## Abstract

In this paper, we generalize the recently developed analytical solutions of the radiation modes problem to the determination of closed-form expressions for the singular value expansion of a number of integral operators that map the boundary velocity of a baffled planar structure onto the acoustic pressure radiated in far-field or intermediate regions. Exact solutions to this problem involve prolate spheroidal wave functions that correspond to a set of independent distributions with finite spatial support and maximal energy concentration in a given bandwidth of the transform domain. A stable solution to the inverse source reconstruction problem is obtained by decomposing the unknown boundary velocity into a number of efficiently radiating singular velocity patterns that correspond to the number of degrees of freedom of the radiated field. It is found that the degree of ill-posedness of the inverse problem is significantly reduced, when considering a hemi-circular observation arc with respect to a linear array of sensors, by a factor scaling on the small angular aperture subtended by the observation line. Estimates are derived from the spatial resolution limits that can be achieved in the source reconstruction problem from the dimension of the efficiently radiating subspace.

## 1. Introduction

Recently, analytical solutions that solve the radiation modes problem have been found, thus providing exact expressions for the radiation efficiencies and shapes of the radiation modes of a baffled beam (Maury & Elliott 2005). Such closed-form expressions would help to enhance the design of spatial radiation filters, such as those implemented in feedback-based active noise control systems, in order to provide a better sensing and a more efficient reduction of the sound power radiated from planar structures (Elliott & Johnson 1993; Gibbs *et al*. 2000). Indeed, the radiation modes are a set of independent optimal velocity distributions defined on the surface of a vibrating structure that best capture the sound power radiated among all admissible velocity patterns; those associated with the largest radiation efficiencies correspond to a set of velocity distributions that have maximal energy concentration in a given radiation bandwidth (Borgiotti 1990). Such non-trivial singular velocity patterns with finite spatial support cannot be band limited in the wavenumber domain. They are sought to solve the so-called concentration problem that was initially considered in the context of communication theory and electrical engineering more than 40 years ago by Slepian and his collaborators (Landau & Pollack 1961; Slepian & Pollack 1961).

The question was then to find a class of band-limited signals, such as those belonging to the natural class of finite energy signals, which encapsulate the largest fraction of energy in a given time slot. An elegant solution has been drawn based on a remarkable commutation result between an integral operator with a symmetric positive definite kernel and a self-adjoint second-order differential operator, which therefore share the same spectral properties. The eigenfunctions of the differential operator are known as prolate spheroidal wave functions (PSWFs); their study arose on the separation of the scalar wave equation in spheroidal coordinate systems (Flammer 1957). Since these investigations, this class of functions has provided useful solutions in different disciplines involving band-limited wave functions, such as multitaper spectral analysis (Slepian 1978; Percival & Walden 1993), optical apodization problems (Slepian 1965), the numerical analysis of hyperbolic partial differential equations based on prolate pseudo-spectral methods (Chen *et al*. 2005) and the achievement of both high spatial and temporal resolutions in magnetic resonance imaging of brain activity (Lindquist *et al*. 2006).

The present study addresses a typical issue at the heart of the acoustic source identification problems, namely the reconstruction of a baffled planar velocity distribution from the knowledge of the measured acoustic pressure radiated in a free field. The estimation of the strength of electromagnetic or acoustic sources from the measurement of their radiated field is an extensive research area that falls within the broad field of inverse source problems. Optical image plane holography has been widely studied for the digital record and the numerical reconstruction of both the amplitude and the phase of the electromagnetic wave reflected or transmitted by an object to be imaged (Gabor 1948; Leith & Upatnieks 1962). Such methods are, respectively, based on Fourier- or Fresnel-transform relationships as approximations of the diffraction integral between the complex field amplitude detected at the hologram plane and the object field (Goodman 2005). Although image processing methods are being developed to enhance the quality of the optical hologram, the resolution is intrinsically limited by the smallest wavelength that can be detected in the object beam for a given size of the digital sensor pixels (Stern & Javidi 2006). This resolution limit can be overcome when dealing with acoustic source reconstruction problems from near-field measurements, i.e. in near-field acoustical holography (NAH). The ability to capture the evanescent wave field decaying away from the source boundary allows the source components, separated by distances much less than one-half an acoustic wavelength, to be resolved (Williams & Maynard 1980).

