Michael B. Muhlestein

On the Evolution of Crackle in Jet Noise from High- Performance Engines

Crackle, the impulsive quality sometimes present in supersonic jet noise, has traditionally been defined in terms of the pressure waveform skewness. However, recent work has shown that the pressure waveform time derivative is a better quantifier of the acoustic shocks believed to be responsible for its perception. This paper discusses two definitions of crackle, waveform asymmetry versus shock content, and crackle as a source or propagation-related phenomenon. Data from two static military jet aircraft tests are used to demonstrate that the skewed waveforms radiated from the jet undergo significant nonlinear steepening and shock formation, as evidenced by the skewness of the time derivative. Thus, although skewness is a source phenomenon, crackle’s perceived quality is heavily influenced by propagation through the near field and into the far field to the extent that crackle is caused by the presence of shock-like features in the waveform.

Evolution of the average steepening factor for nonlinearly propagating waves

Difficulties arise in attempting to discern the effects of nonlinearity in near-field jet-noise measurements due to the complicated source structure of high-velocity jets. This article describes a measure that may be used to help quantify the effects of nonlinearity on waveform propagation. This measure, called the average steepening factor (ASF), is the ratio of the average positive slope in a time waveform to the average negative slope. The ASF is the inverse of the wave steepening factor defined originally by Gallagher [AIAA Paper No. 82-0416 (1982)]. An analytical description of the ASF evolution is given for benchmark cases-initially sinusoidal plane waves propagating through lossless and thermoviscous media. The effects of finite sampling rates and measurement noise on ASF estimation from measured waveforms are discussed. The evolution of initially broadband Gaussian noise and signals propagating in media with realistic absorption are described using numerical and experimental methods. The ASF is found to be relatively sensitive to measurement noise but is a relatively robust measure for limited sampling rates. The ASF is found to increase more slowly for initially Gaussian noise signals than for initially sinusoidal signals of the same level, indicating the average distortion within noise waveforms occur more slowly.

Educational demonstration of a spherically propagating acoustic shock.

Exploding gas-filled balloons are common chemistry demonstrations. They also provide an entertaining and educational means to experimentally verify nonlinear acoustical theory as described by the Earnshaw solution to the lossless Burgers equation and weak-shock theory. This article describes the theory, the demonstration, and the results of a propagation experiment carried out to provide typical results. Data analysis shows that an acetylene-oxygen balloon produces an acoustic shock whose evolution agrees well with weak-shock theory. On the other hand, the pressure wave generated by a hydrogen-oxygen balloon also propagates nonlinearly, but does not approach N-wave-like, weak-shock formation over the propagation distance. Overall, the experiment shows that popular demonstrations of chemical reactions can be extended from chemistry classrooms to a pedagogical tool for the student of advanced physical acoustics.

Evolution of the derivative skewness for nonlinearly propagating waves

The skewness of the first time derivative of a pressure waveform, or derivative skewness, has been used previously to describe the presence of shock-like content in jet and rocket noise. Despite its use, a quantitative understanding of derivative skewness values has been lacking. In this paper, the derivative skewness for nonlinearly propagating waves is investigated using analytical, numerical, and experimental methods. Analytical expressions for the derivative skewness of an initially sinusoidal plane wave are developed and, along with numerical data, are used to describe its behavior in the preshock, sawtooth, and old-age regions. Analyses of common measurement issues show that the derivative skewness is relatively sensitive to the effects of a smaller sampling rate, but less sensitive to the presence of additive noise. In addition, the derivative skewness of nonlinearly propagating noise is found to reach greater values over a shorter length scale relative to sinusoidal signals. A minimum sampling rate is recommended for sinusoidal signals to accurately estimate derivative skewness values up to five, which serves as an approximate threshold indicating significant shock formation.

Experimental evidence of Willis coupling in a one-dimensional effective material element

The primary objective of acoustic metamaterial research is to design subwavelength systems that behave as effective materials with novel acoustical properties. One such property couples the stress–strain and the momentum–velocity relations. This response is analogous to bianisotropy in electromagnetism, is absent from common materials, and is often referred to as Willis coupling after J.R., Willis, who first described it in the context of the dynamic response of heterogeneous elastic media. This work presents two principal results: first, experimental and theoretical demonstrations, illustrating that Willis properties are required to obtain physically meaningful effective material properties resulting solely from local behaviour of an asymmetric one-dimensional isolated element and, second, an experimental procedure to extract the effective material properties from a one-dimensional isolated element. The measured material properties are in very good agreement with theoretical predictions and thus provide improved understanding of the physical mechanisms leading to Willis coupling in acoustic metamaterials.

