Earl G. Williams

Regularization methods for near-field acoustical holography

The reconstruction of the pressure and normal surface velocity provided by near-field acoustical holography (NAH) from pressure measurements made near a vibrating structure is a linear, ill-posed inverse problem due to the existence of strongly decaying, evanescentlike waves. Regularization provides a technique of overcoming the ill-posedness and generates a solution to the linear problem in an automated way. We present four robust methods for regularization; the standard Tikhonov procedure along with a novel improved version, Landweber iteration, and the conjugate gradient approach. Each of these approaches can be applied to all forms of interior or exterior NAH problems; planar, cylindrical, spherical, and conformal. We also study two parameter selection procedures, the Morozov discrepancy principle and the generalized cross validation, which are crucial to any regularization theory. In particular, we concentrate here on planar and cylindrical holography. These forms of NAH which rely on the discrete Fourier transform are important due to their popularity and to their tremendous computational speed. In order to use regularization theory for the separable geometry problems we reformulate the equations of planar, cylindrical, and spherical NAH into an eigenvalue problem. The resulting eigenvalues and eigenvectors couple easily to regularization theory, which can be incorporated into the NAH software with little sacrifice in computational speed. The resulting complete automation of the NAH algorithm for both separable and nonseparable geometries overcomes the last significant hurdle for NAH.

Sound source reconstructions using a microphone array

A square microphone array with 256 elements has been constructed along with interfacing electronics to study low‐frequency (1–5 kHz) sound sources in air using the principles of acoustical holography. The array is used in the nearfield of a radiating object and the sound source structure of that object is reconstructed with an on‐line minicomputer. Reconstruction of the source structure of a point source and an unbaffled, free rectangular plate point excited below its coincidence frequency are presented. The latter shows clear evidence of ’’corner’’ and ’’edge’’ modes in which the respective areas of the plate are the dominant radiating sources. A new imaging process which is not limited in resolution by the wavelength of the radiated sound is introduced.

Generalized nearfield acoustical holography for cylindrical geometry: Theory and experiment

From the measurement of the acoustic pressure on a cylindrical, two‐dimensional contour located close to the surface of an underwater, vibrating cylinder, the complete three‐dimensional sound field can be reproduced (reconstructed) with the aid of a computer. This reconstruction technique, called GENAH (generalized nearfield acoustical holography), is unlike conventional holography because it provides a super resolution image of the sound‐pressure field from the surface of the cylinder to the farfield. At the same time, GENAH reconstructs, from this two‐dimensional measurement, the vector velocity and the vector intensity fields (energy flow) in the nearfield of the source, and identifies modes of surface vibration of the cylinder. Experimental results are provided and the accuracy of GENAH is demonstrated by comparison with the two‐hydrophone technique.

Numerical evaluation of the Rayleigh integral for planar radiators using the FFT

Rayleigh’s integral formula is evaluated numerically for planar radiators of any shape, with any specified velocity in the source plane using the fast Fourier transform algorithm. The major advantage of this technique is its speed of computation—over 400 times faster than a straightforward two‐dimensional numerical integration. The technique is developed for computation of the radiated pressure in the nearfield of the source and can be easily extended to provide, with little computation time, the vector intensity in the nearfield. Computations with the FFT of the nearfield pressure of baffled rectangular plates with clamped and free boundaries are compared with the ‘‘exact’’ solution to illuminate any errors. The bias errors, introduced by the FFT, are investigated and a technique is developed to significantly reduce them.

Fast Fourier transform and singular value decomposition formulations for patch nearfield acoustical holography

Nearfield acoustical holography (NAH) requires the measurement of the pressure field over a complete surface in order to recover the normal velocity on a nearby concentric surface, the latter generally coincident with a vibrator. Patch NAH provides a major simplification by eliminating the need for complete surface pressure scans-only a small area needs to be scanned to determine the normal velocity on the corresponding (small area) concentric patch on the vibrator. The theory of patch NAH is based on (1) an analytic continuation of the patch pressure which provides a spatially tapered aperture extension of the field and (2) a decomposition of the transfer function (pressure to velocity and/or pressure to pressure) between the two surfaces using the singular value decomposition (SVD) for general shapes and the fast Fourier transform (FFT) for planar surfaces. Inversion of the transfer function is stabilized using Tikhonov regularization and the Morozov discrepancy principle. Experimental results show that root mean square errors of the normal velocity reconstruction for a point-driven vibrator over 200-2700 Hz average less than 20% for two small, concentric patch surfaces 0.4 cm apart. Reconstruction of the active normal acoustic intensity was also successful, with less than 30% error over the frequency band.