There appear to have been three main approaches when one attempts to solve NAH problems. The first approach concerns Fourier-based methods, such as the planar or the cylindrical NAH, which rely on the relationship in the spatial wavenumber domain between a given source distribution and its radiated field that, in principle, should be recorded over a complete hologram surface (Williams 1999). This constraint might, however, be alleviated if one uses procedures such as patch NAH for the analytic continuation of the sound field outside the measurement aperture (Williams *et al*. 2003) or the statistically optimized NAH (SONAH) that makes use of a surface-to-surface projection scheme based on the plane wave expansion of the sound field (Steiner & Hald 2001). The SONAH process avoids the use of a spatial Fourier transform and its inherent truncation effects. It may therefore be viewed as part of a second NAH approach based on a convergent expansion of the radiated sound field as a series of orthogonal solutions of the homogeneous Helmholtz equation, such as cylindrical wave functions for the SONAH in cylindrical coordinates (Cho *et al*. 2005) or spherical wave functions for the Helmholtz equation least-squares (HELS) method (Wang & Wu 1997). This approach provides a methodology to transfer the sound field measured on the hologram surface onto another surface, including the source surface and, unlike Fourier-based methods, accounts for non-separable source geometries. The third NAH approach makes use of model-based methods as it requires the inversion of a spatial model of the transmission paths between a source distribution and its radiated field. Such a reconstruction technique is also suitable to recover the pressure or velocity field on an irregularly shaped surface. It includes the inverse boundary-element method (IBEM) and the equivalent source method (ESM). The IBEM uses either a direct (Veronesi & Maynard 1989) or an indirect (Schuhmacher *et al*. 2003) integral representation of the radiated pressure field with an unknown surface velocity interpolated at the nodes of each discretized boundary element. The ESM relies on an indirect formulation of the radiated pressure field in terms of single- or double-layer potentials, respectively, assuming a distribution of velocity monopole-type (Nelson 2001) or pressure dipole-type (Grace *et al*. 1996) source strengths. The ESM with high-order multipolar sources might be considered to improve the accuracy of the reconstructed source strengths. It is then equivalent to a decomposition of the sound field into spherical harmonics (Filippi *et al*. 1988) and is interestingly linked to the HELS technique.

In the literature dealing with model-based inverse approaches, a stable and accurate approximation of the unknown source strength is numerically sought in terms of the singular value decomposition (SVD) of the discretized forward operator that is an ill-posed linear compact operator. The SVD is a powerful tool that has been used in conjunction with the regularization techniques to solve a number of NAH problems, for instance to provide IBEM source reconstruction results robust to the presence of noise in the measured field data (Schuhmacher *et al*. 2003), extract dominant acoustic modes in the HELS method (Zhao & Wu 2005) or extend patch NAH to complex source geometries while avoiding the replication problem of the measurement window (Williams *et al*. 2003).

In the present study, closed-form expressions are found in terms of PSWFs when dealing with a model-based approach for the singular value expansion (SVE) of the radiation kernel for a baffled beam. Analytical approximations are proposed to extend the SVE of the radiation operator to the three-dimensional case of a baffled elastic panel. The theoretical framework herein developed provides a principled method of determining the number of independently radiating velocity patterns required to obtain a stable reconstruction of the boundary velocity. In particular, it is shown how the analytical SVE of the radiation kernel allows to gain further insight into the nature of the singular pressure and velocity modes (Nelson 2001; Williams 2001), the essential dimension of the radiating subspace (Borgiotti 1990) and the resolution properties of the reconstructed velocity distribution (Kim & Nelson 2003). An outline of the paper is as follows. Section 2 presents the acoustic source reconstruction problem and provides an analysis of its physical limitations in terms of accuracy and spatial resolution. How the band-limited PSWFs provide an exact singular function representation of a number of integral operators for the acoustic pressure radiated into far-field and intermediate domains in the two- and three-dimensional cases is shown in §3. The singular function expansions are used in §4 to determine the number of degrees of freedom (d.f.) of the radiated field and to obtain stable estimates of the reconstructed boundary velocity in the presence of noise for a number of regularization schemes. In particular, it is shown how the field dimensionality determines the spatial resolution limits of the acoustic source identification problem.

## 2. Formulation of the problem

### (a) The forward radiation problem

The two-dimensional radiation problem illustrated in figure 1 is considered: an elastic beam of length 2*L*′, set in an infinite baffle, is harmonically excited (with time dependence e^{iωt}) and radiates into a fluid at rest. A normal velocity distribution, *v*(*x*′), is specified over the length of the beam. One assumes that the radiated pressure is measured by a moving microphone at every point of the observation line of extent 2*L* along the *x*-axis and located at a variable height *z* above the baffled beam.

At any given frequency *ω*, the acoustic pressure field radiated in the upper domain *p* can be expressed in terms of the boundary velocity *v* as(2.1)where *G*^{+} is the two-dimensional Green function of the Helmholtz equation satisfying a Neumann boundary condition over the plane (*z*=0), which can be given as(2.2)where *ρ* is the fluid density; *c* is the sound speed in the fluid; *k*=*ω*/*c* is the acoustic wavenumber; and is the Hankel function of second kind and zero order. The distance between an observation point in the radiation domain and a source point on the beam is denoted by .