Comparison of Two Time-domain Measures of Nonlinearity in Near-field Propagation of High-power Jet Noise

Time-domain metrics are used to investigate the nonlinearity of the sound in the vicinity of the F-35 AA-1. The first measure considered is the average steepening factor (ASF), which we define as the inverse of the wave steepening factor and is a ratio of the expectation value of the positive slopes in the waveform to the expectation value of the negative slopes. The second nonlinearity metric is the skewness of the time derivative of the pressure waveform (derivative skewness), which describes the asymmetry of the distribution of slopes in the waveform. Spatial maps of both metrics applied to the F-35 AA-1 data reveal that regions of increasing derivative skewness correspond more closely to the maximum sound radiation area, whereas the largest values for the ASF seem aligned with the regions where the waveform amplitude distributions are most asymmetric. It is proposed that these two metrics reveal different characteristics of the nonlinear propagation of jet noise. The ASF is more representative of the average slopes, which are dominated by high frequencies. Conversely, the derivative skewness identifies of large positive slopes and hence relates to the shock content in the noise.

Reciprocity, passivity and causality in Willis materials

Materials that require coupling between the stress–strain and momentum–velocity constitutive relations were first proposed by Willis (Willis 1981 Wave Motion 3, 1–11. (doi:10.1016/0165-2125(81)90008-1)) and are now known as elastic materials of the Willis type, or simply Willis materials. As coupling between these two constitutive equations is a generalization of standard elastodynamic theory, restrictions on the physically admissible material properties for Willis materials should be similarly generalized. This paper derives restrictions imposed on the material properties of Willis materials when they are assumed to be reciprocal, passive and causal. Considerations of causality and low-order dispersion suggest an alternative formulation of the standard Willis equations. The alternative formulation provides improved insight into the subwavelength physical behaviour leading to Willis material properties and is amenable to time-domain analyses. Finally, the results initially obtained for a generally elastic material are specialized to the acoustic limit.

A micromechanical approach for homogenization of elastic metamaterials with dynamic microstructure

An approximate homogenization technique is presented for generally anisotropic elastic metamaterials consisting of an elastic host material containing randomly distributed heterogeneities displaying frequency-dependent material properties. The dynamic response may arise from relaxation processes such as viscoelasticity or from dynamic microstructure. A Greens function approach is used to model elastic inhomogeneities embedded within a uniform elastic matrix as force sources that are excited by a time-varying, spatially uniform displacement field. Assuming dynamic subwavelength inhomogeneities only interact through their volume-averaged fields implies the macroscopic stress and momentum density fields are functions of both the microscopic strain and velocity fields, and may be related to the macroscopic strain and velocity fields through localization tensors. The macroscopic and microscopic fields are combined to yield a homogenization scheme that predicts the local effective stiffness, density and coupling tensors for an effective Willis-type constitutive equation. It is shown that when internal degrees of freedom of the inhomogeneities are present, Willis-type coupling becomes necessary on the macroscale. To demonstrate the utility of the homogenization technique, the effective properties of an isotropic elastic matrix material containing isotropic and anisotropic spherical inhomogeneities, isotropic spheroidal inhomogeneities and isotropic dynamic spherical inhomogeneities are presented and discussed.

Effective wavenumbers for sound scattering by trunks, branches, and the canopy in a forest

Sound propagation in a forest is often represented as propagation in free space with an effective complex wavenumber, which accounts for scattering and absorption. In this paper, the effective wavenumbers due to sound scattering by trunks, large branches, and the canopy are determined and analyzed based on three-dimensional multiple scattering theory. Trunks and branches are modeled as vertical and slanted finite cylinders, while the canopy is modeled by diffuse scatterers. The results are compared with two-dimensional effective wavenumbers previously used in the literature, which were obtained by approximating the trunk layer as infinite vertical cylinders.

Evolution of the temporal slope density function for waves propagating according to the inviscid Burgers equation

An exact formulation for the evolution of the probability density function of the time derivative of a waveform (slope density) propagating according to the one-dimensional inviscid Burgers equation is given. The formulation relies on the implicit Earnshaw solution and therefore is only valid prior to shock formation. As explicit examples, the slope density evolution of an initially sinusoidal plane wave, initially Gaussian-distributed planar noise, and an initially triangular wave are presented. The triangular wave is used to examine weak-shock limits without violating the theoretical assumptions. It is also shown that the moments of the slope density function as a function of distance may be written as an expansion in terms of the moments of the source slope density function. From this expansion, approximate expressions are presented for the above cases as well as a specific non-Gaussian noise case intended to mimic features of jet noise. Finally, analytical predictions of the propagation of initially Gaussian-distributed noise are compared favorably with plane-wave tube measurements.