Spherical Nearfield Acoustical Holography

Transcription

1 Author: Guillermo Moreno Spherical Nearfield Acoustical Holography Supervisor: Finn Jacobsen Acoustic Technology Building 352 Kongens Lyngby Master s Thesis October 17, 2008

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3 To my family, because they consciously and honestly contributed to making me what I am today.

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5 Abstract Spherical near field acoustical holography (SNAH) is a recently developed technique that allows to reconstruct the sound field inside and in the surroundings of a spherical array. Differently from previous works, this thesis presents the theory for SNAH using a rigid spherical array instead of a transparent one. Rigid spheres are also used in Beamforming which focuses on the far field. Due to the near field nature of SNAH there is potential problem of a solid spherical array producing reflections back to the microphones and therefore influencing the reconstructions. Numerical simulations and experimental measurements carried out along this work, illustrate the possible limitations of SNAH using the modified theory for a rigid spherical array. The results obtained from both simulations and experimental measurements show the minimum influence of the reflections in the reconstruction accuracy. i

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7 Acknowledgements First of all i would like to thank my supervisor Finn Jacobsen, for his priceless support along the whole master thesis and his marvelous Advanced Acoustics course. I would also like to thank Karim Haddad, Jørgen Hald and Jesper Gomes from Bruel & Kjær, for their guidance, and for their help providing the necessary equipment for the experimental. Several people (apart from my beloved bikes) have played a decisive role in ensuring my punctual arrival to meetings, laboratories, exams and lectures along this two years of master degree. These people are; Jordan Diez, Giorgos Kostopoulos and Adrien Roux. Thank you for waking me up on the many occasions it has been necessary during our time living together, without you, this master degree would have taken another extra year. I cannot finish without mentioning Marton, Efren, Tarmo and Julien who accompanied me on the master program together with Adrien and Giorgos during the last two years. Thank you for your support and friendship. iii

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9 Table of Contents List of Figures viii List of Symbols 1 1 Introduction 1 2 Spherical Nearfield Acoustic Holography (SNAH) Spherical waves Spherical Bessel functions Spherical Harmonics Scattering from a sphere Scattering from a rigid sphere Reconstruction Equations Pressure Particle velocity and Intensity Numerical integration Discussion Simulation study Pressure on the 50-microphone rigid spherical array Point sources Single monopole source Two monopole sources Rigid vs. transparent spherical array Particle velocity and Intensity Baffled plate source Rectangular finite simply supported plate Radiation of a baffled rectangular simply supported driven plate Reconstruction without reflections First order reflections simulation Validation of the model Discussion v

10 vi TABLE OF CONTENTS 4 Experimental results Measurement equipment Validation of the measurements Monopole Results Two monopoles Results Vibrating plate Results Discussion Conclusion and perspectives 43 Bibliography 45 Appendix 45 A Matlab codes 47 A.1 Weighting factors A.2 Spherical Harmonics A.3 Spherical Bessel functions A.4 Spherical Hankel functions A.5 Pressure on the 50 microphones from a monopole A.6 Pressure on the 50 microphones from a baffled monopole with reflections.. 53 A.7 Plate displacement A.8 Pressure reconstruction A.9 Particle velocity reconstruction A.10 FFT processing B Baffled vibrating plate 65 C Measurements images 67

11 List of Figures 2.1 Definition of the spherical coordinate system used Spherical Bessel functions, j n (ka) for n=0, 1, 2, 3, 4, Total sound field p t in the surroundings of a solid sphere Pressure on the surface of the rigid spherical array Reconstructed sound pressure on a concentric spherical surface of radius 15cm Reconstructed sound pressure with noise, on a spherical surface of radius 15cm Configuration of the Matlab simulation scenario. One monopole radiating Reconstruction of a monopole sound field at 20cm distance, ka= Reconstruction of a monopole sound field at 20cm distance, ka=3, Reconstruction of a monopole sound field at 1m distance, ka=2, Frequency dependency reconstruction for a source 20cm away Frequency dependency reconstruction for a source 20cm and 30 cm away Configuration of the Matlab simulation scenario. Two monopoles radiating Reconstruction of two monopole sound field at 20cm distance, ka=1, Reconstruction of two monopole sound field at 40cm distance, ka=2, Comparisson between rigid and transparent array Frequency dependency reconstruction of rigid and transparent arrays Intensity reconstruction Normal displacement and normal velocity of a simply supported steel plate Simulated sound pressure level generated by a vibrating steel plate Configuration for the vibrating steel plate simulations Sound field reconstruction of a vibrationg plate withka= Sound field reconstruction of a vibrating plate with ka=2 and Sound field reconstruction of a vibrating plate with ka=7 and Frequency dependence at the origin of coordinates at 20cm distance Frequency dependence at a point further two times the radius of the sphere Backscattered reflections calculation principle Reflections reconstruction. Frequency ka= Reflections reconstruction. Frequency ka= Reflections reconstruction. Frequency ka= Reflections reconstruction. Frequency ka= vii

12 viii LIST OF FIGURES 3.29 Validation model illustration Reconstruction from the model developed for the simulation Reconstruction from the validation model with the two monopoles Spherical rigid array Fifty microphones frequency spectrum Coherence between two different microphones in the array Measurement set up for the single monopole sources case Measured and reconstructed pressure along the y axis at 2000 Hz frequency Measured and reconstructed pressure along the y axis at different frequencies Measured and reconstructed spectrum for monopole source Measurement set up for the two monopole sources case Measured and reconstructed spectrum for a point 10 cm away of the sphere Measurement set up for the vibrating plate source case Measured and reconstructed spectrum for vibrating plate B.1 Baffled plate simulations ka= B.2 Baffled plate simulations ka= C.1 Set up for the measurement of the monopole source C.2 Set up for the measurement of the monopole source C.3 Set up for the measurement vibrating plate C.4 Set up for the measurement vibrating plate

13 List of Symbols a radius of the spherical array [m] A nm coefficient in spherical expansion A p area of the plate [m 2 ] B nm coefficient in spherical expansion C nm coefficient in spherical expansion c speed of sound [m/s] D nm coefficient in spherical expansion d n e i E f F h h 1 n h 2 n h n H n distance for the plate to the point of calculation [m] quadrature weighting factors Young s modulus [Pa] frequency [Hz] force [N] thickness of the plate [m] Spherical Hankel function of the second kind of order n spherical Hankel function of the first kind of order n spherical Hankel function of the first kind of order n cylindrical Hankel function of the first kind of order n I i Sound intensity [W/m 2 ] j complex root j n J n spherical Bessel function of order n cylindrical Bessel function of order n k wave number [m 1 ] Dimension of the plate along x axis [m] L x L y p p i p s p bs p t p T P m n Dimension of the plate along y axis [m] sound pressure [Pa] incident sound field [Pa] scattered sound field [Pa] backscattered sound field [Pa] total sound field [Pa] total sound field with reflections [Pa] Lengendre functions of degree n and order m P nm spherical harmonics coefficients ix

14 x LIST OF FIGURES Q s Q n Q m n r r s source volume velocity [m 3 /s] numerical integration coefficients] Lengendre functions of the second kind of degree n and order m radial distance in spherical coordinate system source radial distance in spherical coordinate system S n area [m 2 ] t time [s] u i u i,r incident particle velocity [m/s] component of the incident particle velocity [m/s] u i,φ component of the incident particle velocity [m/s] u i,θ component of the incident particle velocity [m/s] v n w i normal velocity of the plate at a certain point [m/s] weighting factors w(x, y,.. z) plate s normal displacement [m] x o Y m n y o rectangular coordinate of the driving point for the plate [m] spherical harmonics of degree n and order m rectangular coordinate of the driving point for the plate [m] α Skudrzyk s constant [] δ nm Kronecker delta ρ density [Kgm 3 ] ρ p density of the plate [Kgm 3 ] φ azimuthal angle in spherical coordinate system [radian] φ s source azimuthal angle in spherical coordinate system [radian] θ polar angle in spherical coordinate system [radian] θ s source polar angle in spherical coordinate system [radian] ω angular frequency [radian/s] ω mn natural angular frequencies [radian/s] gradient ν Poisson s number Ω solid angle [sr] ˆ Ξ Orthonormal modes of the plate vector representation of a harmonic variable complex representation of a harmonic variable

15 1 Introduction Beamforming has become an usual technique in noise source identification, and consequently the usage of microphone arrays. More precisely, spherical beamforming has become recently used in several applications due to its advantages in comparison with the planar, which the directions of identifications are limited due to the nature of the array. Although, beamforming is based on far field approximations and therefore is only valid for the far field. Instead, near field acoustical holography (NAH) does not make use of any kind of far field approximation and thus is suitable for the reconstruction of the sound field between the array and the source (inverse and ill-posed problem) and in a volume inside the spherical array (well-posed and forward problem). In the past few years, spherical near field acoustical holography (SNAH) has gained attention from the acoustic community. The publication of remarkable studies in the field [1] had revealed the enormous potential of this new technique yet to be developed. However, it was based on an transparent spherical array assuming not disturbance from the array in the sound field. Obviously, this assumption will not be proper at high frequencies. On the contrary a rigid spherical array exhibit a more robust boundary condition for all frequency range. There is a potential problem in placing the rigid spherical array near the source coming from the reflection that may occur during the measurement. Along this master thesis, the new formulation for the usage of a rigid spherical array for SNAH is presented (section 2). During section 3 multiple simulations will be carried out until finishing with a model that simulates the reflections from an infinite baffled plate is studied. Finally in chapter 4, the conclusions extracted from the simulations are tested under real conditions in some experimental measurements. The rigid spherical array used was provided by the Bruel & Kjær and consisted on a fifty flushed-mounted microphones on a plastic sphere with eleven cameras meant to be used with the beamforming. It is an attractive idea that the same spherical array could be used for both spherical beamforming and holography thus combining far and near field reconstructions in one device. For example, as it is shown in [2] for a transparent sphere or more recently in [3] for a rigid array, the spatial intensity information can be entirely reproduced in a volume outside the array which is the same type of spherical array used in beamforming. 1

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17 Spherical Nearfield Acoustic Holography (SNAH) 2 In the following chapter, an introduction to the theory behind SNAH is presented, together with the necessary tools to understand it, starting from the wave equation in spherical coordinates and ending in the reconstruction equations when a spherical rigid array is used. Most of the equations in the first sections of this chapter were taken from [1]. Therefore not all of the equations will be referenced in the running text. Figure 2.1 Definition of the spherical coordinate system used relative to a Cartesian coordinate system.θ is measured from the polar axis, z.φ is measured in the x, y plane (lined surface) from the x axis. 2.1 Spherical waves In figure 2.1, the spherical coordinate system used along this project is defined. The relation between the Cartesian and spherical coordinate systems is given in the following equations x=r sinθcosφ, y=r sinθsinφ, z=r cosθ, the wave equation can be now rewritten into spherical coordinates as ( 1 r 2 r 2 p ) + r r 1 r 2 sinθ ( sinθ p ) + θ θ 3 (2.1) 1 2 p 1 2 p r 2 sin 2 θ φ 2 c 2 = 0. (2.2) t2

18 4 2. Spherical Nearfield Acoustic Holography (SNAH) Solving equation (2.2) in detail it is well described in the literature (e.g [1] or [4]). Later, in section 2.3, a similar derivation necessary to achieve the reconstruction equation is presented in detail. Thus, the solution is given briefly in this text. The solution of eq. (2.2) is given by separation of variables: which leads to the following differential equations: p(r, θ, φ, t) = R(r)Θ(θ)Φ(φ)T(t), (2.3) ( 1 d sinθ dθ sinθ dθ dθ ( 1 d r 2 dr ) + r 2 dr dr d 2 Φ dφ 2+ m2 Φ=0, (2.4) ] [n (n+1) m2 Θ=0, (2.5) sin 2 θ ) + k 2 R n(n+1) r 2 R=0, (2.6) 1 d 2 T c 2 dt 2 + k2 T= 0. (2.7) In the next section the properties of the solution of these equations shown above will be demostrated. As can be already anticipated, there will be a relation between the angles solutions eq. (2.5) and eq. (2.4), the socalled Spherical Harmonics. The time dependence eq. (2.7), has the simple solution of T(t)=e jωt, and the radius dependence solution eq. (2.6) is given as a function of spherical Bessel and Hankel functions Spherical Bessel functions The solution to the differential equation (2.6) is given by R n (r)= A n r 1/2 J n+1/2(kr)+ B n r 1/2 Y n+1/2(kr), (2.8) where J n+1/2 (kr) and Y n+1/2 (kr) are cylindrical Bessel functions of the first and second kind respectively. The corresponding spherical Bessel functions are defined in terms of these solutions: j n (x)= y n (x)= π 2x J n+1/2(x) π 2x Y n+1/2(x). (2.9) Another important formulation of the Bessel functions are the Hankel functions which are defines as a linear combination of the Bessel functions: h (1) n (x)= j n (x)+ jy n (x) h (2) n (x)= j n (x) jy n (x). (2.10) The Hankel functions of the first and second kind (eq. (2.10)) are used to express an outgoing or incoming wave respectively, depending on the sign convention. Usually in the literature, these functions are referred as radial functions, since they represent the solution of the radial dependent differential equation. Notice that, the spherical Bessel The solution for eq. (2.7) has the form T(ω)=T 1 e jωt + T 2 e jωt. Here was choosen T 1 = 0

