Acousto-optic holography.

Acousto-optic sensing is based on the interaction between sound and light: pressure waves induce density variations, which, in turn, alter the way light propagates in air. Pressure fields are, thus, characterized by measuring changes in light propagation induced by pressure waves. Although acousto-optic sensing provides a way of acquiring acoustic information noninvasively, its widespread application has been hindered by the use of reconstruction methods ill-suited for representing acoustic fields. In this study, an acousto-optic holography method is proposed in which the sound pressure in the near field of a source is captured via acousto-optic sensing. The acousto-optic measurements are expanded into propagating and evanescent waves, as in near-field acoustic holography, making it possible to completely characterize the radiated field noninvasively. An algebraic formulation of the wave expansion enables the use of arbitrary sets of projections. The proposed method is demonstrated experimentally by capturing the acoustic field radiated by a vibrating plate. Accurate holographic reconstructions of the pressure, particle velocity, and intensity fields are obtained using purely optical data. These results are particularly significant for the study of sound fields at mid and high frequencies, where using conventional transducers could perturb the measured field and spatial sampling requirements are challenging.

A Fourier transform formulation for radiation from an unbaffled cylinder.

A formulation based on the Fourier transform and generalized functions, and implemented with a fast Fourier transform, is developed to solve a classic acoustics problem: radiation from an unbaffled cylinder with flat endcaps. The endcaps as well as the cylindrical surface have a specified modal vibration pattern, and the problem is solved using the sum of two independent formulations based on the Fourier transform: (1) a vibrating cylinder with rigid endcaps and (2) a rigid cylindrical tube with vibrating diaphragms at its ends. The resulting nearfield solution correctly models the diffraction effects generated at the sharp ends of the cylinder. Calculation of the farfield radiated pressure follows directly from the nearfield solutions with a slight modification to the standard formulas. Results from the formulations are validated with a boundary element simulation and show excellent agreement with errors of less than 1%.

Bistatic scattering of an elastic object using the structural admittance and noise-based holographic measurements.

This paper presents a method to calculate the bistatic response of an elastic object immersed in a fluid using its structural Green's function (in vacuo structural admittance matrix), calculated by placing the object in a spatially random noise field in air. The field separation technique and equivalent source method are used to reconstruct pressure and velocity fields at the object's surface from pressure measurements recorded on two conformal holographic surfaces surrounding the object. Accurate reconstruction of the surface velocity requires subtraction of the rigid body response computed using a finite element approach. The velocity and pressure fields on the surface lead to the extraction of the in vacuo structural admittance matrix of the elastic object, which is manipulated to yield the farfield bistatic response for a fluid-loaded target for several angles of incidence. This method allows the computation of the scattering properties of an elastic object using exclusive information calculated on its surface (no knowledge of the internal structure required). A numerical experiment involving a cylindrical shell with hemispherical caps is presented, and its bistatic response in water shows excellent agreement with a finite element solution.

A general radiation model for sound fields and nearfield acoustical holography in wedge propagation spaces.

In this work an expression for the solution of the Helmholtz equation for wedge spaces is derived. Such propagation spaces represent scenarios for many acoustical problems where a free field assumption is not eligible. The proposed sound field model is derived from the general solution of the wave equation in cylindrical coordinates, using sets of orthonormal basis functions. The latter are modified to satisfy several boundary conditions representing the reflective behaviour of wedge-shaped propagation spaces. This formulation is then used in the context of nearfield acoustical holography (NAH) and to obtain the expression of the Neumann Green function. The model and its suitability for NAH is demonstrated through both numerical simulations and measured data, where the latter was acquired for the specific case of a loudspeaker on a hemi-cylindrical rigid baffle.

Experimental estimation of in vacuo structural admittance using random sources in a non-anechoic room.

Identification of unexploded ordinance buried in the sediment in the littoral waters throughout the world is a problem of great concern. When illuminated by low-frequency sonar some of these targets exhibit an elastic response that can be used to identify them. This elastic behavior is embodied and identified by a quantity called the in vacuo structural admittance matrix Ys, a relationship between the sonar-induced forces and resulting vibration on its surface. When it is known it can be combined with surface impedances to predict the three-dimensional bistatic scattering in any fluid-like media and for any burial state (depth and orientation). At the heart of this is the measurement of Ys and it is demonstrated in this paper that this can be accomplished by studying the target in a simple (acoustically unaltered) in-air laboratory environment. The target chosen in this study is a thick spherical shell that was illuminated by a nearly spatially isotropic array of remote loudspeakers. Ys is constructed from ensemble averages of the cross-correlations of eight collocated accelerometers and microphones placed on the surface of the object. The structural admittance determined from the data showed excellent agreement with theory.

Characterization of space dust using acoustic impact detection.