We observe that equation (2.1) is a convolution integral between Green's function (2.2) and the boundary velocity continued as a zero-valued function outside (−*L*′, *L*′). By virtue of the convolution and Fourier inversion theorems, the analytic continuation of *p* outside (−*L*, *L*) can be written as(2.3)where *k*_{x} is the spectrum variable, dual of the variable *x* in the wavenumber domain; *w*_{L′} is the space-limiting function equal to unity within its support (−*L*′, *L*′) and zero elsewhere; is the sensitivity function defined as the spatial Fourier transform of Green's function *G*^{+}; and *F*_{x}{*g*} is the spatial Fourier transform of the distribution *g*, which can be written as(2.4)

An expression of the sensitivity function is derived from the Sommerfeld integral representation of the Hankel function of the second kind when evaluated along a suitable integration path in the complex plane (Gradshteyn & Ryzhik 2007). It is given by(2.5)where for |*k*_{x}|<*k* and for |*k*_{x}|>*k*. *S*^{+} is bounded when |*k*_{x}|=*k* if one introduces a small negative imaginary part to the acoustic wavenumber such as . It ensures that Green's function exponentially decays at infinity and therefore complies with the radiation condition. Note that the integral representation (2.3) for the radiated field could be generalized in the complex plane, as shown in the electronic supplementary material, appendix A. However, one has considered only wavenumbers on the real line since it does not modify the main results of the paper.

The Rayleigh integral operator (2.1) can be interpreted in the wavenumber domain as a result of passing the velocity spectrum, which cannot be band limited owing to the finite spatial support of the velocity distribution, through the sensitivity function *S*^{+} that depends only on the physical properties of the fluid domain. The acoustic radiation filter (2.5) divides the wavenumber domain, respectively, into supersonic or propagating wave components and subsonic or evanescent wave components. The high spatial frequencies of the velocity distribution are clearly filtered out due to the exponential decay of the sensitivity function in the subsonic wavenumber region, i.e. when |*k*_{x}|>*k*. This low-pass filtering behaviour is all the more pronounced as the height of the observation domain above the radiator increases so that, when *z* is greater than the acoustic wavelength, the radiated pressure is band limited with bandwidth (−*k*, *k*).

### (b) Asymptotic approximations

Analytical approximations of the integral representation (2.1) for the radiated pressure field are derived that simply relate the acoustic pressure distribution to the beam velocity for a range of values of the separation distance *z* between the observation domain and the source. Assuming *z*≫*L*+*L*′, the Hankel function in (2.2) is asymptotically equivalent to(2.6)where with |*x*|≤*L* and |*x*′|≤*L*′. Substituting the zero-order approximation, *kr*≈*kz*, in the amplitude factor of (2.6), the radiated pressure field (2.1) is expressed as a spatial finite Fresnel transform of the source velocity distribution (Goodman 2005), namely(2.7)If one further assumes that *λz*≫*πL*^{′2} for the expansion of *kr* in the oscillating phase term of (2.6), then one may write with |*x*|≤*L* and |*x*′|≤*L*′, which results in an approximation of the radiated pressure field in terms of the spatial finite Fourier transform of the boundary velocity distribution, as follows:(2.8)where is the distance from an observation point to the origin, as depicted in figure 1. Alternatively, the asymptotic approximation (2.8) can be obtained by the method of stationary phase applied to the Fourier integral (2.3), the integrand of which is rapidly oscillating for large values of *kz*, or in the complex plane by the steepest descent method, as shown in the electronic supplementary material, appendix A.

Instead of considering an observation line of extent 2*L* on which the pressure field bandwidth is necessarily contained within the supersonic radiation bandwidth (−*k*, *k*), one may prefer a hemi-circular observation contour of radius *R*=*L* surrounding the vibrating beam and for which the angular spectrum encompasses all the radiating components, i.e. −*k*≤*k*_{x}≤*k*. It can be seen from figure 2 that the acoustic wavenumber and the observation vector share the same polar angle *θ* with and in which *k*_{x} spans the radiation bandwidth. Assuming *R*≫*L*′, the first-order approximation of the phase term in (2.6) is given as with |*θ*|≤*π*/2 and |*x*′|≤*L*′, so that an approximation of the radiated pressure field over the observation arc is given by(2.9)

Equations (2.8) and (2.9) provide a direct interpretation of the far-field pressure distribution in terms of the wavenumber components of the velocity distribution prescribed on the beam. However, they are valid only in the far-field zone (or radiation zone), i.e. in a region separated from the source by a distance sufficiently large compared with both the source dimensions and the acoustic wavelength, whereas (2.7) holds in an intermediate zone (or pre-radiation zone) that precedes the far zone but is still far enough from the source compared with the acoustic wavelength.

### (c) The acoustic source reconstruction problem

A model-based approach requires the inversion of the radiation operators (2.7)–(2.9) to retrieve the amplitude of the true velocity distribution *v*. These operators are associated with Fredholm linear integral equations of the first kind and their solutions are known to be ill-posed, i.e. to depend discontinuously on the measured data. More precisely, if one attempts to reconstruct the boundary velocity from the noisy pressure distribution , assumed to be measured over an observation line of infinite extent, can be expressed from (2.1) as , where asterisk stands for the convolution product and is assumed to be an additive spatially white noise term with zero mean and variance *σ*^{2}. The Fourier transform of the estimated solution is then given by(2.10)in which is the Fourier transform of *G*^{ −}, the kernel associated with the backward radiation operator. It is given by . In (2.10), represents the reconstructed noise-free velocity in the wavenumber domain and it coincides with the spatial Fourier transform of the true velocity, whatever the observation distance is. This results from the condition , where *w*_{∞} is the characteristic function of the real wavenumber line, which guarantees that the exponential decay in the subsonic components of the forward radiation operator is counterbalanced by their amplification in the backward operator. In practice, the subsonic wavenumber components of the measurement noise are amplified by the unbounded inverse operator *S*^{ −}, so that the cancellation process does not hold anymore. Such ill-posed behaviour of the solution is all the more important as the observation domain is several wavelengths apart from the beam.