19 2.1. Spherical waves 5 j n (ka) n=0 n=1 n=2 n=3 n=4 n= ka Figure 2.2 Spherical Bessel functions, j n (ka) for n=0, 1, 2, 3, 4, 5. With fixed radius of the spherical array a= metres. functions are real values and therefore they could be zero -see fig On the other hand, the spherical Hankel functions are complex values, which cannot become zero. This fact become more important when the rigid sphere is compared against the transparent sphere, discussed in the next section. It is possible to rewrite the equation (2.8) in terms of the spherical Bessel and Hankel functions: R n (r)=r 1 j n (kr)+r 1 y n (kr), R n (r)=r 3 h (1) n (kr)+r 2 h (2) n (kr), (2.11) depending on the location of the sources in the problem under study the proper expression from the equations on (2.11) will be chosen. It is important to mention that the sign convention used in this master thesis is e jωt. Therefore, all the Hankel functions (h n ) mentioned in the following sections are of the first kind, describing an outgoing wave Spherical Harmonics After applying the separation of variables method for solving equation 2.2, four differential equations were derived. In section 2.1.1, the radius dependent equation was solved. In this section the angle dependent differential equations (2.5) and (2.4) will be solved. In this way, both solutions are combined to find a unique expression that represents the dependency of both azimuthal and polar angle. The solutions of equation (2.4) are the 2π periodic functions cos mφ and sin mφ, thus m must be and integer so there is continuity and periodicity, Φ(φ)=Φ 1 cos(mφ)+φ 2 sin(mφ). (2.12)

20 6 2. Spherical Nearfield Acoustic Holography (SNAH) After a variable transformation (γ = cos θ) in the equation (2.5), the associated Legendre differential equation is found -see [4]. Its solutions are the associated Legendre functions P m n (γ) as the is denoted in the equation (2.13). Θ(θ)=Θ 1 P m n (cosθ)+θ 2 Q m n (cosθ). (2.13) The associated Legendre function P m n (x) is defined in terms of the mth derivative of the Legendre polynomial (P n (x)) -see [1] eq. (6.29). For negative m s is defined that P m n (x)=( 1) m (n+m)! (n m)! P m n (x). (2.14) Therefore combining properly both solutions for equations (2.5) and (2.4) the angle functions are expressed now in one function, Y m n (θ,φ), called Spherical Harmonics. Y m n (θ,φ) 2n+1 (n m)! 4π (n+m)! Pm n (cosθ)e jmφ. (2.15) The spherical Harmonics appear as a solution of many important applications. Basically in almost all the spherical geometry related topics. They are orthogonal, which plays an important role when solving the reconstruction equation, in section 2.3, Yn m (θ,φ)yn m (θ,φ) dω=δ nn δ mm, (2.16) 4π where the complex conjugated is indicated with the asterisk symbol ( ). Nevertheless, probably the most important property regarding the Spherical Harmonics is the fact that any arbitrary function on a sphere, f (θ,φ), can be expanded in terms of them. This is called the Spherical Harmonic Series Expansion: f (θ,φ)= n n=0 m= n E nm Y m n (θ,φ), (2.17) where the spherical harmonics coefficients P nm are given by P nm = f (θ,φ)yn m (θ,φ) dω, (2.18) and the solid angleωis defined by 4π dω = sin θdθdφ. (2.19) It is possible at this point to write a solution for the wave equation in spherical coordinates (2.2) as or alternatively n p(r,θ,φ,ω)= (A mn j n (kr)+ B mn y n (kr))yn m (θ,φ)e jωt (2.20) n=0 m= n n p(r,θ,φ,ω)= (C mn h (1) n (kr)+ D mn h (2) n (kr))yn m (θ,φ)e jωt. (2.21) n=0 m= n Q m n, is the Legendre function of the second kind. This solution is discarded because whenγ=±1, Q m n is not finite

21 2.2. Scattering from a sphere 7 In the same way as it was mentioned before about the sign convention and the spherical Hankel functions, from now on the time dependence e jωt will not be mentioned but remain implicit for the equations that it is necessary. 2.2 Scattering from a sphere It is important to study the scattering from spheres for the SNAH theory. In particular, most of the formulation presented in this section will be oriented to the scattering of a rigid sphere which is the case of this master thesis. In general as seen in [5], the total sound field (p t ) with a sphere present on it can be divided into the incident (p i ) and scattered (p s ) sound fields, so p t = p i + p s. (2.22) The incident sound field, correspond to the sound field that would be present if the sphere was not there. Hence it can be characterized from eq. (2.20) as p i (r,θ,φ,ω)= n n=0 m= n A mn j n (kr)y m n (θ,φ), (2.23) where a P 0 factor regarding the nature of the source was included into the A nm coefficients. The scattered field which is the modification of the incident field due to the presence of the sphere can be described as p s (r,θ,φ,ω)= n=0 m= n n C mn h n (kr)yn m (θ,φ), (2.24) so that in the case of a transparent sphere the scattered field will be zero, meaning no disturbance of the sphere in the incident sound field. Consequently, scattering is a boundary problem, therefore it is necessary to apply some boundary conditions to the formulation in order to find the coefficients A mn and C mn needed. Figure 2.3 Total sound field p t in the surroundings of a solid sphere produced by a plane wave coming from infinity. The sphere radius is a=1 metre, and ka=0, 7.

22 8 2. Spherical Nearfield Acoustic Holography (SNAH) Scattering from a rigid sphere For this master thesis a rigid spherical array provided by Bruel & Kjær was used, and for this reason the case of the rigid sphere is studied. As it was mentioned before, the usage of boundary conditions is needed for the scattered field to be accurately described. One boundary condition is that the particle velocity in the surface of a rigid sphere of radius a is equal to zero, so u t (a) p t r (p i+ p s ) r=a r = 0. (2.25) r=a Using equations (2.23), (2.24) and the boundary condition (2.25) an important relationship between the coefficients of the incident and scattered sound fields is obtained in [6] as C mn = A mn j n(ka) h n(ka). (2.26) In this way the expression of the scattered field is rewritten in terms of the incident field as p s (r,θ,φ,ω)= j n(ka) h n(ka) h n(kr) n=0 n m= n and consequently the total sound field as indicated in [6] is expressed as p t (r,θ,φ,ω)= n=0 A mn Y m n (θ,φ), (2.27) ( ) n j n (kr) j n(ka) h n(ka) h n(kr) A mn Yn m (θ,φ). (2.28) m= n An illustration from equation (2.28) is shown in figure 2.3, where the simulation of the total sound field around a sphere of one metre radius was calculated in Matlab. The coefficients A nm were calculated for an incident plane wave coming from infinity - see [6]. More detailed calculation of these coefficients for a spherical wave type of source will be provided in section Reconstruction Equations The basis of SNAH is to reconstruct the sound field as if the sphere was not there, in other words, to reconstruct the incident sound field. Therefore, it is necessary to separate the incident field from the scattered field. For that purpose, the unknown coefficients A mn should be determined. Thus another boundary condition has to be established. The other required boundary condition is derived from the fact that the pressure on the surface of the rigid sphere is known, since is measured by the array. Hence, the pressure on the surface of the sphere with radius a p t (a,θ,φ,ω)= n=0 ( ) n j n (ka) j n(ka) h n(ka) h n(ka) A mn Yn m (θ,φ), (2.29) m= n is known. Because of the orthogonality of the spherical harmonics mentioned in section (see eq. (2.16)) the coefficients can be calculated. Multiplying eq. (2.29) with a complex conjugated spherical harmonic and integrating over the entire solid angle is found that A mn = 2π π 0 0 p t(a,θ,φ)yn m (θ,φ) sinθdθdφ ( ). (2.30) j n (ka) j n(ka) h h n(ka) n(ka)

23 2.3. Reconstruction Equations Pressure Having calculated the coefficients by replacing their value in eq.(2.23) the incident sound pressure becomes p i (r,θ,φ)= n n=0 m= n 2π π 0 0 p t(a,θ,φ)yn m (θ,φ) sinθdθdφ ( ) j n (kr)y j n (ka) j n(ka) h h n m (θ,φ). (2.31) n(ka) n(ka) It is important to be aware of the importance of this last equation, where it is possible to reconstruct the incident sound field without any input regarding the source, once the pressure on the surface of the spherical array is known Particle velocity and Intensity From [7] the relationship between sound particle velocity and pressure is known as p i = jωρû i, (2.32) which together with the fact that the gradient of a function in spherical coordinates becomes f (r,θ,φ)= ( f r, 1 ) f r θ, 1 f r sinφ φ (2.33) gives the incident particle velocity components, u i,r (r,θ,φ)= j ρc u i,θ (r,θ,φ)= j ρc u i,φ (r,θ,φ)= 1 ρc n n=0 m= n n n=0 m= n n n=0 m= n A mn j n(kr)y m n (θ,φ), (2.34) A mn j n (kr) kr A mn j n (kr) kr Yn m (θ,φ), (2.35) θ myn m (θ,φ). (2.36) sinθ Note that the partial derivative of the spherical harmonics from the polar component of the particle velocity in eq is not found straight forward, since requires derivation of the Legendre functions. For this purpose the property of the Legendre functions that states that (1 x 2 ) dpm n (x) dx = (n+1)xp m n (x) (n m+1)p m n+1 (x), (2.37) is used for the Matlab implementation. Finally the incident sound intensity can be calculated from the sound pressure and particle velocity as Î i = Re[ p iû i ]. (2.38) 2

24 10 2. Spherical Nearfield Acoustic Holography (SNAH) 2.4 Numerical integration There are obvious limitations in practice for eq. (2.31). For example, for the coefficients A nm to be determined, knowledge of the continuous pressure distribution on the surface of the rigid sphere is necessary. In this section the fact that the sound pressure is only known for a finite number of discrete positions (e.g, where the fifty microphones are flushed in the surface of the sphere) will be discussed. Thus some kind of estimation is required, for instance in this master thesis, a Gaussian numerical integration is used, - see [8]: Q n (g)= I w i g(x i ), (2.39) i=1 where g(x) is an arbitrary function on the sphere. Making use of eq.(2.39) together with the Spherical Harmonics series expansion property, the approximated Spherical Harmonics coefficients (2.18) can be estimated as P nm = I e i p(ω i )Yn m (Ω i ). (2.40) i=1 The weight factors e i and the associated microphones angles were provided by Bruel & Kær. They provided perfect exact numerical integration for Spherical Harmonics up to N= 5 degree - see [9]. This data can be seen in the appendix A.1. Finally, the incident sound pressure reconstruction equation (2.31) becomes ( N n Ii=1 e i p(θi,φ i )Yn m (θ i,φ i ) ) p i (r,θ,φ= ( ) j n (kr)y n=0 m= n j n (ka) j n(ka) h h n m (θ,φ), (2.41) n(ka) n(ka) by replacing the continuous pressure with the estimation found from the numerical integration in eq. (2.40). Observe that now, the Jacobian does not appear in the equation when replacing the integral by a weighted sum of samples because each sample represents a certain part of the solid angle. Also notice that the limit for the discrete summation is truncated up to N= 5, as mentioned before. 2.5 Discussion It has been demonstrated that if the sound pressure is known on the surface of a rigid sphere, the entire incident sound field can be theoretically reconstructed. In practice, the perfect reconstruction cannot be achieved. Due to the discrete integration, the summation to infinity in the reconstruction equations has to be truncated for N=5 degree of spherical harmonics, leading to a limited resolution. Moreover, the reconstruction inside the sphere is a forward problem, but outside the sphere the problem becomes inverse and ill-posed. In the following chapters, the error coming out of this facts mentioned before will be analyzed. Either from Matlab simulations (section 3) or practical measurements (section 4) the limitations of SNAH using a rigid spherical array will be shown. In this case the Jacobian refers to the sinθ that appears in eq. (2.19)

25 3 Simulation study This section is mainly focused on the development of simulation tools and cases in order to study the limitations, advantages and disadvantages of SNAH. In section 3.1 the derivation needed to establish the coefficients A nm regarding the nature of the source are first given and hence allowing the sound pressure on the surface of the sphere to be estimated. Special attention will be paid to the monopole source simulation, which is the basis of this simulation study. Understanding the monopole source case, many different cases can be simulated in Matlab, and eventually in section 3.3 a model for a baffled plate source including first order reflections will be developed. 3.1 Pressure on the 50-microphone rigid spherical array As mentioned in section 2.3, the boundary condition that permits the reconstruction equation to be derived is the fact that the pressure on the surface of the spherical array is known. Therefore the first step of a simulation study is to model the pressure on the surface of the spherical array at the position of the given microphones. For this purpose the incident field has to be simulated as well, thus from [1] it is known that a incident plane wave can be expanded as, p plane (r, θ,φ) e jkr = 4π j n j n (kr) n=0 n m= n Y m n (θ,φ)y m n (θ s,φ s ). (3.1) In the same way, an incident spherical wave from a monopole can be expanded as, p monopole (r,θ,φ) jρωq se jkr n = ck 2 ρq s h n (kr) Yn m (θ,φ)yn m (θ s,φ s ), (3.2) 4πr where Q s is the volume velocity of the source, and the position of the source in spherical coordinates is defined by r s,θ s,φ s. In accordance with section 2.2, the coefficients A nm in the equation (2.28) for the total sound field produced by the presence of a rigid sphere, can be recalculated for the two different incident sound fields (i.e. p i ) in eq. (3.1) and eq. (3.2). j n 4π j n (kr s )Yn m (θ s,φ s ) if incident plane wave A nm = ck 2 ρq s h n (kr s )Yn m (θ s,φ s ) (3.3) if monopole source The equation of the pressure on the surface of a rigid sphere of radius a in a presence of a sound field created by a monopole source at the position (r s,θ s,φ s ) can be written as, p t (a,θ,φ,ω)=ck 2 ρq s n=0 n=0 m= n ( ) n h n (kr s ) j n (ka) j n(ka) h n(ka) h n(ka) Yn m (θ,φ)yn m (θ s,φ s ). (3.4) m= n 11

