1.2 Literature study
1.2.3 Near-field Acoustic Holography techniques
NAH is an inverse array technique used to reconstruct acoustic parameters by measuring sound pressure with an array of microphones, in a parallel and near to the sound source. Basic principle of NAH methods is described as a flow chart and shown in Fig. 1.8. The current research provides study of four most popular methods of NAH technique and same methods are employed in the reconstruction of sound sources. Four NAH techniques considered in the present study are:
Fourier NAH (FNAH)
Statistically Optimized NAH (SONAH)
Equivalent Source Model (ESM)
Inverse Boundary Element Method (IBEM)
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Sound radiated from the noise source can be measured in terms of sound pressure on a plane near to source surfaces with a set of microphones arranged in a suitable pattern. These measured pressures are related to unknown source strength by transfer matrix. Calculation of the elements of Transfer Matrix (TM) varies for different NAH methods. In Statistically Optimized NAH (SONAH) method, TM can be formulated using elementary wave functions.
For Equivalent Source Method (ESM), TM can be calculated using free space Green’s function and for Inverse Boundary Element Method (IBEM), TM can be obtained using Helmholtz integral equation. As all these NAH methods are ill-posed problems, regularization is necessary to overcome the same and for reconstructing accurate results.
Figure 1.8: Flowchart of the procedure involved in NAH reconstruction techniques.
NAH method was first proposed by Maynard and Williams in the 1980’s based on discrete Fourier transform [63]. Primary objective of using this method is to reconstruct sound pressure, particle velocity on / near to the actual source surface. By using these two quantities, active intensity can be calculated and thus total radiated sound power can be obtained. A good overview of the development of NAH techniques is described by Bai et al. [64], Magalhaes and Tenenbaum [65] and Chelliah et al. [66]. Methods based on measurement of particle velocity to reconstruct the sound field were also developed and were found to be more beneficial, if the particle velocity is reconstructed [67, 68]. Main drawback of this method is difficult to practically match the velocity sensors, unlike pressure sensors. Different NAH methods vary with the model used to correlate sound pressure measured at different microphones to acoustic quantities on the source surface.
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Fourier NAH (FNAH) method uses Discrete Fourier Transform (DFT) in Cartesian coordinates to reconstruct sound field on the surface parallel to measurement plane [63, 69, 70]. This DFT transforms the sound pressure from spatial to wavenumber domain (k-space).
Here, the sound field is expressed as propagating and evanescent waves. Multiplying this wave component with inverse propagator function yields sound field in wave number domain on the required plane. Inverse Fourier Transformation (IFT) is applied to obtain reconstructed acoustic quantities in spatial domain. FNAH method can also be implemented in cylindrical and spherical coordinates [70]. It is efficient and more suitable for planar sources, but main drawback is the size of measurement surface, which needs to be at least two times larger than source leading to tedious measurement process. To overcome these limitations, a patch holography technique such as Statistically Optimized NAH method [71] is proposed.
In SONAH method, acoustic quantities are reconstructed on the mapping surfaces near the source. These quantities are calculated using the transfer matrix comprising set of propagating and evanescent waves with appropriate weightage. J. Hald [71], described detailed mathematical formulation for reconstruction of sound pressure and particle velocity. Gomes [72] and Hald et al. [73, 74] investigated applicability of SONAH method on different cases such as non-planar surfaces and cabin environment using two-layer array. Similar to SONAH, another method called HELS (Helmholtz Equation Least Squares) is developed which uses spherical wave functions instead of plane wave functions (used in SONAH method) [75-77].
In this method, a number of elementary waves are always less than or equal to number of microphones.
IBEM has been developed to obtain accurate reconstructions in NAH for arbitrarily shaped vibrating surfaces. Firstly, this method is developed based on the Helmholtz integral equation, which relates sound pressure in field plane (exterior to the source) to sound pressure and velocity on the source surface [78, 79]. Later, IBEM is formulated based on use of continuous distribution of virtual sources on the boundary to represent radiated sound field [80, 81]. If partial measurements are available, then conventional IBEM can’t be used. However, an approximation method, called as patch IBEM, has been developed. The advantage of this method is, it can be implemented on large source [82] while major drawback being the chances of reconstruction errors to be high, due to ignoring of partial source region in the measurements.
Equivalent source method is also treated as an alternative method to BEM since BEM requires large computational resources for complex structures [83]. This method uses a set of virtual
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sources such as monopole, dipole or combinations to represent sound radiation from source and are placed inside the vibrating body. Actually, this method is developed for the forward progression problem to predict the radiated sound field [84, 85]. Later on, this method was used to solve inverse problems also [86, 87]. The basic idea of this method is to estimate the source strengths of distributed simple sources by mapping the model with measured sound pressure. Using these source strengths, sound field on the source can be estimated. Main advantage of ESM is, the computational time required is much lesser than other NAH methods. Essential benefits of ESM and patch NAH methods are combined and developed new near-field acoustic holography surface decomposition method by Valdivia et al. [88].
Selection of appropriate stand-off distance (SD, distance between actual source plane to virtual plane) is important to obtain accurate reconstruction results. M.R. Bai et al. [89]
described the selection of optimal SD and suggested that optimum distance varies from 0.4- 0.5 of microphone spacing for planar sources and for spherical sources it may vary from 0.8- 1.7 times of average spacing.
Inverse techniques are also implemented on aero-acoustic sources to visualize sound field.
Lee et al. used NAH technique based on virtual coherence method for reconstruction of sound field for small ducted fan assembly [90]. Kim and Nelson [91] described estimation of acoustic source strengths using measured radiated sound field within the cylindrical duct.
They proved that accuracy of estimating source strength and spatial resolutions can be achieved by improving conditioning of the transfer matrix (frequency response function matrix). Lowis and Joseph [92] proposed a new method, based on modified Green’s function, for estimating broadband source strength over the ducted rotor using sound pressure measured at duct walls. Zhang et al. [93] developed a generalized NAH method based on boundary element method to reconstruct the noise source inside a ducted fan. In this method, aero- acoustic sources are represented as point sources such as monopole and dipoles. Sound field on the duct surface was measured and utilized to reconstruct noise sources.
As these NAH techniques are inverse methods, reconstructed results are ill-posed due to presence of evanescent waves. Regularization methods are used to overcome ill-posedness of the problem [94]. Main purpose is to avoid amplification of the noise while solving inverse problem. Tikhonov regularization is the most commonly used technique for NAH methods [95, 96]. Conjugate gradient approach is another method used in inversion [97, 98]. The choice of proper regularization parameter is most important to obtain better inversion results.
Different methods available to select regularization parameters are: L-curve method [99, 100], Generalized Cross Validation (GCV) method [101] and Morozov discrepancy principle [102].
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