If one considers a far-field observation domain, a minimum norm-regularized solution is obtained if one restricts the support of the backward sensitivity function *S*^{ −} to the range of *S*^{ +}, i.e. to the radiation bandwidth, so that the inverse Fourier transform of (2.10) is given by(2.11)where . It follows that the boundary velocity reconstructed from far-field data is a noisy band-limited approximation of the true velocity with the second term on the r.h.s. of (2.11) bounded by the noise power *σ*^{2} contained within the radiation bandwidth. If the far-field pressure is due to a point monopole source, the reconstructed velocity in the noise-free case is given by , which is first zero-valued when . This corresponds to the well-known Rayleigh resolution limit, *R*_{0}=*λ*/2, for which source details that are separated by less than a distance of one-half the acoustic wavelength cannot be recovered from far-field pressure data.

Details beyond the Rayleigh limit can be resolved if the sensors are located in the near-field of the beam, typically at a distance smaller than the acoustic wavelength, so that the high spatial frequency information, conveyed by the evanescent waves, can be sensed. This can be achieved despite the deleterious effect of the subsonic wavenumber components in the measurement noise. Indeed, from (2.10), the power spectrum of the velocity takes the form(2.12)where Exp is the expectation operator and for |*k*_{x}|≥*k*. The term |*S*^{−}|^{2} exponentially magnifies the subsonic spatial frequencies of the noise. However, this behaviour is moderated by introducing filtered approximations of the source distribution so that *σ*|*S*^{−}| stays below a certain error tolerance *δ* on the velocity amplitude outside the radiation bandwidth. A proper choice of the cut-off wavenumber, *k*_{c}, is critical to an accurate and well-resolved reconstruction of the velocity. This is achieved for(2.13)where *W*_{0} is the principal branch of the Lambert function *W* (Corless *et al*. 1996) that satisfies the functional equation and takes real values for real *u*≥−e^{−1}. A simple regularized solution is obtained after truncating the wavenumber content of the data field outside the cut-off bandwidth so that the reconstructed velocity reads like (2.11), in which *k* is substituted by *k*_{c}. It can be interpreted as a noisy band-limited estimate of the true velocity for which *k*_{c} should be chosen in such a way that the solution captures most of the wavenumber content of the source. The resolution distance *R*_{c} is now given by *π*/*k*_{c}. It decreases either when *kz* decreases or the signal-to-noise ratio (SNR) *δ*/*σ* increases. In the near-field limit when *kz*<1, *R*_{c} can be approximated by , which is of the order of the resolution law characteristic of the NAH, i.e. , as shown by Williams (1999). In the far-field limit when *kz*→∞, *kz* increases faster than , so that *k*_{c}→*k* and *R*_{c} tends towards the classical Rayleigh resolution limit *R*_{0}.

In practice, there are often limiting factors such as acoustic diffraction effects on the sensors, signal distortion or harsh environments, which limit the source–sensor separation distance to far-field or intermediate regions. A key point is then to determine the number of independent velocity source components that can be recovered accurately from band-limited pressure data acquired in these configurations. Section 3 shows how the PSWFs provide exact solutions to this class of acoustic source reconstruction problems.

## 3. Exact solutions to the inverse source reconstruction problem

The problem is to find an exact decomposition of the radiation operators (2.7)–(2.9) onto the corresponding sets of singular radiation and velocity patterns. The invariance properties satisfied by the PSWFs provide a solution to this problem since the radiation integral operators are directly related to the spatial finite Fourier (respectively Fresnel) transforms of the source velocity distribution, up to a scale and amplitude factor. The closed-form expressions are also provided for the SVE of the radiation operators extended to the three-dimensional case. Note that the theoretical background for the SVE of the radiation operators is found in the electronic supplementary material, appendix B. A number of relevant mathematical properties satisfied by the PSWFs are included in the electronic supplementary material, appendix C.

### (a) Exact singular function representation of the radiation operators

Let *R* denote the far-field (respectively intermediate) radiation operators such that the boundary velocity *v* and its radiated field *p* satisfy *p*=*Rv*. From eqn (B.1) in the electronic supplementary material, the singular radiation and velocity patterns, respectively, *u*_{n} and *v*_{n}, are solutions of , with *σ*_{n} as the corresponding singular value.

Observation line in the far-field region.