26 12 3. Simulation study The Matlab simulations relay on eq.(3.4) for estimating the pressure on the surface of the spherical array. For the monopole source cases, the volume velocity Q s is assumed to be one. In the case of the baffled vibrating plate simulation, the volume velocity is calculated from another Matlab function. (a) ka=0.09 (b) ka=1 Figure 3.1 Pressure on the surface of the rigid spherical array due to a monopole at 5 metres distance of the center of the array. Subfigure 3.1a corresponds to ka = 0.09 while subfigure 3.1b is for ka=1. The radius of the spherical array situated in the origin of coordinates is a= metres. Figure 3.1 shows an example of the pressure level simulated on the surface a spherical array where a monopole source was placed at 5 metres distance from the center the with an azimuthal angle of zero radians and a polar angle ofπ/2 radians. It is apparent in subfigure 3.1a that at lower frequencies, the disturbance of such a small sphere in the sound field is minimal. Consequently at higher frequencies, as shown in subfigure 3.1b, the disturbance in the sound field created by the sphere becomes more critical. The simulations in figure 3.1 have only illustrative purposes, and therefore were carried out for much higher number of points than microphones flushed in the surface of the real array used and also for a higher order than N= 5 of the spherical harmonics. In further sections, when the reconstruction equation is implemented, the pressure on the surface of the sphere is only calculated at the fifty points where the microphones are mounted on the real array for N= Point sources Through out the following section, different cases using a monopole source were simulated. The sound field created on the surface of the spherical array by a monopole source was simulated by means of equation (3.4). Consequently, when more than one monopole should be simulated two different sound field on the surface of the sphere were simulated and later added Single monopole source As proved in section 2.3, knowing the pressure on the surface of the rigid sphere, the sound field can,(in principle) ideally, be reconstructed. Later in section 3.1 the pressure on the surface of the array was simulated, within certain limitations. The pressure was discretized to the fifty microphones positions on the array. The order of spherical harmonics was limited to N=5

27 3.2. Point sources 13 due to the numerical integration carried out with the weights of the microphones positions. No further regularization was attempted. Figure 3.2 shows an example of reconstruction of the sound pressure in a concentric sphere of 15cm, that is, 5cm outside the spherical array. (a) Level (b) Phase Figure 3.2 Sound field generated by a monopole at a distance of 40cm from the center of the sphere at ka = 1. Reconstructed sound pressure on a concentric spherical surface of 15cm radius. Note that the pressure difference between the front and the back of the sphere is close to the decay rule 1/r, where r is the distance to the source. The phase also appears to be reasonable, where all the points seem to be delayed with respect of the position (θ,φ)=(π/2, 0) that is the closest to the source in this case. Thus, the performance of this part of the simulation is satisfactory. The previous result in figure 3.2 comes from an ideal simulation, but in a real measurement, the pressure on the surface of the array would be affected by noise and imperfectly matched microphones. Therefore, as in [10] the pressure in the microphones was modified with rectangularly distributed amplitude mismatch and rectangularly distributed phase mismatch with standard deviations of 0.2 db and 1.1 degree, respectively. The result of such modification can be seen in the figure 3.3.

28 14 3. Simulation study (a) Level (b) Phase Figure 3.3 Sound field generated by a monopole at a distance of 40cm from the center of the sphere at ka = 1. Reconstructed sound pressure, with added noise, on a concentric spherical surface 15cm of radius. The reconstruction is mainly affected by the noise when it is carried out outside the spherical array (e.g. in figure 3.3), where the error becomes more critical due to the ill-posed characteristic. Inside the spherical array, the error in the reconstruction presents not important differences with the reconstruction simulated without noise. It is difficult to establish the limits of the reconstruction outside the sphere in the three dimensional plots shown before. Hence, another kind of representation will be adopted. Firstly, the case for one simple monopole radiating will be simulated. The monopole is situated in positive part of the x axis at a variable distance from the origin of coordinates where the spherical array is placed. An illustration of this configuration is shown in figure 3.4. The pressure will be simulated along the x axis for a real monopole, and at the same time the pressure on the surface of the spherical array is calculated so that the reconstruction of the pressure can be compared with the true pressure. Figure 3.4 Configuration of the Matlab simulation scenario. One monopole radiating at the position (θ, φ) = (π/2, 0). Figure 3.5 shows the result of a reconstruction of the sound field generated by a monopole at 20 cm from the center of the spherical array. The frequency now is ka=2. The reconstruction is very good inside the sphere whereas outside the spherical array the reconstruction lose accuracy. As can be seen the phase reconstruction also follows the same trend as the pressure, therefore from now on it will be omitted from the reconstruction figures from 3.6. It

29 3.2. Point sources Monopole Reconstructed 3 2 Monopole Reconstructed Level [db ref 20µPa] Phase [rad] Distance to the source (m) (a) Level Distance to the source (m) (b) Phase Figure 3.5 Sound field generated by a monopole 20 cm away from the sphere at ka=2. True and reconstructed sound pressure level and phase along a line through the monopole and the centre of the spherical array is apparent how moving toward the source in the reconstruction is less accurate that moving away from it. From figure 3.5 it seems acceptable accuracy for the reconstruction is found at distances up to twenty centimeters from the centre of the spherical array. Level [db ref 20µPa] Monopole reconstructed Level [db ref 20µPa] Monopole Reconstructed Distance to the source (m) (a) ka= Distance to the source (m) (b) ka=3 Figure 3.6 Sound field generated by a monopole 20 cm away from the sphere at ka=5 and ka=3. True and reconstructed sound pressure level along a line through the monopole and the centre of the spherical array The reconstruction in figure 3.6 correspond to the same configuration as in figure 3.5 for higher frequencies. Subfigure 3.6a relates to ka = 5 and subfigure 3.6b to ka = 3. It is apparent that at higher frequencies the reconstruction it is not satisfactory outside the spherical array. From the study of the simulations can be estimated that above ka=3 frequencies, the reconstruction outside the sphere is not acceptable. At lower frequencies the reconstruction is acceptable up to twice the radius of the sphere. It is not shown in this chapter, but similar results to figure 3.5 are obtained for simulations at lower frequencies than ka=2. In figure 3.7 the simulation was carried out for one metre distance to the source. Notice that there are some marker in figure 3.7 delimiting the value of one and two times the radius of the spherical array in both directions. The results appears to agree with what was observed

30 16 3. Simulation study Level [db ref 20µPa] Monopole Reconstructed Level [db ref 20µPa] Monopole Reconstructed Distance to the source (m) (a) ka= Distance to the source (m) (b) ka=3 Figure 3.7 Sound field generated by a monopole 1 m away from the sphere at ka=2 and ka=3. True and reconstructed sound pressure level along a line through the monopole and the centre of the spherical array before in figure 3.6. At higher frequencies than ka = 3 the reconstruction is only acceptable within the spherical array, whereas at lower frequencies reconstruction up to twice the radius of the spherical array is possible. Another way to prove the statements made until now would be to plot the reconstructed sound for a fixed reconstruction point as a function of frequency. In this way the frequency dependence can be more clearly studied Level [db ref 20µPa] Reconstructed Monopole ka (a) Level [db ref 20µPa] Reconstructed Monopole ka (b) Figure 3.8 Sound pressure level generated by a monopole 20cm away from the center of the sphere. True and reconstructed pressure level for different reconstruction points. At the center (a) and at a point 10cm toward the source (b). Following the same configuration represented in figure 3.4, for a distance to the source of 20 cm, figure 3.8 confirms what was found before. Reconstruction inside the spherical array is possible up to ka=3 - see sub figure 3.8b. From figure 3.9, it is also confirmed that it is possible to reconstruct the sound field with an acceptable accuracy at distances twice the radius of the spherical array at frequencies up to ka=2, meaning that reconstruction 2 cm away from the source is possible up to such frequencies. Notice how the accuracy was improved as the source is moved 30 cm away from the array

31 3.2. Point sources 17 - see sub figure 3.9b. But, in this last case, the reconstruction point is 18 cm away from the source instead of 2 cm. As seen before in the reconstruction figures along a line through the monopole and the center of the spherical array (e.g figure 3.5), reconstructing toward the source is less accurate. The closer to the source, the less accuracy in the reconstruction. Level [db ref 20µPa] Reconstructed Monopole ka (a) Level [db ref 20µPa] Reconstructed Monopole ka (b) Figure 3.9 Sound pressure level generated by a monopole 20cm (a) and 30cm (b) away from the center of the sphere. True and reconstructed pressure level for a reconstruction point 18cm toward the source direction Two monopole sources The following simulations were carried out for two coherent monopoles configured as shown in figure The distances of the two monopole to the centre of the array is the same for both sources. Figure 3.10 Configuration of the Matlab simulation scenario. One monople radiating at the position (θ, φ) = (π/2, 0), and the other one at (θ, φ) = (π/2, π/2). In figure 3.11 it can be seen that the simulations agree with what was shown for the single monopole source case. At lower frequencies, the reconstruction is acceptable at distances up to twice the radius of the sphere. At higher frequencies, the sound field becomes complicated and therefore more difficult to be reconstructed outside the spherical array. Due to the truncation in the spherical harmonics order, the oscillations in the sound field at higher frequencies become more difficult to follow in the reconstruction.

32 18 3. Simulation study Monopoles Reconstructed Monopoles Reconstructed Level [db ref 20µPa] Level [db ref 20µPa] Distance to the source (m) (a) ka= Distance to the source (m) (b) ka=5 Figure 3.11 Sound field generated by two monopoles 20 cm away from the sphere at ka=1 and ka=5. True and reconstructed sound pressure level along a line through the monopole and the centre of the spherical array At longer distances to the source the simulations appear to be in accordance with what was mentioned in the single monopole source case - see figure Monopoles Reconstructed Monopoles Reconstructed Level [db ref 20µPa] Level [db ref 20µPa] Distance to the source (m) (a) ka= Distance to the source (m) (b) ka=5 Figure 3.12 Sound field generated by two monopoles 40 cm away from the sphere at ka=2 and ka=5. True and reconstructed sound pressure level along a line through the monopole and the centre of the spherical array Rigid vs. transparent spherical array Recent investigations showed that there is advantages in the usage of rigid spherical array other than transparent -see [11]. From [1], the coefficients A nm in eq. (2.30) needed in case of transparent sphere are known. In fact, with a simple modification in the denominator of eq. (2.30) the reconstruction equation can be implemented for a transparent spherical array. This modification is simple a variation of the boundary conditions, where obviously the scattered field is zero for a transparent sphere. Therefore, the coeffients in A nm can be rewritten for a transparent spherical array as

33 3.2. Point sources 19 2π π 0 0 A mn = p t(a,θ,φ)yn m (θ,φ) sinθdθdφ. (3.5) j n (ka) Notice that for the transparent the spherical Bessel functions are the only function appearing in the equation. As mentioned in section 2.1.1, the Bessel functions have zeros while the Hankel functions are complex and therefore never zero. Thus, at certain values of ka the spherical Bessel function value is zero and hence the denominator of the coefficients A nm is zero as well, producing an obvious instability that affects the accuracy of the reconstruction. On the other hand, for the case of the rigid sphere, the coefficients A nm are function of the spherical Bessel and Hankel functions. This fact will prevent the denominator to become zero at any frequency. Looking at figure 2.2, the frequencies (i.e. values of ka) where the spherical Bessel functions cross zero value can be found. In figure 3.13, the reconstruction of a sound field generated by two monopoles at two different frequencies was carried out. The reconstruction for a rigid spherical array, transparent spherical array and the true sound field can be compared Monopole Rigid Transparent Monopole Rigid Transparent Level [db ref 20µPa] Level [db ref 20µPa] Distance to the source (m) (a) ka= Distance to the source (m) (b) ka=3.1 Figure 3.13 Sound field generated by two monopoles 20 cm away from the sphere at ka= 2.6 and ka=3.1. True and reconstructed sound pressure level along a line through the monopole and the centre of the spherical array for the case of rigid (blue line) and transparent (red line) spherical arrays with noise introduced in the reconstructions. It is clear from figure 3.13, that for a value ka=3.1 where the spherical Bessel function cross zero (see figure 2.2) the transparent sphere has a poor performance compared with the rigid sphere. In contrast, out ot the values where the Bessel functions cross zero, the performance of both arrays is very similar - see figure 3.13a. Due to practical reasons, the artificial noise generated for the transparent and rigid sphere reconstructions was different, preventing better agreement of the simulations out of the problematic frequencies. For the ideal simulations where no noise was introduced, the agreement between the different types of array is almost perfect. Although, the scattering from the microphones in the transparent spherical array at high frequencies is a another negative factor ignored here, that definely will not enhance the accuracy of the transparent spherical array. This phenomena can easily be observed when the frequency reconstruction is plotted against frequency as done in section From figure 3.14 the instability of the transparent sphere at the frequencies, before ka=10, where the spherical Bessel function cross the zero value is apparent. Note that the reconstruction in figure 3.14 is carried out in the center of

