispectrum,⁸ and the Morfey-Howell Q/S.⁹ Nonlinearity indicators have been widely used to identify cumulative finite-amplitude propagation effects in jet^10–13 and rocket^14–16 noise analysis. However, the use of these indicators has often been qualitative, which has sometimes resulted in debate regarding their validity, utility, and interpretation.^11,17

To improve physical understanding and interpretation, Reichman et al.¹⁸ reformulated the Q/S spectrum by expressing the frequency-domain Burgers equation in terms of the change in sound level with respect to range. In this context, the Q/S spectrum at a given location in the sound field is interpreted as the local spatial rate of change in level of a spectral component due to nonlinearity. In addition to absorptive and geometric spreading losses, a spectral component can experience extra loss (or gain) in level through the nonlinear energy exchange with other spectral components, the extent of which is quantified by Q/S.

In this Letter, we provide an alternative, but equally intuitive interpretation of Q/S based on the familiar concept of specific acoustic impedance. Both the prior level and the current impedance-based formulations of this indicator demonstrate that its interpretation is rather straightforward, and thus they would serve to promote the wider use of Q/S in nonlinear acoustics pedagogy and systems analysis.

The formulation here follows from recognizing that (a) for a progressive sound wave, impedance encapsulates how the waveform is modified locally via changes in amplitude (absorption) and phase (dispersion) of its Fourier components, and hence (b) nonlinearity, being one of waveform distortion mechanisms, could be cast within the framework of impedance, specifically in the form of effective absorption and dispersion.

We start with the generalized Burgers equation for plane progressive waves,¹⁹ 

where p is the acoustic pressure, x is the propagation distance, τ is the retarded time, $ρ0$ is the density, $c0$ is the small-signal sound speed, β is the coefficient of nonlinearity, and $Lτ(p)$ is a linear, retarded time operator that describes the absorption and dispersion properties of the medium. The corresponding spectral version of Eq. (1) is

where $p̃$ and $q̃$ are the Fourier transforms of the acoustic pressure and the squared acoustic pressure, respectively, ω is the angular frequency, and $k=ω/c0$ is the wave number. Here, the Fourier equivalent of the absorption/dispersion operator can be written as

where α and δ are frequency-dependent absorption and dispersion coefficients, respectively. The generalized spectral Burgers equation then becomes

Within the second-order approximation theory,^20,21 absorption and dispersion are assumed weak on the scale of wavelength such that $α/k∼O(ε̃)$ and $δ/k∼O(ε̃)$⁠, where $ε̃$ is a small, generic ordering parameter. Given $p̃∼O(ε̃)$⁠, the absorption/dispersion term $−(α+jδ)p̃$ is $O(ε̃2)$⁠. Equation (4) is thus a consistent $O(ε̃2)$ wave equation, in which all the terms, including $∂p̃/∂x$⁠, are $O(ε̃2)$ small. This $O(ε̃2)$ consistency ought to be maintained while manipulating Eq. (4).

Recasting the generalized spectral Burgers equation [Eq. (4)] in terms of specific acoustic impedance

requires the spectral version of the momentum equation that connects the pressure gradient $∂p̃/∂x$ and the particle velocity $ũ$⁠. We begin with the following one-dimensional momentum equation for Newtonian fluids in nonretarded time t, which is exact up to $O(ε̃2)$⁠,²⁰ 

where u is the particle velocity, μ_B is the bulk viscosity, μ is the shear viscosity, and $L=12ρ0u2−p2/2ρ0c02$ is the Lagrangian density. To rewrite Eq. (6) in retarded time τ for plane progressive waves [for which $L=0$ at $O(ε̃2)$ upon substitution of the $O(ε̃)$ relation $p=ρ0c0u+O(ε̃2)$], consider the coordinate transformation²² 

where $x1$ is the slow scale corresponding to the retarded time frame τ. Partial derivatives in the transformed coordinates $(x1,τ)$ are then

Substituting Eq. (8) into Eq. (6) and retaining terms up to $O(ε̃2)$ yields

By replacing $ε̃(∂p/∂x1)$ in Eq. (9) with $∂p/∂x$⁠,²² the momentum equation in retarded time τ is obtained,

Equation (10) is a consistent $O(ε̃2)$ momentum equation for plane progressive waves, in which $∂p/∂x∼O(ε̃2)$⁠, $(c0−1(∂p/∂τ)−ρ0(∂u/∂τ))∼O(ε̃2)$⁠, and $(μB+43μ)c0−2(∂2u/∂τ2)∼O(ε̃2)$⁠, respectively. [To see this, consider that $p=ρ0c0u+O(ε̃2)$ for plane progressive waves, and coefficients μ_B and μ are assumed $O(ε̃)$⁠.] The Fourier transform of Eq. (10) gives the spectral version of the momentum equation