Referring to the geometry shown in figure 1, the singular system of the far-field radiation operator (2.8) is deduced from the self-reproducing property (C.3) of the PSWFs after the scale change of variables . It is given by(3.1)where *ψ*_{n} is the *n*th normalized PSWF; is the angular space-bandwidth parameter; and *z* is the height of the observation line above the beam. Simulations have been performed at 2 kHz, assuming a baffled elastic beam of length 2*L*′=4*λ* in harmonic motion and a rectilinear observation domain of extent 2*L*=8*λ*, located at a height *z*=100*λ* above the radiator. Figure 3 shows the exponential decay (on a logarithmic scale) of the first singular values associated with the far-field radiation operator (2.8) and determined from the analytic expression (3.1). Their values coincide with those computed from an SVD of the acoustic transfer matrix that relates the amplitudes of the far-field pressures obtained at 81 observation points evenly spaced over the linear array of microphones to the strengths of 81 monopoles uniformly distributed over the beam length.

This number of field and source points clearly overestimates the minimum number of points set by the Nyquist sampling theorem to recover the pressure and velocity distributions from their sampled values, which is equal to the integer part of , denoted by . It would correspond to 17 field points and 9 source points at 2 kHz. A larger number of points have been chosen for the numerical solution to achieve convergence on the first five SVD-based singular functions with a mean-square spatial relative error less than 2%. This can be seen from figure 4, which shows the first five normalized singular velocity and radiation patterns obtained from (3.1), together with their SVD-based numerical approximations that remarkably superimpose onto the distribution of the corresponding analytical singular functions. Owing to the small values taken by *γ*_{R}, one observes that the first three normalized singular functions capture most of the source and field information contents.

Hemi-circular observation arc in the far-field region.

As a further consequence of the invariance property (C.3), the singular system of the far-field radiation operator (2.9) for a hemi-circular observation arc is found to be(3.2)where . The singular velocity patterns *v*_{n} are the so-called ‘radiation modes’, solutions of the eigenvalue problem, , where *μ*_{n} is the radiation efficiency of the *n*th radiation mode and *R*^{*} is the adjoint operator of *R*. The radiation modes are known to maximize the radiation efficiency ratio *μ*_{n} of the radiator among all band-limited surface velocity distributions (Maury & Elliott 2005). They may also be regarded as a set of singular functions that retain a certain amount of ‘information’ under the forward radiation transformation (2.9), which relates the pressure radiated in a far-field hemi-circular domain to the source velocity distribution. In this sense, the radiation efficiency of the vibrating structure quantifies the extent to which a square-integrable boundary velocity distribution and its spatial Fourier transforms can be simultaneously concentrated.

One observes when comparing (3.2) here and eqn (4.7) in Maury & Elliott (2005) that the inverse radiation problem over a far-field hemi-circular arc and the radiation modes problem share the same angular space-bandwidth parameter *γ*_{H}. Indeed, left-iterating the Fourier operator (2.9) leads to the integral operator associated with the bilinear form of the active sound power *W*(*γ*_{H}) radiated by a baffled beam of boundary velocity *v* and is given by with the scalar product defined in the electronic supplementary material, appendix B. It also corresponds to the sound power radiated within a far-field hemi-circular arc, given as , due to the property , where and . Physically, the angular-space bandwidth parameter scales on the number of efficiently radiating singular velocity patterns that significantly contribute to the net flow of acoustic energy integrated over a far-field hemi-circular arc surrounding the baffled beam or, equivalently, over the source line. As expected, it does not depend on the radius of the far-field observation arc.

Figure 5*a–f* depicts the beaming properties of the first six supersonic radiation patterns, as given by the exact expression of *u*_{n} in (3.2). They closely agree with their numerical approximations, i.e. the left singular vectors of the transfer matrix between the acoustic pressure sampled over 81 points positioned at equal increments of sin *θ* in the far-field arc of radius *R*=100*λ* and the strengths of 81 monopoles uniformly distributed over the beam of length 2*L*′=4*λ*. It is shown in figure 5 that each supersonic radiation pattern beams in a particular direction within each quadrant, with the least radiating modes beaming towards grazing angles from the source distribution. One infers that, for compact source distributions, it would be sufficient to use a hemi-circular array of microphones with an angular aperture limited to the largest beaming direction covered by the finite number of supersonic radiation patterns, as determined by the angular space-bandwidth parameter *γ*_{H}. As expected, the number of secondary lobes increases as the radiation pattern order increases. It reflects the high spatial frequency content of the high-order radiation patterns.

Observation line in the intermediate region.

In the intermediate zone, the pressure radiated (2.7) is a finite Fresnel transform of the source velocity distribution. Identification of (C.5) with (2.7) leads to the following singular system for the radiation operator in the intermediate region:(3.3)where , the same angular space-bandwidth parameter as for the far-field rectilinear case. The SVD has been calculated from the transfer matrix between 81 field and source points associated with a discrete form of the asymptotic approximation (2.7) of the radiation operator in the pre-radiation zone. Figure 6 shows that both the real and imaginary parts of the first five left and right singular vectors agree well with the exact values (3.3) of the corresponding singular velocity and radiation functions. The variations of the singular radiation patterns *u*_{n} along the observation line can be interpreted as the pressure distribution across *z* of a diverging cylindrical wave with its axis on *z*=0, with wavefront curvature 1/*λz* and whose amplitude is modulated by the corresponding PSWF *ψ*_{n}. A similar interpretation holds for the variations of the singular velocity patterns *v*_{n} in terms of the source distribution across *z*=0 of converging cylindrical waves originating from quota *z* on the observation line.