34 20 3. Simulation study coordinates, where the reconstruction is very accurate. Another important fact that can be extracted from figure 3.14 is that after ka=10 for both of the arrays the reconstruction accuracy breaks down even inside the spherical array Rigid Transparent Monopole Level [ref 20µPa] ka Figure 3.14 Sound field generated by a monopoles 20 cm away from the sphere. True and reconstructed sound pressure level at the center for a rigid (blue dashed line) and transparent (red dashed line) spherical array Particle velocity and Intensity Up to now the study was mainly focused on pressure information, but as it was shown in the theory presented in section 2, the particle velocity vector and thus the intensity vector can be calculated from the reconstruction equation. In principle, since the intensity is function of the pressure and particle velocity, the same limitations seen for the pressure reconstruction should apply also to the intensity reconstruction. For this reason a simple simulation case consisting of a monopole in front of the spherical array, similar to the configuration presented in figure 3.4 was studied. The results are shown in figure From the monopole equation, it is straight forward that the only component of the particle velocity is the radial component, and hence only radial intensity component can be found. Consequently, the azimuthal and polar components of the intensity must be zero. Figure 3.15a shows the result of the reconstructed and simulated radial intensity along a line between the monopole and the center of the sphere. The monopole was situated at 40 cm distance in this case. Like the pressure simulations, shown before, the reconstruction is acceptable up to twice the radius of the sphere for a frequency ka=3. In figure 3.15a the frequency dependence is shown for a similar source configuration. The reconstruction point is 20 cm away from the center of the sphere toward the source direction. Again the results are in accordance to the pressure simulations. In general the pressure reconstruction limitations can be extrapolated to the intensity reconstruction as far as seen in more simulations not shown here. It is important to mention that in the intensity reconstruction, in agreement with the theory, the azimuthal and polar vector components were found to be zero. Apart from this simple monopole configuration, no other cases were studied for the intensity reconstruction. Thus a more extensive investigation is needed where more complicated intensity representation of more complicated cases should be considered.

35 3.3. Baffled plate source 21 Level [db ref 10pW/m 2 ] Monopole Reconstructed Distance to the source [m] (a) Level [db ref 10pW/m 2 ] Monopole Reconstructed ka (b) Figure 3.15 Sound radial intensity magnitude simulated (solid line) and reconstructed (dashed line) for a real monopole at 40 cm distance. Subfigure 3.15a shows the distance dependence reconstruction for a frequency ka=3 while subfigure shows 3.15b the frequency dependence. 3.3 Baffled plate source Rather ideal sources have been simulated until now. These models are helpful to understand the possible limitations of SNAH. But in a real situation sound sources are more complicated, and therefore a more complex model is necessary to be implemented. Vibrating plates are common sources of noise in practical problems, besides more complicated sound fields are provided from such sources. Within this section, a model for a vibrating plate will be studied in Matlab Rectangular finite simply supported plate Finite vibrating plates represent a realistic test case, which is not difficult to implement. Together with the condition of infinite rigid baffle and the support called, the simply supported boundary condition, the solution of the equation of motion leads to a rather simple solution. The formulation extracted from [1] is described in the following. The normal displacement of a point in the plate w(x, y,ω) can be expressed as w(x, y,ω)= F ρ p m=1 n=1 Ξ mn (x 0, y 0 )Ξ mn (x, y) ω 2 ω 2, (3.6) mn where the point that excites the plate is defined by the coordinates x 0 and y 0 with a force magnitude F. The plate is defined by the density of the materialρ p and its thickness h. The set of orthonormal modes,ξ mn, are defined in terms of the finite dimensions of the panel L x and L y as Ξ mn (x, y)= 2 Lx L y sin ( mπx L x ) sin ( nπy L y ), n=1, 2, m=1, 2. (3.7) The Eigenfrequency correspondent to each mode (m, n) are ω nm =α 2 [ (mπ/l x ) 2 + ( nπ/l y ) 2 ], (3.8) The force is given by F(w)e jwt

36 22 3. Simulation study where α is named in the literature as Skudrzyk s constant. In terms of the Young s modulous (E) and Poission s number (ν) of the material together with the plate parameters mentioned in eq. (3.7),αis given by ( Eh 2 ) 1/4 α= 12ρ p (1 ν 2. (3.9) ) As an example, the displacement and particle velocity were calculated for a steel plate of 40 cm by 40 cm and 5 mm thickness. The material properties of steel used for this master thesis were; densityρ p = 7800Kg/m 3, Young s modulus E= 0.2 GPa and Poisson s number ν=0.3. The results obtained from the simulation of the plate with such parameters can be seen in figure (a) Displacement (b) Normal velocity Figure 3.16 Normal displacement (a) and normal vibrational velocity at the surface (b) of a simply supported asymmetrically driven steel plate of 40 by 40 cm and 5 mm thickness. Frequency of excitation is f=2000 Hz Radiation of a baffled rectangular simply supported driven plate The sound pressure radiated from a vibrating plate can be calculated if the vibrational velocity is known. Dividing the panel into small rectangles that act as monopoles on a baffle, and adding all the pressures form each of the monopoles in the plate the pressure can be estimated in a point g=(r,θ,φ). At a fixed frequency the vibrational velocity is just the displacement time jω. In this master thesis the effect of reflections in the reconstruction is under study, thus it seems reasonable to baffle the plate. From the theory (see [5]) it is known that the amplitude of and infinite baffled monopole is doubled, thus the pressure generated by a baffled monopole is p(g)= jωρ 2π Qejkr r. (3.10) With the area (S n ) and the particle velocity v n information of each division (or rectangle), the approximation to the Rayleigh s integral can be used to calculated the pressure radiated from the baffled vibrating plate - see [1]. Being; P the number of divisions per side of the plate, d n is the distance from each monopole to the calculation point g and S n = A p /(1 P) 2 where A p is the total area of the plate, the radiated pressure from the baffled plate becomes

37 3.3. Baffled plate source 23 ˆp(g)= jωρ 2π PxP n=1 v n S n e jkd n d n. (3.11) Figure 3.17 shows an example of pressure radiated from a baffled simply supported point driven plate. The calculation point is 10 cm away in front of the plate. The dimensions of the steel plate used for the simulation in figure 3.17 were 20 by 20 cm and a thickness of 5 mm. 150 Level [db ref 20µPa] (1,1) 1000 (2,1) 2000 (2,2) (3,1) (3,2) Frequency (Hz) Figure 3.17 Simulated sound pressure level generated by a baffled vibrating steel plate at a 10 cm distance point. The plate is 20 by 20 cm and 5 mm thickness. In figure 3.17 it is apparent that the peaks in the frequency spectrum correspond to the first five natural frequencies of the plate. These frequencies had been marked in the frequency axis and named after the mode they represent. Hence, it is correct to conclude that the function implemented to calculate the displacement of the plate (see appendix A.7) is working properly. For the explanatory figure 3.17, no artificial noise was included, but when the function is used in further section to calculate the sound field generated in the sphere due to the plate radiating, noise was introduced in the pressures of the fifty microphones Reconstruction without reflections In this section the reliability of the reconstruction using rigid spherical array from a point driven simply supported baffled vibrating plate will be studied. Using the model developed in section together with the reconstruction functions implemented for section 3.2.1, it is possible to simulated the reconstruction of the sound field generated from the baffled vibrating plate. Initially the effect of the reflections from the baffled will not be introduced. In this way a similar study as in section 3.2 is carried out. In later sections the effect of the reflections will be included in the model and compared with the results obtained here. The configuration of the simulation carried out in Matlab can be seen in figure As in section 3.3.2, the dimensions of the aluminum plate are 20 by 20 cm, 5 mm thickness and the amplitude of the force driving the plate is 10 N. Similar three-dimensional simulations as shown in section 3.2 can be obtained here regarding the pressure on the surface of the sphere or the reconstructed incident sound field in a concentric spherical surface. However, not very significant information can be extracted from such plots. Thus, only the two-dimensional graphs are presented. Unlike it was done before for the point sources, the reconstruction line

38 24 3. Simulation study Figure 3.18 Configuration for the vibrating steel plate simulations. The plate is assumed to be baffled compared in the following figures is the y axis which, looking at the simulation configuration in figure 3.18, corresponds to a parallel line to the vibrating plate Level [db ref 20µPa] Baffled plate Reconstructed Phase [rad] Baffled plate Reconstructed Y axis [m] Y axis [m] (a) Level (b) Phase Figure 3.19 Sound field reconstruction of a simply supported asymmetrically driven steel plate of 20 by 20 cm and 5 mm thickness. Frequency of excitation is ka=1. Distance to the plate is 20cm. The reconstruction of the sound field from a baffled vibrating plate compared with the calculated sound field at 20 cm distance from the source can be seen in figure These results verify what it was also seen for the point sources study, that the reconstruction inside the sphere and also at distances twice the radius of the sphere is accurate at low frequencies. Similar results not shown, have been obtained at lower excitation frequencies than ka = 1. Notice also the agreement in the phase reconstruction - see sub figure 3.19b. Phase reconstruction will be omitted in the rest of the figures because it follows the same trends as the pressure. In figure 3.20, results again agree with what it was found in section 3.2. The reconstruction up to frequencies ka=3 is fairly accurate for distances (from the origin of coordinates) twice the radius of the sphere. Thus, with no effect of the reflections included, the reconstruction of the sound field on the surface of the source is possible and fairly accurate. Notice how the reconstruction begins to oscillate after reaching distances over twice the radius of the sphere

39 3.3. Baffled plate source 25 for ka=2. The same occurs when ka=3, despite that the oscillations start a bit before reaching the point of twice the radius of the array (0.18 cm) but still the error is not critical Level [db ref 20µPa] Level [db ref 20µPa] Baffled plate Reconstructed Y axis [m] (a) ka=2 70 Baffled plate Reconstructed Y axis [m] (b) ka=3 Figure 3.20 Sound field reconstruction of a simply supported asymmetrically driven steel plate of 20 by 20 cm and 5 mm thickness. Frequency of excitation is ka=2 (a) and ka=3 (b). Distance to the plate is 20cm Level [db ref 20µPa] Baffled plate Reconstructed Y axis [m] Level [db ref 20µPa] Baffled plate Reconstructed Y axis [m] (a) ka=7 (b) ka=9 Figure 3.21 Sound field reconstruction of a simply supported asymmetrically driven steel plate of 20 by 20 cm and 5 mm thickness. Frequency of excitation is ka=7 (a) and ka=9 (b). Distance to the plate is 20cm. Examining figure 3.21, it is noted that unlike the point sources, for the plate source the reconstruction inside the sphere is not acceptable for ka=9 (see sub figure 3.21b). From subfigure 3.21a, where ka=7, the reconstruction inside the sphere is still acceptable but the deterioration of the reconstruction accuracy closer to the radius of the array becomes noticeable. This fact can be confirmed in figure 3.22, where the reconstruction frequency dependence is studied in the origin of coordinates at 20 cm distance from the source. It is apparent that after ka=8, the reconstruction breaks down. In contrast with the study of the point sources where the reconstructions inside the array broke down after ka=10, for more complicated sources this limit is reduced to ka=8. In figure 3.23 the reconstruction point is twice the radius distance further from the origin of coordinates but still at 20 cm from the vibrating plate.again, the limit for the accurate reconstruction appears to go down from ka=3 for the point sources to ka=2 for the vibrating plate. Note that even when the reconstruction s accuracy is not acceptable still follows the trend of the true sound field, tending to underestimate it. No figures from further than 20 cm to the source were included. As seen for the point

40 26 3. Simulation study Level [db ref 20µPa] Baffled plate Reconstructed ka= ka Figure 3.22 Frequency dependence at the origin of coordinates at 20cm distance from the source Baffled plate Reconstructed Level [db ref 20µPa] ka Figure 3.23 Frequency dependence at a point two times the radius of the sphere further from the origin of coordinates. The distance to the source is 20 cm. sources, the results from the simulation agree also for further distances. Besides, they are not relevant to be included since the main concern for SNAH is the near field. Additional simulations from further distances to the vibrating pate are included in appendix B First order reflections simulation As mentioned during the introduction of this master thesis, one of the potential problems that the rigid sphere might have, is the reflections from the source back to the microphones in the surface of the array and therefore possibly variating the reconstructed incident field from the real one. Thus it is important to include the effect of the reflections created from the rigid spherical array, in the model developed in section For this purpose an image rigid array was simulated. For each vibrating monopole, in which the plate was divided into, the backscattered field from the image source on the surface of the real rigid spherical array is calculated and added to the total field that the same monopole is creating. In figure 3.24 is shown an explanatory drawing of what was just explained. From section 2.2 it is possible to calculate the scattered sound field for a rigid sphere in