Manipulation of Eq. (11) leads to an expression containing the dimensionless impedance $Z¯=Z/ρ0c0$⁠:

where $αv=(μB+43μ)k2/2ρ0c0$ is the absorption coefficient due to viscosity.²³ Furthermore, the term within the braces in Eq. (12) can be expressed as, via binomial expansion in $ΔZ¯=Z¯−1$⁠,

Here, expansion to leading order would suffice, because terms of $O(ε̃2)$ in Eq. (13) produce $O(ε̃3)$ terms upon substitution to Eq. (12). Therefore, the final form of the spectral momentum equation at $O(ε̃2)$ is given by

Now substitute Eq. (14) into the generalized spectral Burgers equation [Eq. (4)] to obtain

Multiplying Eq. (15) by $p̃*$ (the complex conjugate of $p̃$⁠) gives

Ensemble-averaging Eq. (16) leads to

where $Spp=E[|p̃|2]$⁠, $Qpp2=Im(E[p̃*q̃])$⁠, and $Cpp2=Re(E[p̃*q̃])$ are referred to as the autospectral density (or power spectral density), quadspectral density, and cospectral density, respectively,²⁴ and the symbol $E[ ]$ denotes ensemble averaging. Finally, dividing Eq. (17) by $Spp$ and rearranging terms yield the desired impedance form of the generalized Burgers equation

A few observations about Eq. (18) are noteworthy at this juncture. First, Eq. (18) is a consistent $O(ε̃)$ equivalent to the $O(ε̃)2$⁠, generalized Burgers equation [Eq. (4)]. The reduction in order [$O(ε̃)2→O(ε̃)$] is simply brought by the successive multiplication and division by $p̃*$ and $Spp$⁠, respectively. Second, for a typical medium in which more than one loss mechanism is present, viscous absorption $αv$ is much smaller than absorption due to other accompanying loss mechanisms such as molecular relaxation, at relatively low frequencies.^23,25 Because $αv$ makes up only a small fraction of the total absorption $α$⁠, it can be ignored (i.e., $α−2αv≈α$⁠) in Eq. (18) for a wide frequency range of practical interest. Splitting Eq. (18) into the imaginary and real parts (with $αv$ ignored) gives

and

where $ε=pc/ρ0c02∼O(ε̃)$ is the acoustic Mach number based on the characteristic amplitude $pc$ of the pressure-time waveform. The two dimensionless $O(1)$ quantities Q/S and C/S are defined by

The choice of $pc$ depends on the type of waveform. For a transient pulse, the peak pressure amplitude is a convenient choice, whereas the root-mean-square pressure $prms$ may be more appropriate for continuous random noise. With $pc=prms$⁠, the Q/S in Eq. (21) reduces to the Morfey-Howell Q/S.⁹ 

For plane progressive waves in linear acoustics, the specific acoustic impedance [Eq. (5)] can be construed as a medium property that a wave sees at any given point. For instance, a wave propagating in an ideal, lossless medium would see the specific acoustic impedance given by the characteristic impedance $ρ0c0$ (or unity in dimensionless impedance),

Any deviation of the dimensionless impedance from unity then signifies the presence of loss mechanisms such as absorption and dispersion. If the medium is lossy, a progressive wave sees the dimensionless impedance different from unity by the amount commensurate with the strength of linear absorption and dispersion. This is immediately apparent from Eqs. (19) and (20) without the Q/S and C/S terms,

Note that absorption (⁠$α/k$⁠) is associated with the imaginary part of impedance [$Im(Z¯)$], which represents the overall spatial rate of change in power spectral density, or

[Equation (24) is obtained by combining $p̃*dp̃/d(kx)+c.c.=d(p̃p̃*)/d(kx)=dSpp/d(kx)$ and Eq. (14) with the $αv$ term ignored, where the symbol c.c. stands for complex conjugate.] Dispersion (⁠$−δ/k$⁠) is then related to the real part of impedance [$Re(Z¯)−1$], which denotes the overall spatial rate of change in phase, or

where ϕ is defined by $p̃=|p̃|ejϕ$ with respect to dimensionless retarded time ωτ. [To arrive at Eq. (25) combine $p̃*dp̃/d(kx)−c.c.=2j|p̃|d2ϕ/d(kx)$ with Eq. (14).]