A few observations are in order. The influence of the height of the observation line above the elastic beam results in a greater angular space-bandwidth parameter *γ*_{R} in the intermediate case with respect to the far-field case. This scales on the inverse ratio of the respective source–receiver separation distances. As illustrated in figure 3, a greater number of singular radiation (respectively velocity) functions efficiently contribute to the representation of the acoustic pressure (respectively boundary velocity) field in the intermediate case with respect to the far-field case. Moreover, these singular functions are band limited of bandwidth *γ*_{R}, as shown by (C.6). Therefore, they capture a broader range of spatial frequencies in the intermediate case with respect to the far-field case with the highest spatial frequencies being obtained from far-field pressure data acquired over a hemi-circular arc.

The condition number of the radiation transfer matrix, defined as the ratio of the largest singular value to the smallest, is an indicator of the degree of ill-posedness of the inverse source reconstruction problem. Numerically, the lowest condition number is found for a hemi-circular observation domain and the difference in the condition numbers between the hemi-circular and rectilinear cases is more accentuated in the low-frequency range. To gain further insight into this point, analytical approximations are derived from the exact expressions (3.1) and (3.2) of the transfer matrix singular values when *γ*_{H}≪*π*/2, i.e. *λ*≫4*L*′, for the ratio of the condition number in the hemi-circular case to that in the rectilinear case, namely *C*_{n,H}/*C*_{n,R}, where *n* is the number of source points. It can be written as(3.4)since when *γ*≪*π*/2 and , where *P*_{n} is the Legendre polynomial of order *n*, with *γ*=*γ*_{R} or *γ*_{H}. *κ*_{0n}(*γ*) is the joining factor introduced in the electronic supplementary material, appendix C. It can be expanded as a power series of *γ*, as detailed by Maury & Elliott (2005) so that the leading-order approximation of the condition number ratios is given by(3.5)

For compact sources with extent 2*L*′≪*λ*/2, given a rectilinear domain with extent 2*L* and height *z* above the source line, such that *z*≫*L*, and a hemi-circular domain of radius *R*=*z*, the ratio between the condition numbers in either case scales on the ratio between the corresponding space-bandwidth parameters and thus on the small angular aperture subtended by the rectilinear observation domain, so that . Therefore, under far-field conditions, the degree of ill-posedness of the inverse source problem can be significantly reduced when considering a hemi-circular observation domain with respect to a rectilinear one. One should however bear in mind that the radiation operators (2.7)–(2.9) stay severely ill-posed due to the exponential decay of their singular values to zero as a consequence of the analyticity of the associated kernel *K*.

### (b) Extension to the three-dimensional case

Following the derivation presented in §2*b*, it can be shown that the acoustic pressure radiated in the intermediate zone by a baffled panel with dimensions over a planar observation domain with dimensions 2*L*_{x}×2*L*_{y} is given by(3.6)with the corresponding far-field approximation in terms of the two-dimensional Fourier transform of the panel velocity(3.7)where is the panel space-limiting function and . The acoustic pressure field approximated over a far-field hemi-spherical observation domain is obtained from (3.7) with the following change of variables .

Owing to the separability of the Fourier kernel in Cartesian coordinates, one obtains, from (3.1), the singular system for the far-field radiation operator (3.7),(3.8)where the angular space-bandwidth parameter for an observation plane and for a hemi-spherical domain, in which case the normalized PSWF *ψ*_{m} and *ψ*_{n} for the singular function *u*_{mn} in (3.8) are functions of sin *θ* cos *φ* and sin *θ* sin *φ*, respectively. From (3.3), one finds the singular system for the radiation operator (3.6) in the intermediate region,(3.9)where and . These singular functions form a basis for the reconstruction of the velocity of a baffled panel radiating onto a planar or hemi-spherical grid of far-field or intermediate microphones.

## 4. Resolution properties of acoustic source reconstruction problems

### (a) The number of d.f. of the radiated field

A truncated approximation *v*_{N} to the true velocity *v* is extracted from (B.4). It can be written as(4.1)where *N* is the number of d.f. of the noise-free radiated pressure field *p*, i.e. the minimum number of independent singular components required to represent the band-limited radiated pressure field (Borgiotti 1990). It is shown in the electronic supplementary material, appendix C, that for any given value of the angular space-bandwidth parameter *γ*, the number of singular radiation patterns whose energy lies within the radiation bandwidth equals ⌊2*γ*/*π*⌋. Analytical expressions are then obtained for the number of d.f. of the pressure field radiated over the far-field and intermediate lines, namely , and over the far-field hemi-circular arc, namely . In the three-dimensional case, the number of d.f. is given by , respectively .