41 3.3. Baffled plate source 27 Figure 3.24 Backscattered reflections calculation principle. a certain point - see eq.(2.27). The expression for the total sound field on the surface of the rigid array is known from eq.(3.4). Therefore, combining both expressions, the total sound field on the surface of the sphere due to the backscattered reflection can be calculated. In this way, the reflections can be included in the Matlab model and hence their influence on the reconstruction accuracy can be studied. The result from the mentioned combination is a new total backscattered sound field as p bs (a,θ,φ)= j n=0 ( )( ) j n (ka) j n(ka) j h n(ka) h n(ka) n(ka) h n(ka) h n(kr sct )... n m= n A nm Y m n (θ,φ)y m n (θ s,φ s ) Y m n (θ sct,φ sct ), (3.12) where the coefficients A nm were defined in eq.(3.3) for a monopole. The variablesθ sct,φ sct and r sct represent the spherical coordinates of the calculation point seen from the origin of coordinates of the image source. Consequently a new total sound field on the surface of the sphere can be defined as p T = p t + p bs. (3.13) The following figures are simulations in which the backscattered reflections were included and excluded respectively. Figures 3.25 and 3.26 correspond to frequencies ka=0.2 and ka=2 respectively, both at a distance of 20 cm for the source and again following the same configuration as in the previous subsection. The sphere is not disturbing the sound fields at low frequencies and, as expected, from figure 3.25 the influence of the reflections is insignificant. Surprisingly at higher frequencies, as seen in figure 3.26 the influence of the backscattered reflections is very small. The coefficients found in section 3.1 were for a non baffled monopole source, therefore a factor of 2π must be added in this case

42 28 3. Simulation study Level [db ref 20uPa] Plate Reconstructed with reflections Reconstructed without reflections Y axis [m] Figure 3.25 Sound field reconstruction of a simply supported asymmetrically driven steel plate of 20 by 20 cm and 5 mm thickness. Frequency of excitation is ka=0.2. Distance to the plate is 20cm Level [db ref 20uPa] Plate Reconstructed with reflections Reconstructed without reflections Y axis [m] Figure 3.26 Sound field reconstruction of a simply supported asymmetrically driven steel plate of 20 by 20 cm and 5 mm thickness. Frequency of excitation is ka=2. Distance to the plate is 20cm. Figure 3.27 present the same kind of results as the figures before but at a higher frequency ka=7. Greater influence of the backscattered reflections would be expected at this frequency. Instead, the influence of the reflections is again minimal even smaller than in the case of ka=2 showed in figure This kind of behavior was observed in other simulations results not showed here. In all the cases the influence of the reflections is not critical, and tends to underestimate the sound field inside the spherical array. The results presented in figure 3.28 were carried out in the same conditions as the ones in figure 3.26, with the only difference that the spherical array was placed 20 cm further from the source, thus 40 cm away from it. The frequency was kept at the same value of ka=2. In accordance with the theory, it is observed that the effect of the reflection is diminished by placing the array further from the source. Nontheless, as mentioned before, it seems that the reflections do not represent a big problem in any case. In general, because of the limited influence of the reflections, most of the results are very similar to the ones obtained in the subsection where the backscattered reflections were

43 3.3. Baffled plate source Level [db ref 20uPa] Plate Reconstructed with reflections Reconstructed without reflections Y axis [m] Figure 3.27 Sound field reconstruction of a simply supported asymmetrically driven steel plate of 20 by 20 cm and 5 mm thickness. Frequency of excitation is ka=7. Distance to the plate is 20cm Level [db ref 20uPa] Plate Reconstructed with reflections Reconstructed without reflections Y axis [m] Figure 3.28 Sound field reconstruction of a simply supported asymmetrically driven steel plate of 20 by 20 cm and 5 mm thickness. Frequency of excitation is ka=2. Distance to the plate is 40cm. not taken into account. Thus no more results of this kind are shown here, since the differences with what it has already been shown are minimal Validation of the model Considering the results from the simulations of the reflections, and given their importance, validation is needed. Experimental measurements confirming the results from the last section could be one possibility -see section 4. Here instead, an approximation model for high frequencies is proposed in order to validate the results form the backscattered reflections given before. The main idea of the validation carried out can be seen in figure The vibrating plate is reduced to a very small size and therefore one only monopole can be considered. At such high frequencies it is assumed that the image sphere is infinitely large and

44 30 3. Simulation study Figure 3.29 Validation model illustration therefore a infinite baffle. In this way, it can be considered that a image monopole radiating with equal strength as the original baffled monopole. Thus reconstructing the sound field assuming two different baffled monopoles and comparing the results with the model developed for the reflections, the validation of the results obtained previously can be achieved. Figure 3.30 is the result of a simulation with a very small plate using backscattered reflection model while figure 3.31 was obtained with a similar configuration as shown in figure The attenuation mentioned before was applied to the image monopole simulated in figure Observe from both figures that the simulations output similar results even though completely different Matlab codes were used for each of them. Hence, the proper performance of the model can be ensured Level [db ref 20µPa] Plate Reconstructed with reflections Reconstructed without reflections Y axis [m] Figure 3.30 Reconstruction from the model developed for the simulation of the backscattered reflection. The distance to the plate is 20 cm at a frequency ka= Discussion A study of the reconstruction behavior has been conducted in this chapter. Regarding the point sources, the reconstruction inside the sphere performs very accurately up to frequencies ka=10. This fact changes if more complicated and realistic sources are under study, thus the

45 3.4. Discussion Level [db ref 20µPa] Plate Reconstruction with reflections Reconstruction without reflections Y axis [m] Figure 3.31 Reconstruction from the validation model with the two monopoles. The distance to the plate is 20 cm at a frequency ka=8 limit is lowered then to ka=8. Accurate reconstruction up to distances twice the sphere s radius are possible very close to the osurce, up to frequencies ka=3. For the case of a more complicated source this limit is still reachable with a small degree of inaccuracy. It has been also shown how reconstructing toward the source a greater error is produced. Ultimately, a model for the study of the backscatter reflections from the rigid array was created. Apparently the reflections do not have a big influence in the reconstruction precision. A validation model was implemented to assure the proper working of the reflection model with positive results. It was seen that the attenuation suffered due to the backscattering and scattering again in the original array weaken severely the reflections and therefore they become a minor problem. In fact, more influence of the reflections was noticed at middle frequencies (e.g. ka=2) than at higher frequencies. The double scattering suffered by the sound field, together with the rapid decay due to distance that is experienced at high frequencies, attenuates them more than lower frequencies. Thus, the influence of the reflections in the reconstruction at high frequency is not as critical as could have been expected.

46

47 4 Experimental results In previous chapters, the theory and simulations have been presented regarding SNAH. This chapter present the results from some experimental measurements done in the laboratory at Technical University of Denmark. The main goal of the measurements was to confirm with real measurements the results obtained from the Matlab simulations presented in section Measurement equipment The measurements were carried out in the small anechoic chamber of the acoustic laboratory at DTU, building 354. The volume of it is approximately 60 m 3 and it has a lower limiting frequency of 100 Hz. The source used as a monopole was the Bruel & Kjær OmniSource (B&K 4295), a loudspeaker mounted in an inverted horn of hard plastic, and radiating through a small opening with a long hose attached and a volume velocity adapter mounted at the end of it [12]. For the measurements with two monopole sources, as a secondary monopole a loudspeaker was used. Figure 4.1 Spherical array produced by Bruel & Kjær with fifty microphones flushedmounted and eleven cameras. Radius of the sphere is 9.75 cm. The spherical array was produced by Bruel & Kjær with fifty flush-mounted microphones and eleven small cameras on a plastic sphere with a radius of 9.75 cm - see figure 4.1. The data acquisition device that was used together with the spherical rigid array was a multichannel front end provided by Bruel & Kær. The PULSE analyzer program was used to interface with the front end and the array channel detection procedure. For the pressure measurements 33

48 34 4. Experimental results of the sound field without the sphere a B&K 4192 microphone was used. Additional photographic material from the measurements set ups and devices used can be seen in appendix C. 4.2 Validation of the measurements The data output obtained from the PULSE analyzer program was the fifty pressure time signals from each microphone in the rigid spherical array. Hence extra signal processing was needed to get the input necessary for the reconstruction. Each time dependent microphone signal was first windowed with a Hanning window then Fourier transformed and finally averaged over the rest of the windowed segments in the frequency domain. For the phase information the cross spectra was used, so a complex number was built with the windowed averaged amplitude and the phase extracted from the cross spectra. The matlab function implemented to process the data can be consulted in appendix A.10. A correction factor of 1.5 was introduced in the processing due to the Hanning windowing - see [13]. The amplitude frequency spectrum measured by the fifty microphones can be seen in figure Level [db ref 20µPa] Frequency [Hz] Figure 4.2 Frequency spectrum measured from all the fifty microphones in the spherical array. Notice the fact that at low frequencies the fifty signals agree, after at higher frequencies the signals differences arise. This is a good indicator of the correct functioning of the array. None of the signals appear to be out of the normal trend, thus all of the microphones appear to be working properly. However a verification of all the microphones was done to ensure this. Besides, during the channel recognition process needed to set up the array the microphones were also double checked. In the same way as done for figure 4.2, the phase information was checked. Selecting the closest microphone to source as a reference, the rest of the phases were delayed with respect to the reference. The phase figure is not shown here, but successful results were observed as mentioned before. Coherence in between the microphones was also monitored during the measurements, and example is given in figure 4.3. In agreement with what was expected, the coherence becomes other than one only at high frequencies where the the disturbance of the rigid array is stronger.

49 4.3. Monopole Coherence Frequency [Hz] Figure 4.3 Coherence between two different microphones in the array during a measurement. 4.3 Monopole The first scenario measured in the anechoic chamber consisted on one simple source that could be assumed to behave as a monopole. For this purpose the OmniSource was used as monopole. The set up measured in the laboratory is illustrated in figure 4.4. The spherical array was hanged from the celling of the anechoic chamber with a aluminum pole to make sure of the stillness of the measurement. The source was placed at the same height of the spherical array s center, and along the y axis. The definition of the cartesian axis was established by the coordinates microphones position and weights provided by Bruel & Kjær. The distance d defined in figure 4.4 was modified to perform the measurements at different distances. The gray blurry spots marked in figure 4.4 are the positions where measurements without the spherical array were taken. Therefore, a comparison between the reconstructed field and the measured could be accomplished. Figure 4.4 Measurement configuration with the OmniSOurce in the anechoic chamber Results The following results in figure 4.5 show the results form the measurements compared with the true estimated sound field along the y axis. Due to time constrains for the availability of

50 36 4. Experimental results the spherical array the estimated true sound field was just interpolated from the single point measurements assuming that the source is a monopole and therefore follows the decay rule of 1/r Reconstructed Measured Reconstructed Measured Level [db ref 20µPa] Level [db ref 20µPa] Y axis [m] (a) Y axis [m] (b) Figure 4.5 Measured and reconstructed pressure along the y axis at 2000 Hz frequency. Subfigure 4.5a is for a distance to the source of 20 cm while subfigure 4.5b corresponds to 40 cm. The results presented in figure 4.5 correspond to a frequency of 2000 Hz that is approximately equivalent to ka=3.5. As mentioned on the simulation results, for this frequency values the reconstruction is only acceptable inside the sphere. The same type of results are shown in figure 4.6, where the distances are the same (20 and 40 cm) but the frequencies are lower. Also in accordance with the simulations results, the reconstruction is acceptable up to twice the radius of the spherical array. From these experimental results it seems that the simulation assumptions are applicable for real measurements Reconstructed Measured Reconstructed Measured Level [db ref 20µPa] Level [db ref 20µPa] Y axis [m] (a) 1000 Hz Y axis [m] (b) 500 Hz Figure 4.6 Measured and reconstructed pressure along the y axis at 1000 and 500 Hz frequency. Subfigure 4.6a is for a distance to the source of 20 cm and frequency of 100 Hz. Subfigure 4.6b corresponds to 40 cm distance and a frequency of 500 Hz Looking in the frequency domain it is seen that because the long tube of the source the frequency response does not look like a monopole one, the resonances from the tube can be seen in the response. Figure 4.7 shows the frequency response measured and reconstructed for

51 4.4. Two monopoles 37 a point in the center of the spherical array and 10 cm away from the center toward the source. The reconstruction in the center is very precise below 1500 Hz and acceptable in the entire measured frequency range up to 3200 Hz, although a tendency to underestimation at hight frequencies can be seen. When the reconstruction is carried for a point 10 cm away from the sphere the reconstruction is acceptable up to approximately 1200 Hz, after the underestimation of the spectrum is also observed in the reconstruction Reconstructed Measured Level [db ref 20µPa] Frequency [Hz] (a) 10 cm away from the center Reconstructed Measured Level [db ref 20µPa] Frequency [Hz] (b) Center of the spherical array Figure 4.7 Measured and reconstructed spectrum for a point in the center of the sphere 4.7a and 10cm away from the center in the direction of the source 4.7b. In both cases the source was placed at 40 cm distance from the sphere 4.4 Two monopoles Another measurement with an extra source disturbing the sound field was done. The secondary source used for this case was a loudspeaker placed at a sufficient distance to be considered as a monopole. The configuration carried out for this measurement can be seen in figure 4.8. The measurement points without the spherical array are marked in the figure with an irregular gray

52 38 4. Experimental results spot. Figure 4.8 Measurement configuration with the OmniSOurce and a loudspeaker in the anechoic chamber Results Due to time constrains the measurement of the sound field along a line was not done. Bear in mind that in this case the sound field can not be estimated from a single measurement due to the disturbance of the secondary source. More precise and amount of measurement points are needed. For this reason only frequency results are presented in this section. Figure 4.9 shows the frequency responses reconstructed and measured for a point 10 cm away form the center. As before with the single monopole source, the agreement is acceptable until 1000 Hz, and after there is a tendency to underestimate the sound field. Level [db ref 20µPa] Reconstructed Measured Frequency [Hz] Figure 4.9 Measured and reconstructed spectrum for a point 10 cm away from the center. The main source was placed at 30 cm distance from the sphere Inspecting figure 4.9 it is noticed how the sound field become more complicated due to the presence of the secondary source. Although, no big variations are observed from the single monopole case presented in the section 4.3. No other results are shown in here because they do not contribute with new information. The results seem to agree with the case of the single monopole