Continuation of acoustic near-fields.

This paper deals with the analytic continuation of a coherent pressure field specified on a finite sheet located close to and conformal to the surface of a vibrator. This analytic continuation is an extension or extrapolation of the given (measured) field into a region outside and tangential to the original finite sheet, and is based on the Greens function (the transfer function) relating acoustic quantities on the two conformal surfaces. The continuation of the measured pressure field is an inverse problem that requires the use or regularization theory, especially when noise is present in the data. An iteration algorithm is presented that is successful in continuing the pressure field into the tangential sheet. The results are accurate close to the original boundary and taper (decay) toward zero with distance away from it. The algorithm is tested on numerical and experimental data from a point-driven rectangular plate. Results show the successful extrapolation (continuation) of this data into an area nearly double that of the original pressure field. This algorithm is not limited to planar surfaces and can be applied to arbitrarily shaped surfaces.

Interior near-field acoustical holography in flight

In this paper boundary element methods (BEM) are mated with near-field acoustical holography (NAH) in order to determine the normal velocity over a large area of a fuselage of a turboprop airplane from a measurement of the pressure (hologram) on a concentric surface in the interior of the aircraft. This work represents the first time NAH has been applied in situ, in-flight. The normal fuselage velocity was successfully reconstructed at the blade passage frequency (BPF) of the propeller and its first two harmonics. This reconstructed velocity reveals structure-borne and airborne sound-transmission paths from the engine to the interior space.

Conformal generalized near‐field acoustic holography for axisymmetric geometries

The normal velocity field on the boundary σ of a radiating source is reconstructed with high resolution by using pressure data, measured in the very near field to include evanescent components, fast decaying with the distance. The measurement points belong to a surface conformal to σ. The reconstruction is performed by identifying the boundary normal velocity distribution generating the pressure field best mean‐square fitting the measurement data. The ensuing normal equations are solved by resorting to the singular value decomposition (SVD) of the transformation from the boundary normal velocity to the pressure at the measurement points. The reconstructed boundary normal velocity is represented as a linear combination of basis functions, each associated with a singular value (SV). The errors in the estimation of the coefficients of the linear combination are proportional to the inverses of the corresponding SVs. To achieve robustness, the SVD is truncated to retain the terms corresponding to a limited dyn...

Study of the comparison of the methods of equivalent sources and boundary element methods for near-field acoustic holography

Boundary element methods (BEM) based near-field acoustic holography (NAH) has been used successfully in order to reconstruct the normal velocity on an arbitrarily shaped structure surface from measurements of the pressure field on a nearby conformal surface. An alternative approach for this reconstruction on a general structure utilizes the equivalent sources method (ESM). In ESM the acoustic field is represented by a set of point sources located over a surface that is close to the structure surface. This approach is attractive mainly for its simplicity of implementation and speed. In this work ESM as an approximation of BEM based NAH is studied and the necessary conditions for the successful application of this approach in NAH is discussed. A cylindrical fuselage surface excited by a point force as an example to validate the results is used.

Nearfield acoustical holography using an underwater, automated scanner

A computer‐controlled, three‐axis Cartesian scanning facility has been constructed in a large water tank to provide accurate preprogrammed contouring with hydrophone probes. As the scanner moves on the preprogrammed contours, usually located very close to a radiating object, the pressure field is sampled at discrete points until a two‐dimensional (2D) pressure map is obtained. This pressure map is essentially a hologram containing amplitude and phase information which can be processed with a computer using a technique called nearfield acoustical holography (NAH). This processing provides the pressure, vector velocity, and vector intensity anywhere in the space from the surface of the source to the farfield. Backward projection (reconstructing the field at the source surface) provides surface velocity and pressure (fluid loading) on the actual source and thus, through the continuity of the normal velocity, provides both the amplitude and phase of the structural vibration of the source. Through a series of ...