This paper describes studies leading to the development of an acoustic instrument for measuring properties of micrometeoroids and other dust particles in space. The instrument uses a pair of easily penetrated membranes separated by a known distance. Sensors located on these films detect the transient acoustic signals produced by particle impacts. The arrival times of these signals at the sensor locations are used in a simple multilateration calculation to measure the impact coordinates on each film. Particle direction and speed are found using these impact coordinates and the known membrane separations. This ability to determine particle speed, direction, and time of impact provides the information needed to assign the particle's orbit and identify its likely origin. In many cases additional particle properties can be estimated from the signal amplitudes, including approximate diameter and (for small particles) some indication of composition/morphology. Two versions of this instrument were evaluated in this study. Fiber optic displacement sensors are found advantageous when very thin membranes can be maintained in tension (solar sails, lunar surface). Piezoelectric strain sensors are preferred for thicker films without tension (long duration free flyers). The latter was selected for an upcoming installation on the International Space Station.

On band gap predictions for multiresonant metamaterials on plates.

Recently wide frequency band gaps were observed in an experimental realization of a multiresonant metamaterial for Lamb waves propagating in thin plates. The band gaps rose from hybridization between the flexural plate (A0 Lamb waves) and longitudinal resonances in rods attached perpendicularly. Shortly thereafter a theory based on considering a one-dimensional periodic array of rods and the scattering matrix for a single rod successfully described the observations. This letter presents an alternative simpler theory, arguably accurate at high rod density, that treats the full two-dimensional array of rods and makes no assumption of periodicity. This theory also fits the measurements.

Convolution formulations for non-negative intensity.

Previously unknown spatial convolution formulas for a variant of the active normal intensity in planar coordinates have been derived that use measured pressure or normal velocity near-field holograms to construct a positive-only (outward) intensity distribution in the plane, quantifying the areas of the vibrating structure that produce radiation to the far-field. This is an extension of the outgoing-only (unipolar) intensity technique recently developed for arbitrary geometries by Steffen Marburg. The method is applied independently to pressure and velocity data measured in a plane close to the surface of a point-driven, unbaffled rectangular plate in the laboratory. It is demonstrated that the sound producing regions of the structure are clearly revealed using the derived formulas and that the spatial resolution is limited to a half-wavelength. A second set of formulas called the hybrid-intensity formulas are also derived which yield a bipolar intensity using a different spatial convolution operator, again using either the measured pressure or velocity. It is demonstrated from the experiment results that the velocity formula yields the classical active intensity and the pressure formula an interesting hybrid intensity that may be useful for source localization. Computations are fast and carried out in real space without Fourier transforms into wavenumber space.

Prediction of a body's structural impedance and scattering properties using correlation of random noise.

This paper derives a method to estimate the structural or surface impedance matrix (or equivalently the inverse of the structural Green's function) for an elastic body by placing it in an encompassing and spatially random noise field and cross-correlating pressure and normal velocity measurements taken on its surface. A numerical experiment is presented that utilizes a cross-correlation method to determine the structural impedance matrix for an infinite cylindrical shell excited by a spatially random noise field. It is shown that the correlation method produces the exact analytic form of the structural impedance matrix. Furthermore, using standard impedance formulations of the scattered and incident pressure fields at the object surface that are based on the equivalent source method and using this estimated structural impedance, a prediction of the scattered acoustic field at any position outside of the object can be made for any given incident field. An example is presented for a point (line) source near a cylindrical shell and when compared with the analytical result, excellent agreement is found between the scattered fields at a radius close to the shell.

Surface decomposition method for near-field acoustic holography.

Near-field acoustic holography reconstruction of the acoustic field at the surface of an arbitrarily shaped radiating structure from pressure measurements at a nearby conformal surface is obtained from the solution of a boundary integral equation. This integral equation is discretized using the equivalent source method and transformed into a matrix system that can be solved using iterative regularization methods that counteract the effect of noise on the measurements. This work considers the case when the resultant matrix system is so large that it cannot be explicitly formed and iterative methods of solution cannot be directly implemented. In this case the method of surface decomposition is proposed, where the measurement surface is divided into smaller nonoverlapping subsurfaces. Each subsurface is used to form a smaller matrix system that is solved and the result joined together to generate a global solution to the original matrix system. Numerically generated data are used to study the use of subsurface extensions to increase the continuity of the global solution, and investigate the size of the subsurfaces, as well as the distance between the measurement and the vibrating surface. Finally a vibrating ship hull structure is considered as a physical example to apply and validate the proposed methodology.

Vector intensity reconstructions in a volume surrounding a rigid spherical microphone array.

A technique is described to image the vector intensity in the near field of a spherical array of microphones flush mounted in a rigid sphere. The spatially measured pressure is decomposed into Fourier harmonics in order to reconstruct the volumetric vector intensity outside the sphere. The theory for this reconstruction is developed in this paper. The resulting intensity images are very successful at locating and quantifying unknown exterior acoustic sources, ideal for application in noise control problems in interior spaces such as automobiles and airplanes. Arrays of varying numbers of microphones and radii are considered and compared and errors are computed for both theory and experiment. It is demonstrated that this is an ill-posed problem below a cutoff frequency depending on array design, requiring Tikhonov regularization below cutoff. There is no low frequency limit on operation, although the signal-to-noise ratio is the determining factor for high-spatial resolution at low frequencies. It is shown that the upper frequency limit is set by the number of microphones in the array and is independent of noise. The accuracy of the approach is assessed by considering the exact solution for the scattering of a point source by a rigid sphere. Several field experiments are presented to demonstrate the utility of the technique. In these experiments, the partial field decomposition technique is used and holograms of multiple exterior sources are separated and their individual volumetric intensity fields imaged. In this manner, the intensity fields of two uncorrelated tube sources in an anechoic chamber are isolated from one another and separated intensity maps are obtained from over a broad frequency range. In a practical application, the vector intensity field in the interior of an automobile cabin is mapped at the fundamental of the engine vibration using the rigid sphere positioned at the driver's head. The source regions contributing to the interior cabin noise are identified.