53 4.5. Vibrating plate Vibrating plate The last step of the measurements process was to implement a much more complicated source that could produced reflections from the spherical array. The theoretical case of an infinite baffled simply supported point driven plate studied in the simulations sections is rather difficult to implement in practice. Thus, a more simple set up was done. Therefore a box of dimensions 44 by 44 by 44 cm was used. The box was made of fiberboard, but the top surface was an aluminum plate of 1 mm thickness driven by an inertial exciter in the middle of it. The center of the array was placed 15.5 cm above the vibrating plate as it is shown in figure A plate of such a dimension should be capable of reflect sound field back to the sphere. The measurement points where the spectrum was measured without the spherical array are marked in figure 4.10 as in previous sections. Figure 4.10 Measurement configuration with the aluminum plate in the anechoic chamber Results As it happened for the two point sources measurements, no detailed sampling of the sound field was measured without the sphere, therefore no results comparing the measured and reconstructed sound field along a line are presented here. In figure 4.11 the results from the frequency spectrum measured and reconstructed can be seen. Subfigure 4.11a shows the reconstruction in a point 5 cm above the plate. The results are accurate below 1 khz and acceptable until 2 khz. For the reconstructions in the center of the sphere seen in subfigure 4.11b it is noted that the results match rather good for all the frequency range measured, although small inaccuracy can be detected at high frequencies No effect from the reflections coming from the array is detected in the results presented neither in the rest of the results not presented here. It seems that the actual measurement agree with the simulation about the minimal influence of the reflections due to the rigid sphere. Although for the case presented here, is not possible to compare the results with and without reflections due to the missing measurement of the true sound field. From the results presented before it is possible to see the similarities with the simulations results. In addition it

54 40 4. Experimental results Reconstructed Measured Level [db ref 20µPa] Frequency [Hz] (a) 5 cm above the plate 60 Reconstructed Measured Level [db ref 20µPa] Frequency [Hz] (b) Center of the spherical array Figure 4.11 Measured and reconstructed spectrum for a point in the center of the sphere 4.11b and 5 cm above the vibrating plate 4.11a.

55 4.6. Discussion 41 must be added that no influence of the reflections was seen in the reconstruction in the center of the sphere where the results were special accurate. 4.6 Discussion The results obtained out of the experimental measurements were presented and discussed in this section. A good agreement with the simulations results in section 3 has been seen. However, no point further from 1 cm out of the surface sphere was measured in the laboratory in any of the different scenarios, thus no possible comparison with the reconstruction was implemented. This was without a doubt a mistake by this thesis author. In spite of this fact, no influence from the reflection have been noticed in the reconstructions, which for a general rule followed the tendency seen in the simulation of the vibrating panel.

56

57 Conclusion and perspectives 5 A study of spherical near field acoustical holography using a rigid spherical microphone array has been presented. Formulation has shown that, as it was also demonstrated with the transparent sphere [1], if the pressure is known on the surface of the array the entire incident sound field can be reconstructed within some limitations. Due to the finite measurement points in the surface of the array, numerical integration using the spherical harmonic expansion property leads to a limit for the exact reconstruction in the spherical harmonics products up to N=5. From the simulation study it has been found that acceptable results can be reconstructed up to distances twice the radius of the sphere at frequencies up to ka=2. At frequencies above ka=3 the acceptable reconstruction area is limited to within the spherical array. This last fact, has been confirmed for point sources and more complicated ones, as infinite baffled panel single point driven fairly close to the sphere. Nevertheless, at upper frequency limit for the reconstruction in a volume inside the sphere has been found at ka=8 for the simulations, and ka 4 from the experimental results. Still remains the comparison of the frequency limit at a distance twice the radius of the sphere, due to a lack of measurement points. In any case, it must be noted that in the experimental measurements the array was placed at a closer distance to the source than in the simulations, and therefore a bigger inaccuracy is expected as seen in section 3. Hence, lower frequency limits are expected from the experimental measurement, besides the fact that a good agreement have been seen during the whole project between the simulations and the measurements. In practice the boundary conditions for a rigid array are better defined than assuming a transparent array as such for the entire frequency range. Additionally, it was proved that even under ideal conditions the rigid spherical array produces more robust results in the reconstruction. On the other hand, the usage of a rigid array, could produce the scattered field to be reflected back toward the source and thus distort the incident sound field. However, it was shown that the potential problem of the reflections from the rigid array disturbing the reconstruction it does not affect critically to the reconstruction. Both, simulation and experimental results support this conclusion. Taking into account the results obtained along this master thesis regarding the reflections from the rigid sphere it is very promising that fact that the same array can be used for both beamforming and SNAH. Therefore a unique device could be used for both, near and far field mapping. It was not deeply studied here, but as shown in section 3 the particle velocity and thus the intensity can be also reconstructed. From [3] the utility of such a property in the near filed noise source identification through volumetric acoustic vector imager (VAIM) has been shown. Thus, the rigid spherical array could be used as a possible alternative to the two microphone intensity probe. Note that the intensity reconstructed with the spherical rigid array is a three-dimensional intensity vector and thus enhancing the results from the two opposed microphones intensity probe. Further work to study the limitations in intensity reconstructions should be done. A priori, since intensity is function of particle velocity and pressure, the same 43

58 44 5. Conclusion and perspectives limitations are applicable to intensity reconstruction.

59 Bibliography [1] E. G. Williams, Fourier Acoustics: Sound radiation and nearfield acoustical holography. San Diego: Academic Press, [2] E. G. Williams, N. Valdivia, and P. C. Herdic, volumetric acoustic vector intensity imager, Journal of the Acoustical Society of America, vol. 120, pp , [3] E. G. Williams, K. Takashima, and H. Nakagawa, Vector intensity measurent with a rigid spherical microphone array in a vehicle cabin, in Acoustics, (Paris), [4] N. H. Asmar, Partial diffrential equations with Fourier series and boundary value problems. Pearson Prentice Hall, second ed., [5] F. Jacobsen and P. Juhl, Radiation of sound, Acoustic Technology, Technical University of Denmark, April [6] E. Fernandez, Nearfield acoustic holography using a rigid sphere microphone array, tech. rep., Denmark Technical University, [7] F. Jacobsen, Sound intensity and its measurement and applications., Acoustic Technology, Technical University of Denmark, August [8] I. H. Sloan and R. S. Womersley, Extremal systems of points and numerical integration on the sphere, Advances in Computational Mathematics, vol. 21, pp , [9] J. Hald, J. Mørkholt, and J. Gomes, Efficient interior nsi based on various beamforming methods for overview and conformal mapping using sonah holography for details on selected panels, SAE international, [10] J. Gomes, F. Jacobsen, and M. Bach-Andersen, Statistically optimised near field acoustic holography and the helmholtz equation least squares method: a comparison, in Proceedings of 8th International Conference on Theoretical and Computational Acoustics, (Heraklion, Crete, Greece), pp , [11] M. Marchall and J. Jourdan, Comparison of beamforming techniques on a sphere, tech. rep., Derpartment of electrical engineering, Technical University of Denmark, [12] F. Jacobsen and Y. Luan, A method of measuring the greens function in a enclosure, Journal of the Acoustical Society of America, vol. 123, pp , [13] R. Randall, Frequency Analysis. Bruel & Kjær, third ed.,

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61 Appendix A Matlab codes A.1 Weighting factors f u n c t i o n [ P s t s p h e r i c a l ]= M i c r o p h o n e s p o s i t i o n s ( r ) M i c r o p h o n e s p o s i t i o n s G e n e r a t e s mat f i l e s w i t h t h e p o s i t i o n s o f t h e microphones i n t h e t h e 50 mic r i g i d s p h e r e. Also p r o v i d e s t h e l a s t column o f t h e m a t r i x w i t h t h e Gaussain n u m e r i c a l i n t e g r a t i o n w e i g h t i n g c o e f f i c i e n t s. I n p u t s : r r a d i u s o f t h e s p h e r e i n use O u t p u t s : P s t s p h e r i c a l M a t rix w i t h t h e s p h e r i c a l c o o r d i n a t e s t ( r, t h e t a, p h i ) i n t h e f i r s t t h r e e columns and t h e w e i g h t i n g f a c t o r s i n t h e l a s one. N u m I n t e g r c o e f f i c i e n t s. mat column w i t h n u m e r i c a l i n t e g r a t i o n c o e f f i c i e n t s. Author : G u i l l e r m o Moreno A p r i l 2008; e m a i l : k i t x i n i t g m a i l. com BEGIN CODE M a t rix p r o v i d e d by B r u e l & k a e j e r w i t h t h e p o s i t i o n s o f t h e 50 microphones i f t h e s p h e r e were 1 meter r a d i u s, and t h e w e i g h t i n g f a c t o r s i n t h e l a s t column M=[

62 48 A. Matlab codes ] ; save ( m i c p o s i t i o n s w i t h c o e f i c i e n t s 1 m, M ) ; P s t = z e r o s ( 5 0, 3 ) ; 0,0975 m e t e r s i s t h e r e a l r a d i u s o f t h e 50 mic sphere, t h e r e f o r e u s i n g Thales, and c i l i n d r i c a l c o o r d i n a t e s p r o p e r t i e s. f o r n=1:50 rho1=s q r t (1ˆ2 M( n, 3 ) ˆ 2 ) ; p r o y e c t i o n o f t h e r a d i u s=1 i n t h e p l a n e xy P s t ( n, 3 ) = M( n, 3 ) r ; t r a n s f o r m z c o o r d i n a t e rho2=s q r t ( r ˆ2 P s t ( n, 3 ) ˆ 2 ) ; p r o y e c t i o n o f t h e new r a d i u s i n t h e p l a n e xy P s t ( n, 1 ) = M( n, 1 ) rho2/rho1 ; t r a n s f o r m x c o o r d i n a t e P s t ( n, 2 ) = M( n, 2 ) rho2/rho1 ; t r a n s f o r m y c o o r d i n a t e end i n case we need t h e c a r t e s s i a n c o o r d i n a t e s ( P s t ) f o r t h e new r a d i u s s p h e r e i s always good t o c a l c u l a t e t h e s p h e r i c a l i n t h i s way. A=M( :, 4 ) ; save ( N u m I n t e g r c o e f f i c i e n t s, A ) ; t r a n s f o r m a t i o n i n t o s p h e r i c a l c o o r d i n a t e s d e f i n i t i o n o f t h e m a t r i x P s t s p h e r i c a l=z e r o s ( 5 0, 4 ) ; f o u r t h column, w e i g h t i n g f a c t o r s P s t s p h e r i c a l ( :, 4 ) = M( :, 4 ) ; f i r s t column, t h e r a d i u s second column, e l e v a t i o n t h i r d column, a z i m u t h f o r n=1:50 t h e r a d i u s i s gonna be always t h e same, b u t i t was a good way t o chek t h a t t h e c o n v e r t i o n was done i n t h e r i g h t way P s t s p h e r i c a l ( n, 1 ) = s q r t ( P s t ( n, 1 ) ˆ2+ P s t ( n, 2 ) ˆ2+ P s t ( n, 3 ) ˆ 2 ) ; P s t s p h e r i c a l ( n, 2 ) = acos ( P s t ( n, 3 )/P s t s p h e r i c a l ( n, 1 ) ) ;