The number of d.f. can alternatively be deduced from physical arguments based on the limited aperture of both the source and the observation lines that introduce a cut-off in the spatial frequency bandwidth of the wave field radiated by the source (Bravo & Maury 2006). Indeed, the largest bandwidth generated by the boundary velocity is reached at the central microphone position and is equal to . According to Shannon's sampling theorem, the pressure distribution can be determined through its samples taken at a rate of(4.2)which also corresponds to the number of d.f. In the far-field limit, when *z*≫*L*, (4.2) reduces to *N*_{R}. When the extent of the observation domain *L* tends to infinity, the number of d.f. (4.2) tends towards *N*_{H}, the number of d.f. over a far-field hemi-circular arc.

Equation (4.2) constitutes a criterion on the number of independent components from which one can extract information about the unknown source velocity distribution and hence obtain a stable regularized solution to the inverse source problem under consideration. On the other hand, ensuring a stable solution may alter the ability to resolve details in the restored source velocity distribution. Indeed, the fine details in the resolution information are contained in the singular functions associated with the smallest singular values and are likely to be filtered out in the stable solution (4.1). Therefore, a criterion is sought on the spatial resolution limit that can be achieved for the reconstruction of a monopole point source, which is the worst case scenario. From (4.1) and (4.2), a stable estimate of a monopole source located at and retrieved from noise-free pressure data is expressed as(4.3)The average distance between two consecutive zeros of the highest order singular function *v*_{N} retained in (4.3) is chosen as a measure of the spatial resolution limit. *v*_{N} has exactly *N* zeros within (−*L*′, *L*′). Assuming that the distance between two consecutive zeros and of *v*_{n}(*γ*, *x*′) is approximately constant, which can be observed in figure 4, an estimate of the spatial resolution limit that can be achieved in the noise-free case is given by(4.4)which reduces to , when the data are acquired from a far-field or intermediate line of extent 2*L* placed at a height *z* above the source plane, and to when the data are acquired from a far-field hemi-circular arc.

### (b) Acoustic source reconstruction from noisy pressure data

In practice, the measured pressure data are perturbed by unavoidable noise components, so that , the range of *R*. Therefore, does not satisfy Picard's condition referred to in the electronic supplementary material, appendix B, since the variance of the coefficients level off, for small |*σ*_{n}|, at the noise power spectrum . As a result, the participation factors associated with the smallest singular values are greatly amplified in an attempt to reconstruct the boundary velocity. This lack of stability, already quoted in §2*c*, is mitigated by introducing filter factors, such as when *α*→0, so that the contribution to the solution of the smallest singular values is effectively damped out. A filtered approximation to the reconstructed velocity is given by(4.5)where when using a direct Tikhonov regularization scheme with an appropriate choice of the parameter *α*>0. Iterative regularization methods such as the Landweber iteration can be favourable alternatives for source and field distributions discretized over a large number of points or elements. The filter factors for a Landweber iteration are given as , in which *κ* is a positive convergence parameter and the iteration number *m* plays the role of a regularization parameter. Assuming an observation line of infinite extent, an exact expression is derived from (2.13) for the cut-off wavenumber *k*_{c} of the filtered approximation (4.5). It corresponds to the index for the singular values, which can be given as(4.6)where for a Tikhonov regularization and for a Landweber iteration. The regularization parameters *α* and *m* are determined from a suitable parameter-choice technique (Hansen 1998). The discrepancy principle of Morozov involves choosing the parameter such that the residual norm of the regularized solution equals the noise variance *σ*^{2} and is only used if *σ*^{2} is known *a priori*. When no prior knowledge about the level of noise contamination is available, the L-curve and the generalized cross-validation techniques are recommended.

A step-like distribution of the singular values associated with the far-field and intermediate radiation operators was shown in §3*a*. In this case, a regularization scheme based on the truncated SVE of the operator appears to be well suited to provide a stable reconstruction of the boundary velocity from noisy pressure data,(4.7)where *M* is an appropriate truncation parameter. From (4.7), the variances of the SVE coefficients are related through(4.8)It follows that one can reconstruct only the boundary velocity components for which the variance is greater than the variance in the reconstruction of the noise, . Therefore, the parameter *M* should be chosen such that , where *δ* is a given tolerance on errors in the reconstructed velocity. Using an argument similar to the noise-free case, a measure of the spatial resolution limit is given by , where *M* corresponds to an ‘effective’ number of d.f., which is a function of the SNR *δ*/*σ*. It is given by (4.6) with *β*=*δ*/*σ* and is termed the essential dimension of the radiation operator. It provides an informative measure of the field dimensionality in the presence of noise.

Figure 7 shows the boundary velocity distribution of a harmonic piston-like source of unit amplitude when reconstructed from (4.7) over a broad frequency range (from 2 to 16 kHz) from far-field pressure data acquired either over a line or a hemi-circular arc. One observes that, for each type of observation domain, the spatial resolution of the reconstructed velocity improves as frequency increases, due to an increase in the frequency of the number of d.f. of the radiated field. Moreover, at any given frequency, figure 7*a*,*b* shows that the reconstruction from a hemi-circular arc of radius *R* enables far greater resolution information about the source distribution to be revealed than reconstruction from a rectilinear array of quota, *z*=*R*, with a necessary limited aperture, 2*L*/*z*. This quantity is linked to the resolution loss with respect to the Rayleigh resolution limit, i.e. , the limit being achieved with far-field data acquired over a hemi-circular arc, which corresponds to figure 7*b*. Figure 7*c* shows the effect of noise contamination on the reconstruction of the piston-like boundary velocity. As the singular values decay exponentially beyond *N*_{H}, the essential dimension *M* of the radiation operator is at most equal to *N*_{H}+1. In the simulations, the equality is reached with a SNR of 10^{3}. The presence of noise then leads to almost the same resolution as in the noise-free case; however, this is at the expense of the accuracy in the reconstruction, as can be appreciated when comparing figure 7*b*,*c*.