Approximations of inverse boundary element methods with partial measurements of the pressure field.

Boundary element methods (BEMs) based near-field acoustic holography (NAH) requires the measurement of the pressure field over a closed surface in order to recover the normal velocity on a nearby conformal surface. There are practical cases when measurements are available over a patch from the measurement surface in which conventional inverse BEM based NAH (IBEM) cannot be applied directly, but instead as an approximation. In this work two main approximations based on the indirect-implicit methods are considered: Patch IBEM and IBEM with Cauchy data. Patch IBEM can be applied with a continuation procedure, which as its predecessor patch NAH (a well known technique that can be used on separable geometries of the wave equation) continues the pressure field using an iterative procedure, or it can be applied by a direct procedure. On the other hand, IBEM with Cauchy data requires measurements over two conformal patches and it will be shown that this technique will be reliable regardless of the position of the source. The theory behind each method will be justified and validated using a cylindrical surface with numerical data generated by point sources, and using experimental data from a cylindrical fuselage excited by a point force.

Violin f-hole contribution to far-field radiation via patch near-field acoustical holography.

The violin radiates either from dual ports (f-holes) or via surface motion of the corpus (top+ribs+back), with no clear delineation between these sources. Combining "patch" near-field acoustical holography over just the f-hole region of a violin with far-field radiativity measurements over a sphere, it was possible to separate f-hole from surface motion contributions to the total radiation of the corpus below 2.6 kHz. A0, the Helmholtz-like lowest cavity resonance, radiated essentially entirely through the f-holes as expected while A1, the first longitudinal cavity mode with a node at the f-holes, had no significant f-hole radiation. The observed A1 radiation comes from an indirect radiation mechanism, induced corpus motion approximately mirroring the cavity pressure profile seen for violinlike bowed string instruments across a wide range of sizes. The first estimates of the fraction of radiation from the f-holes F(f) indicate that some low frequency corpus modes thought to radiate only via surface motion (notably the first corpus bending modes) had significant radiation through the f-holes, in agreement with net volume changes estimated from experimental modal analysis. F(f) generally trended lower with increasing frequency, following corpus mobility decreases. The f-hole directivity (top/back radiativity ratio) was generally higher than whole-violin directivity.

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.

The vibro-acoustic response and analysis of a full-scale aircraft fuselage section for interior noise reduction.

The surface and interior response of a Cessna Citation fuselage section under three different forcing functions (10-1000 Hz) is evaluated through spatially dense scanning measurements. Spatial Fourier analysis reveals that a point force applied to the stiffener grid provides a rich wavenumber response over a broad frequency range. The surface motion data show global structural modes (approximately < 150 Hz), superposition of global and local intrapanel responses (approximately 150-450 Hz), and intrapanel motion alone (approximately > 450 Hz). Some evidence of Bloch wave motion is observed, revealing classical stop/pass bands associated with stiffener periodicity. The interior response (approximately < 150 Hz) is dominated by global structural modes that force the interior cavity. Local intrapanel responses (approximately > 150 Hz) of the fuselage provide a broadband volume velocity source that strongly excites a high density of interior modes. Mode coupling between the structural response and the interior modes appears to be negligible due to a lack of frequency proximity and mismatches in the spatial distribution. A high degree-of-freedom finite element model of the fuselage section was developed as a predictive tool. The calculated response is in good agreement with the experimental result, yielding a general model development methodology for accurate prediction of structures with moderate to high complexity.

Krylov subspace iterative methods for boundary element method based near-field acoustic holography.

The reconstruction of the acoustic field for general surfaces is obtained from the solution of a matrix system that results from a boundary integral equation discretized using boundary element methods. The solution to the resultant matrix system is obtained using iterative regularization methods that counteract the effect of noise on the measurements. These methods will not require the calculation of the singular value decomposition, which can be expensive when the matrix system is considerably large. Krylov subspace methods are iterative methods that have the phenomena known as "semi-convergence," i.e., the optimal regularization solution is obtained after a few iterations. If the iteration is not stopped, the method converges to a solution that generally is totally corrupted by errors on the measurements. For these methods the number of iterations play the role of the regularization parameter. We will focus our attention to the study of the regularizing properties from the Krylov subspace methods like conjugate gradients, least squares QR and the recently proposed Hybrid method. A discussion and comparison of the available stopping rules will be included. A vibrating plate is considered as an example to validate our results.

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 Green's 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.