63 A.2. Spherical Harmonics 49 end P s t s p h e r i c a l ( n, 3 ) = atan2 ( P s t ( n, 2 ), P s t ( n, 1 ) ) ; END OF CODE A.2 Spherical Harmonics f u n c t i o n [Y]= s p h e r i c a l h a r m o m n i c s ( n,m, t h e t a, p h i ) T h i s f u n c t i o n c a l c u l a t e s t h e s p h e r i c a l harmonics f o r one v a l u e o f n and m. The f u c t i o n i s d i v i d e d i n t o two main c o n d i t i o n s f o r c a l c u l a t i n g t h e p o s i t i v e and n e g a t i v e t e r m s s e p a r e t e l y due t o Matlab r e s t r i c t i o n s f o r t h e Legendre p o l y n o m i a l s. S y n t a x : [ Yt ]= s p h e r i c a l H a r m o m n i c s ( n, m, t h e t a, p h i ) I n p u t s : n v a l u e f o r t h e c u r r e n t l y n p o s i t i o n i n t h e loop m t h e t a row v e c t o r o f e l e v a t i o n a n g l e s o f t h e s p h e r e p h i column v e c t o r o f a z i m u t h a n g l e s t h e s p h e r e O u t p u t s : Y v e c t o r w i t h t h e S p h e r i c a l Harmonics c a l c u l a t e d f o r one n and m v a l u e s. Other m f i l e s r e q u i r e d : none S u b f u n c t i o n s : none MAT f i l e s r e q u i r e d : none Author : G u i l l e r m o Moreno A p r i l 2008; e m a i l : k i t x i n i t h o t m a i l. com BEGIN CODE Y=0;Yp=0;Yn=0; i n i c i a l i z a t i o n i n s i d e t h e loop g e r e n a t i o n o f Legendre f u n c t i o n s f o r computing t h e s p h e r i c a l harmonics Pnm=l e g e n d r e ( n, cos ( t h e t a ) ) ; g e n e r a t e s n+1 p o l i n o m i a l s i f (m>=0) Only t h e p o s i t i v e ones t h e n e g a t i v e s are done i n t h e n e x t sigma i s t h e c o e f f i c i e n t o f t h e S p h e r i c a l harmonics end sigma=s q r t ( ( ( 2 n+1) f a c t o r i a l ( n m) )/ ( 4 pi f a c t o r i a l ( n+m) ) ) ; Yp=sigma. exp ( i m p h i ) Pnm (m+ 1, : ) ; Y=Yp ; Computes t h e p o s i t i v e t o t a l summation t e r m s i n t h i s loop, o n l y t h e n e g a t i v e m i n d e x e s are c o n s i d e r e d... e x c l u d i n g m=0 which was done i n t h e p r e v i o u s loop i f (m<= 1) mm=abs (m) ; sigma i s t h e c o e f f i c i e n t o f t h e S p h e r i c a l harmonics sigma=s q r t ( ( ( 2 n+1) f a c t o r i a l ( n mm) )/ ( 4 pi f a c t o r i a l ( n+mm) ) ) ; t h e n e g a t i v e s i g n i n t h e e x p o n e n t i a l r e p r e s e n t s t h e c o n j u g a t e o f t h e s p h e r i c a l harmonic ( f o u r i e r a c o u s t i c s book ) Yn=(( 1) ˆmm) sigma exp ( i mm p h i ) Pnm (mm+ 1, : ) ; f o r t h e s p h e r e term Y=Yn ; Computes t h e p o s i t i v e t o t a l summation t e r m s end

64 50 A. Matlab codes BEGIN CODE A.3 Spherical Bessel functions f u n c t i o n [ j ] = s p h b e s s e l ( k, n, r ) s p h b e s s e l C a l c u l a t e s t h e s p h e r i c a l b e s s e l f u n t i o n s ( from t h e c i l y n d r i c a l ones ) o f f i r s t k i n d f o r a g i v e n n v a l u e. S y n t a x : [ h ] = s p h h a n k e l ( k, n, r ) I n p u t s : k wave number n v a l u e f o r t h e c u r r e n t l y n p o s i t i o n i n t h e loop r r a d i o u s O u t p u t s : h s p e h r i c a l b e s s e l o f f i r s t k i n d f o r a g i v e n n Other m f i l e s r e q u i r e d : none S u b f u n c t i o n s : none MAT f i l e s r e q u i r e d : none Author : G u i l l e r m o Moreno A p r i l 2008; e m a i l : k i t x i n i t h o t m a i l. com BEGIN CODE J= b e s s e l j ( n+ ( 1/2), ( k r ) ) ; c l y l i n d r i c a l B e s s e l j=s q r t ( pi./(2 k r ) ). J ; s p h e r i c a l b e s s e l END OF CODE A.4 Spherical Hankel functions f u n c t i o n [ h ] = s p h h a n k e l ( k, n, r ) SPH HANKEL C a l c u l a t e s t h e s p h e r i c a l h a n k e l f u n t i o n s ( from t h e c i l y n d r i c a l ones ) o f f i r s t k i n d f o r a g i v e n n v a l u e. S y n t a x : [ h ] = s p h h a n k e l ( k, n, r ) I n p u t s : k wave number n v a l u e f o r t h e c u r r e n t l y n p o s i t i o n i n t h e loop r r a d i o u s O u t p u t s : h s p e h r i c a l h a n k e l o f f i r s t k i n d f o r a g i v e n n Other m f i l e s r e q u i r e d : none S u b f u n c t i o n s : none MAT f i l e s r e q u i r e d : none Author : G u i l l e r m o Moreno A p r i l 2008;

65 A.5. Pressure on the 50 microphones from a monopole 51 BEGIN CODE H = b e s s e l h ( n+ ( 1/2), ( k r ) ) ; C y l i n d r i c a l Hankel h =s q r t ( pi/ ( 2 k r ) ). H; S p h e r i c a l h a n k e l END CODE A.5 Pressure on the 50 microphones from a monopole f u n c t i o n [ P r e s s u r e 5 0 m i c s p h e r i c a l ]= PressureOnThe50micSphere ( k, r0, t h e t a 0, phi0, p o s i t i o n s ) PressureOnThe50micSphere S i m u l a t e s t h e p r e s s u r e on t h e s p h e r e and e x t r a c t t h e v a l u e o f t h i s p r e s s u r e f o r t h e p o s i t i o n s where t h e 50 microphes are s i t u a t e d. I n p u t s : k wave number r0 d i s t a n c e t o t h e s o u r c e t h e t a 0 e l e v a t i o n o f t h e s o u r c e phi0 a z i m u t h o f t h e s o u r c e p o s i t i o n s v e c t o r w i t h t h e p o s i t i o n s o f t h e mic i n t h e s p h e r e O u t p u t s : P r e s s u r e 5 0 m i c s p h e r i c a l p r e s s u r e i n t h e 50 microphones on t h e r i g i d s p h e r e (4 t h column ) along w i t h t h e s p h e r i c a l c o o r d i n a t e s ( r, t h e t a, p h i ) o f them. Other m f i l e s r e q u i r e d : s p h b e s s e l. m, s p h h a n k e l. m, s p h e r i c a l h a r m o n i c s.m MAT f i l e s r e q u i r e d : none See a l s o : M i c r o p h o n e s p o s i t i o n s. m, Author : G u i l l e r m o Moreno A p r i l 2008; e m a i l : k i t x i n i t h o t m a i l. com BEGIN CODE G e t t i n g t h e c o o r d i n a t e s o f t h e microphone a= p o s i t i o n s ( 1, 1 ) ; t a k i mg t h e v a l u e f o r t h e r a d i u s o f t h e s p h e r e t h e t a = p o s i t i o n s ( :, 2 ) ; p h i = p o s i t i o n s ( :, 3 ) ; t h i s i s a column v e c t o r f o r p r a c t i c a l m a t t e r i n t h e p l o t t i n g, and c a l c u l a t i o n s. ka=k a ; Order t r u n c a t i o n N=20; Main loop P s p h e r i c a l=0; P p l a n e = 0 ; f o r n=0:n Yt=0; i n i c i a l i z a t i o n o f t h e v a r i a b l e f o r t h e n e x t loop f o r m= n : n S p h e r i c a l harmonics c a l c u l a t i o n Ynm = s p h e r i c a l h a r m o n i c s ( n, m, t h e t a, p h i ) ; S p h e r i c a l Harmonics f o r t h e s o u r c e Yo= s p h e r i c a l h a r m o n i c s ( n, m, t h e t a 0, phi0 ) ;

66 52 A. Matlab codes end Yt = Yt + Ynm. conj ( Yo ) ; Product o f t h e e x p r e s i o n o f t h e r e c o n s t r u c t i o n c o m p u t a t i o n o f t h e B e s s e l and Hankel j a = s p h b e s s e l ( k, n, a ) ; b e s s e l needed f o r t h e d e r i v a t i o n jm 1 = s p h b e s s e l ( k, n 1, a ) ; d e r i v a t i o n o f B e s s e l f u n c t i o n d j = jm 1 (( n+1)/ka ) j a ; Hankel f o r t h e s o u r c e and t h e s p h e r e ho= s p h h a n k e l ( k, n, r0 ) ; ha= s p h h a n k e l ( k, n, a ) ; needed f o r t h e d e r i v a t i o n hm 1= s p h h a n k e l ( k, n 1, a ) ; d e r i v a t i o n o f Hankel f u n c t i o n dh=hm 1 (( n+1)/ka ) ha ; R a d i a l f u n t i o n s p l a n e wave ( a t r=a ) Rplane =((( i ) ˆ n ) ( ja ( d j/ dh ) ha ) ) ; s p h e r i c a l wave ( a t r=a w i t h t h e o r i g i n i n r0 ) Rsphere =(( i ) ho ( ja ( d j/ dh ) ha ) ) ; Rsphere =( ho ( ja ( d j/ dh ) ha ) ) ; Pressure p l a n e wave P p l a n e=p p l a n e+yt 4 pi Rplane ; s p h e r i c a l wave P s p h e r i c a l=p s p h e r i c a l+yt 4 p i Rsphere ; P s p h e r i c a l=p s p h e r i c a l+yt Rsphere ; end a m p l i t u d e o f t h e monopole f o r t h e case o f comparisson w i t h green f u n c t i o n P s p h e r i c a l=p s p h e r i c a l k ; Q= ; rho= 1. 2 ; c=343; a m p l i t u d e o f t h e monopole f o r t h e case o f comparisson w i t h r e a l monopole b a f f l e d P s p h e r i c a l=p s p h e r i c a l ( ( rho c k ˆ 2 ) ) Q; we t a k e t h e v a l u e s o f t h e p r e s s u r e f o r t h e 50 p o s i t i o n s where t h e microphones are which c o r r e s p o n d s t o t h e d i a g o n a l o f t h e m a t r i x. P s p h e r i c a l 5 0 m i c s=0; P p l a n e 5 0 m i c s=0; f o r end f=1:50 P p l a n e 5 0 m i c s ( f )=P p l a n e ( f, f ) ; P s p h e r i c a l 5 0 m i c s ( f )= P s p h e r i c a l ( f, f ) ;

67 A.6. Pressure on the 50 microphones from a baffled monopole with reflections 53 c r e a t i n g a m a t r i x w i t h t h e c o o r d i n a t e s and v a l u e s f o r t h e p r e s u r e s a t t h e p o s i t i o n o f t h e 50 mics. s p h e r i c a l wave case P r e s s u r e 5 0 m i c s p h e r i c a l ( :, 1 : 3 ) = p o s i t i o n s ( :, 1 : 3 ) ; p o s i t i o n s o f t h e microphones P r e s s u r e 5 0 m i c s p h e r i c a l ( :, 4) = P s p h e r i c a l 5 0 m i c s ; l a s t column c o r r e s p o n d t o t h e p r e s s u r e r e a l v a l u e p l a n e wave case P r e s s u r e 5 0 m i c p l a n e ( :, 1 : 3 ) = p o s i t i o n s ( :, 1 : 3 ) ; p o s i t i o n s o f t h e microphones P r e s s u r e 5 0 m i c p l a n e ( :, 4) = P p l a n e 5 0 m i c s ; l a s t column c o r r e s p o n d t o t h e p r e s s u r e r e a l v a l u e END OF CODE A.6 Pressure on the 50 microphones from a baffled monopole with reflections f u n c t i o n [ P r e s s u r e 5 0 m i c s p h e r i c a l, S c a t t e r e d F i e l d ] = P a n e l S o u n d F i e l d 5 0 m i c R e f l e c t i o n s R e s c a t t e r e d ( k, xp, yp, zp, Lx, Ly, x0, y0, p o s i t i o n s, p o s R e f l e c t i o n s, P ) P a n e l S o u n d F i e l d 5 0 m i c R e f l e c t i o n s R e s c a t t e r e d C a l c u t e s t h e s o u n f i e l d produced by t h e b a f f l e d p a n e l i n t h e s p h e r e. t a k e s i n t o a c c o u n t s c a t t e r i n g and r e f l e c t i o n s o f t h e s p h e r e from t h e b a f f l e. S y n t a x : [ P r e s s u r e 5 0 m i c s p h e r i c a l, S c a t t e r e d F i e l d ] = P a n e l S o u n d F i e l d 5 0 m i c R e f l e c t i o n s ( k, xp, yp, zp, Lx, Ly, x0, y0, p o s i t i o n s, P ) I n p u t s : f f r e q u e n c y xp P o i n t t o c a l t u l a t e t h e d i s p l a c e m e n t i n t h e x c o o r d i n a t e yp P o i n t t o c a l t u l a t e t h e d i s p l a c e m e n t i n t h e y c o o r d i n a t e zp P o i n t t o c a l t u l a t e t h e d i s p l a c e m e n t i n t h e z c o o r d i n a t e Lx Dimension along t h e one c o o r d i n a t e o f t h e p l a t e Ly Dimension along t h e a n o t h e r c o o r d i n a t e o f t h e p l a t e P Number o f d i v i s i o n s i n t h e p l a t e per edge x0 x c o o r d i n a t e o f t h e d r i v i n g p o i n t y0 y c o o r d i n a t e o f t h e d r i v i n g p o i n t O u t p u t s : P r e s s u r e 5 0 m i c s p h e r i c a l p r e s s u r e i n t h e 50 microphones on t h e r i g i d s p h e r e (4 t h column ) along w i t h t h e s p h e r i c a l c o o r d i n a t e s ( r, t h e t a, p h i ) o f them. S c a t t e r e d F i e l d I s t h e s c a t t e r e d f i e l d from t h e i m a g i n a r y s p h e r e ( r e f l e c t i o n s from t h e r e a l s p h e r e ) Other m f i l e s r e q u i r e d : none S u b f u n c t i o n s : P l a t e d i s p l a c e m e n t S i n g l e M o n o MAT f i l e s r e q u i r e d : none See a l s o : P l a t e d i s p l a c e m e n t. Author : G u i l l e r m o Moreno e m a i l : k i t x i n i t h o t m a i l. com JuLy 2008; BEGIN CODE G e t t i n g t h e c o o r d i n a t e s o f t h e microphone i n t h e s p h e r e rho= 1. 2 ; c=343; a= p o s i t i o n s ( 1, 1 ) ; t a k i mg t h e v a l u e f o r t h e r a d i u s o f t h e s p h e r e t h e t a = p o s i t i o n s ( :, 2 ) ; p h i = p o s i t i o n s ( :, 3 ) ; t h i s i s a column v e c t o r f o r p r a c t i c a l m a t t e r