## 5. Conclusions

A formal analogy has been pointed out between the problem of determining the singular radiation and velocity patterns of a baffled planar vibrating structure and the invariance properties satisfied by the PSWFs under Fourier and Fresnel integral transforms. Analytical expressions have been obtained for the SVE of the radiation operators that map the boundary velocity of the baffled structure onto the pressure distribution radiated over a number of observation domains in the far-field and intermediate regions. In particular, it is shown that the radiation modes of a baffled planar structure correspond to the singular velocity patterns of the operator governing the acoustic radiation over a far-field hemi-circular arc surrounding the structure.

The closed-form expressions of the singular system are found to correlate well with the numerical solutions obtained from an SVD of associated radiation matrices, thus providing a reference case against which the effect of spatially sampling the source and field data can be assessed in terms of convergence of the singular system of the discrete problem. Moreover, the condition number of the far-field radiation matrix is found to be significantly reduced when considering a hemi-circular array with respect to a rectilinear one, by a factor scaling on the small angular aperture subtended by the observation line.

Stable estimates of the reconstructed boundary velocity in the presence of noise have been obtained for Tikhonov and Landweber regularization schemes, with an exact expression of the cut-off wavenumber for the corresponding filter factors in terms of the regularization parameter. A regularization scheme based on the truncated SVE is well suited for source reconstruction from far-field or intermediate regions, and a stable solution is found by decomposing the unknown boundary velocity into a finite number of efficiently radiating singular velocity patterns. They correspond to the number of d.f. of the radiated field for which exact expressions have been deduced from the singular system of the radiation operators in the far-field and intermediate regions. This number scales on the spatial frequency bandwidth of the radiated pressure field. Its inverse yields to an estimate of the spatial resolution limits that can be achieved in the inverse source reconstruction problem for different configurations of the observation domains. In particular, the smallest resolution loss with respect to the Rayleigh resolution limit is reached when one attempts to reconstruct the boundary velocity from pressure data acquired over a hemi-circular arc. This result is shown to be weakly influenced by the presence of spatially white noise in the far-field data.

Although the closed-form singular systems derived in §3 optimally describe the far-field or intermediate pressure radiated by baffled beams or panels, the PSWFs are *a priori* not adapted to decompose the pressure radiated by unbaffled vibrating structures of arbitrary geometry and for which a convergent expansion as a series of cylindrical or spherical wave functions would be more appropriate. This is a practical limitation of the proposed PSWFs expansion for which convergence holds only if the source and field distributions are Fourier or Fresnel transforms of each other. For instance, such closed-form expressions would provide an exact solution to the inverse problem of reconstructing aero-acoustic broadband sources at the origin of the noise radiated from turbojet engines when using a polar correlation technique (Fisher *et al*. 1977). Indeed, this technique is based on a spatial Fourier transform that relates the cross-spectral density between pressure measurements acquired over a far-field hemi-circular domain to the unknown cross-spectral density between the strengths of a line source distribution on the jet flow axis. The exact solution would thus give further insight into practical methods of aero-acoustic source characterization for jet engine applications.

As already observed by Veronesi & Maynard (1989), a formal similarity, but not a strict equivalence, exists between the SVD- and Fourier-based approaches for noise source reconstruction problems, as it is observed between (4.8) and (2.12) as well as between (B.3) and the relationship , deduced from (2.3) to (2.5) and which linearly relates the Fourier components of the radiated pressure field to those of the source velocity for any given spatial frequency. The SVD approach is based on a decomposition of the spatially bounded source and field distributions as a convergent series of the singular functions of the radiation operator. The Fourier-based approach assumes a representation of the source and field distributions of infinite extent (but necessarily truncated in practice) over a continuum of supersonic and subsonic spatial frequencies with Fourier modes as the corresponding singular functions (Williams 1999). If one assumes that the radiator and the observation domains are spatially unbounded (so that *γ*_{R} tends to infinity) or with periodic boundary conditions, the radiation operator is translation invariant and spatial homogeneity occurs. The SVE of the operator then coincides with its Fourier representation, so that the SVD approach can indeed be considered as a generalization of Fourier-based NAH methods for the reconstruction of the vibrating surface of any boundary-value problem when the field satisfies a Helmholtz wave equation.

## Acknowledgments

T.B. is supported by a Marie-Curie Intra-European Fellowship granted by the European Commission under contract MEIF-CT-2006-022579.

## Footnotes

Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2007.0369 or via http://journals.royalsociety.org.

- Received December 16, 2007.
- Accepted February 20, 2008.

- © 2008 The Royal Society