Now what if the finite-amplitude effects are taken into account? The significance of Eqs. (19) and (20) is that they provide a framework within which the quantities Q/S and C/S can be interpreted as the additional change in impedance due to nonlinearity. It follows from Eqs. (19) and (20) that Q/S and C/S represent the parametrically induced change in impedance in the form of additional absorption and dispersion. Here, the passage of a finite-amplitude wave alters the apparent medium property (i.e., absorption and dispersion), the extent of which can be used to quantify the strength of nonlinearity.

A caveat: in light of Eqs. (19)–(21), one must exercise caution when using Q/S and C/S to judge “nonlinearity” of a wave. The way in which Q/S and C/S are defined [i.e., they are nondimensionalized by the characteristic amplitude $pc$ in Eq. (21)] renders these nonlinearity indicators dependent on the “shape” of the wave only. For example, two waves with the same waveform but at different levels, say 40 dB and 120 dB, would be equally nonlinear under Q/S and C/S. Nonlinearity due to the “size” of the wave is instead reflected by the acoustic Mach number $ε$ as shown in Eqs. (19) and (20), the values of which for the two waves are many orders of magnitude different from each other, and so are their contributions to the parametrically induced change in impedance. Ultimately, Q/S and C/S are statements about the temporal shape of the wave, in which some waveforms, regardless of the amplitude, are more nonlinear than others.

Finally, a companion nonlinearity indicator C/S is introduced for the first time. Examination of Eqs. (19) and (20) indicates that Q/S and C/S are complementary (i.e., C/S is to dispersion as Q/S is to absorption), and together, they constitute a complete set of nonlinearity indicators for finite-amplitude waves in fluids with general absorption and dispersion laws. Note that only absorption is considered in the original derivation of the Morfey-Howell Q/S.⁹ 

For dispersion-dominant systems, it is recommended that C/S be used in place of Q/S as a nonlinearity indicator. For example, consider a wave system governed by the Korteweg-de Vries (KdV) equation²⁶ 

where d is the dispersion parameter. In Eq. (26) absorption is assumed to be zero (⁠$α=0$⁠), and dispersion exhibits a cubic dependence on frequency (⁠$δ∝ω3$⁠). Wave systems with KdV-type dispersion include incompressible waves on the liquid surface²⁶ and sounds in bubbly liquids.^27,28 To demonstrate a potential problem with Q/S and the utility of C/S in a dispersion-dominant system, we consider the bubbly liquid of Kuznetsov et al.²⁸ with parameters $ρ0=1167 kg/m3$⁠, $c0=107 m/s$⁠, $d=2.5×10−11 s3/m$⁠, and $β=111$⁠. When applied to a soliton solution [Fig. 1(a)] for which there is no change in power spectrum [$Im(Z¯)=0$⁠; recall Eq. (24)], Q/S ought to be identically zero [dashed line in Fig. 1(b)] in the absence of linear absorption (⁠$α/k=0$⁠), according to Eq. (19). It would nonetheless be wrong to suggest from $Q/S=0$ that there is no nonlinearity. The corresponding C/S, which is a quadratic function of frequency [solid line in Fig. 1(b)], can indeed capture the interplay between nonlinearity and dispersion through Eq. (20), where nonlinearity (⁠$C/S∝ω2$⁠) offsets the innate dispersion of the medium (⁠$−δ/k∝ω2$⁠) to result in a wave of permanent form traveling at a constant speed [$Re(Z¯)−1=constant$⁠; see Eq. (25)].

In an attempt to further demystify the Morfey-Howell nonlinearity indicator Q/S, the generalized spectral Burgers equation is recast in terms of specific acoustic impedance, which is comprised of the normalized quadspectral and cospectral densities Q/S and C/S in addition to linear absorption and dispersion coefficients. This allows an impedance-based interpretation in which Q/S and C/S signify nonlinearity-induced absorption and dispersion, respectively.

“Blessed be Fourier,”²⁹ for an arbitrary wave can be represented as a sum of sinusoidal waves. Then there are in effect only two ways to modify a wave at any given location: via changes in amplitude (absorption) and phase (dispersion) of each sinusoidal component. Therefore, any complex wave process, be it finite-amplitude wave motion, refraction, or diffraction, could simply be couched in extra absorption and/or dispersion, adding to the inherent medium property.

W.-S.O. gratefully acknowledges the generous support from the Department of Physics and Astronomy, Brigham Young University. The authors thank Brent O. Reichman for helpful discussions.