68 54 A. Matlab codes i n t h e p l o t t i n g, and c a l c u l a t i o n s. Taking t h e c o o r d i n a t e s f o r t h e s p h e r e from t h e i m a g i n a r y one R e f l e c t r= p o s R e f l e c t i o n s ( :, 1 ) ; R e f l e c t t h e t a= p o s R e f l e c t i o n s ( :, 2 ) ; R e f l e c t p h i = p o s R e f l e c t i o n s ( :, 3 ) ; ka=k a ; t r a n s f o r m i n g t h e c a r t e s s i a n c o o r d i n a t e s o f t h e s o u r c e monopole r0=s q r t ( xp ˆ2+ yp ˆ2+ zp ˆ 2 ) ; t h e t a 0=acos ( zp/ s q r t ( xp ˆ2+ yp ˆ2+ zp ˆ 2 ) ) ; phi0=atan ( yp/xp ) ; c a l c u l a t i n g t h e i n c i d e n t p r e s s u r e r a d i e a t e d from t h e b a f f l e d p l a t e. Order t r u n c a t i o n N=20; Main loop P s p h e r i c a l=0; P s c a t t=0; P r e s c a t t=0; f o r n=0:n Yt=0; i n i c i a l i z a t i o n o f t h e v a r i a b l e f o r t h e n e x t loop Ysc=0; i n i c i a l i z a t i o n o f t h e v a r i a b l e f o r t h e n e x t loop Ynm sct=0; i n i c i a l i z a t i o n o f t h e v a r i a b l e f o r t h e n e x t loop Ys=0; i n i c i a l i z a t i o n o f t h e v a r i a b l e f o r t h e n e x t loop Y t r e s c a t t=0; f o r m= n : n end S p h e r i c a l harmonics c a l c u l a t i o n Ynm = s p h e r i c a l h a r m o n i c s ( n, m, t h e t a, p h i ) ; S p h e r i c a l Harmonics f o r t h e s o u r c e Yo= s p h e r i c a l h a r m o n i c s ( n, m, t h e t a 0, phi0 ) ; Yt = Yt + Ynm. conj ( Yo ) ; Product o f t h e e x p r e s i o n o f t h e r e c o n s t r u c t i o n Product o f t h e s c a t t e r e d f i e l d from t h e i m a g i n a r y s p h e r e I s implemented t h i s way due t o t h e d i f f e r e n t r a d i u s o f each p o i n t needed l a t e r Y t r e s=0; f o r s c t=1: l e n g t h ( t h e t a ) S p h e r i c a l harmonics c a l c u l a t i o n Ynm sct ( s c t ) = s p h e r i c a l h a r m o n i c s ( n, m, R e f l e c t t h e t a ( s c t ), R e f l e c t p h i ( s c t ) ) ; Ys ( s c t ) = Ynm sct ( s c t ). conj ( Yo ) ; t a k i n g t h e harmonics f o r l a t e r m u l t i p l i c a t i o n o f t h e s c a t t e r e d f i e l d Y t r e s ( s c t )= Yt ( s c t, s c t ) ; end Ysc=Ysc+Ys ; Y t r e s c a t t=y t r e s c a t t+ Y t r e s ; c o m p u t a t i o n o f t h e B e s s e l and Hankel j a = s p h b e s s e l ( k, n, a ) ;

69 A.6. Pressure on the 50 microphones from a baffled monopole with reflections 55 b e s s e l needed f o r t h e d e r i v a t i o n jm 1 = s p h b e s s e l ( k, n 1, a ) ; d e r i v a t i o n o f B e s s e l f u n c t i o n d j = jm 1 (( n+1)/ka ) j a ; Hankel f o r t h e i m a g i n a r y s p h e r e hs= s p h h a n k e l ( k, n, R e f l e c t r ) ; Hankel f o r t h e s o u r c e and t h e s p h e r e ho= s p h h a n k e l ( k, n, r0 ) ; ha= s p h h a n k e l ( k, n, a ) ; needed f o r t h e d e r i v a t i o n hm 1= s p h h a n k e l ( k, n 1, a ) ; d e r i v a t i o n o f Hankel f u n c t i o n dh=hm 1 (( n+1)/ka ) ha ; R a d i a l f u n t i o n s s p h e r i c a l wave ( a t r=a w i t h t h e o r i g i n monopole a t r0 ) Rsphere =( ho ( ja ( d j/ dh ) ha ) ) ; Pressure s p h e r i c a l wave, adding a l s o t h e r e f l e c t i o n s from t h e b a f f l e P s p h e r i c a l=p s p h e r i c a l+yt 4 pi Rsphere ; s p h e r i c a l wave, o n l y t h e s c a t t e r e d f i e l d c a l c u l a t e d i n t h e o r i g i n a l s p h e r e f o r s s=1: l e n g t h ( p h i ) end Hankel f o r t h e i m a g i n a r y s p h e r e hs ( s s )=s p h h a n k e l ( k, n, R e f l e c t r ( s s ) ) ; r a d i a l f u n c t i o n R s c a t ( s s )=( ( i ) ho ( ( d j/ dh ) hs ( s s ) ) ) ; P s c a t ( s s )=Ysc ( s s ) 4 pi R s c a t ( s s ) ; P s c a t t= P s c a t t + P s c a t ; r e s c a t t e r i n g t h e r e f l e c t i o n s i n t h e o r i g i n a l s p h e r e P r e s c a t t=p r e s c a t t+p s c a t t ( ja ( d j/ dh ) ha ). Y t r e s c a t t ; end a m p l i t u d e o f t h e monopole f o r t h e case o f comparisson w i t h monopole w i t h r e a l volume v e l i c i t y. C a l c u l a t i o n o f t h e volume v e l o c i t y [W,V] = P l a t e d i s p l a c e m e n t S i n g l e M o n o ( k, xp, yp, Lx, Ly, x0, y0 ) ; Q=V ( Lx Ly/ ( P 1) ˆ 2 ) ; volume v e l o c i t y P s p h e r i c a l=p s p h e r i c a l ( ( rho c k ˆ 2 )/ ( 2 pi ) ) Q; P r e s c a t t=p r e s c a t t ( ( rho c k ˆ 2 )/ ( 2 pi ) ) Q; we t a k e t h e v a l u e s o f t h e p r e s s u r e f o r t h e 50 p o s i t i o n s where t h e microphones are which c o r r e s p o n d s t o t h e d i a g o n a l o f t h e m a t r i x.

70 56 A. Matlab codes P s p h e r i c a l 5 0 m i c s=0; P s c a t 5 0 m i c s= P r e s c a t t ; f o r f=1:50 adding up t h e s c a t t e r e d f i e l d form t h e i m a g i n a r y s o u r c e and t h e r e f o r e s i m u l a t i n g t h e r e f l e c t i o n i n t h e s p h e r e P s p h e r i c a l 5 0 m i c s ( f )= P s p h e r i c a l ( f, f )+ P r e s c a t t ( f ) ; end c r e a t i n g a m a t r i x w i t h t h e c o o r d i n a t e s and v a l u e s f o r t h e p r e s u r e s a t t h e p o s i t i o n o f t h e 50 mics. s p h e r i c a l wave case P r e s s u r e 5 0 m i c s p h e r i c a l ( :, 1 : 3 ) = p o s i t i o n s ( :, 1 : 3 ) ; p o s i t i o n s o f t h e microphones P r e s s u r e 5 0 m i c s p h e r i c a l ( :, 4) = P s p h e r i c a l 5 0 m i c s ; l a s t column c o r r e s p o n d t o t h e p r e s s u r e r e a l v a l u e S c a t t e r e d F i e l d ( :, 1 : 3 ) = p o s i t i o n s ( :, 1 : 3 ) ; p o s i t i o n s o f t h e microphones S c a t t e r e d F i e l d ( :, 4) = P s c a t 5 0 m i c s ; l a s t column c o r r e s p o n d t o t h e p r e s s u r e r e a l v a l u e END CODE A.7 Plate displacement f u n c t i o n [W,V] = P l a t e d i s p l a c e m e n t S i n g l e M o n o ( k, xp, yp, Lx, Ly, x0, y0 ) P l a t e d i s p l a c e m e n t S i n g l e M o n o C a l c u t e s t h e d i s p l a c e m e n t f o r a f i n i t e p l a t e d r i v e n a t a r i g i d s i n g l e p o i n t. The p l a t e i s s i m p l y s u p p o r t e d. Only c a l c u l a t e s f o r one o f t h e d i v i s i o n o f t h e p l a t e. we assume t h a t t h e p l a t e i s a t a f i x d i s t a n c e from t h e s p h e r e and n o t t i l t e d. S y n t a x : [W, V] = P l a t e d i s p l a c e m e n t S i n g l e M o n o ( k, xp, yp, Lx, Ly, P, x0, y0 ) I n p u t s : f f r e q u e n c y xp P o i n t t o c a l t u l a t e t h e d i s p l a c e m e n t i n t h e one c o o r d i n a t e yp P o i n t t o c a l t u l a t e t h e d i s p l a c e m e n t i n t h e one c o o r d i n a t e Lx Dimension along t h e x c o o r d i n a t e Ly Dimension along t h e y c o o r d i n a t e P Number o f d i v i s i o n s i n t h e p l a t e x0 x c o o r d i n a t e o f t h e d r i v i n g p o i n t y0 y c o o r d i n a t e o f t h e d r i v i n g p o i n t O u t p u t s : W D i s p l a c e m e n t o f t h e p o i n t s V V e l o c i t y Other m f i l e s r e q u i r e d : none S u b f u n c t i o n s : none MAT f i l e s r e q u i r e d : none Author : G u i l l e r m o Moreno e m a i l : k i t x i n i t h o t m a i l. com May 2008; BEGIN CODE PARAMETERS t h i c k n e s s o f t h e p l a t e

71 A.8. Pressure reconstruction 57 h = ; d e n s i t y o f t h e p l a t e rhop = 7800; Young modulus o f t h e p l a t e E = 2 10ˆ11; Poisson s r a t i o nu = 0. 3 ; Skudrzyk s c o n s t a n t squared ( F o u r i e r a c o u s t i c s book, page 56) skudct = s q r t ( ( E h ˆ 2 )/ ( 1 2 rhop (1 nu ˆ 2 ) ) ) ; i n p u t Force F = 1 0 ; c=343; f= ( k c )/ ( 2 pi ) ; omega=2 pi f ; i m p l e m e n t a t i o n o f e q u a t i o n i n F o u r i e r a c o u s t i c s book W=0; f o r m=1:20 f o r n=1:20 omega mn=skudct ( (m pi/ Lx ) ˆ2+( n pi/ Ly ) ˆ 2 ) ; PHI 0 = ( 2/ s q r t ( Lx Ly ) ) s i n (m pi ( x0 )/ Lx ) s i n ( n pi ( y0 )/ Ly ) ; PHI = ( 2/ s q r t ( Lx Ly ) ) s i n ( n pi ( yp )/ Ly ) s i n (m pi ( xp )/ Lx ) ; W=W+PHI 0 PHI/ ( omegaˆ2 omega mn ˆ 2 ) ; end end W= F/ ( rhop h ) W; D i s p l a c e m e n t V= j omega W; V e l o c i t y return END OF CODE A.8 Pressure reconstruction f u n c t i o n [ P Recons ]= SNAH recon ( t h e t a, phi, Rrec, f r e q, S p h e r e p r e s s u r e ) SNAH recon T h i s f u n c t i o n i m p l e m e n t s t e h SNAH r e c o n s t r u c t i o n e q u a t i o n f o r a r i g i d s p h e r e w i t h 50 mic f l u s h e d i n t h e s u r f a c e and a f i x r a d i o u s a. S y n t a x : [ P r s s r e c o n s ]= SNAH recon ( Rrec, f r e q, S p h e r e p r e s s u r e ) I n p u t s : Rrec R e c o n s t r u c t i o n r a d i u s f r e q C u r r e n t f r e q u e n c y o f s t u d y S p h e r e p r e s s u r e M a t rix w i t h t h e s p h e r i c a l c o o r d i n a t e s o f t h e microphones i n t h e f i r s t 3 columns ( r, e l e v a t i o n, a z i m u t h ), and t h e p r e s s u r e v a l u e i n t h e l a s t columm t h e t a v a l u e or v e c t o r w i h t t h e e l e v a t i o n a n g l e/s d e s i r e d f o r t h e c a l c u l a t i o n p h i v a l u e or v e c t o r w i h t t h e a z i m u t h a l a n g l e/s d e s i r e d f o r t h e c a l c u l a t i o n O u t p u t s : P Recons v e c t o r w i t h t h e complex v a l u e f o r t h e p r e s s u r e r e c o n t r u c t e d Other m f i l e s r e q u i r e d : PressureOnThe50micSphere. m, s p h b e s s e l. m, s p h e r i c a l h a r m o n i c s. m, s p h h a n k e l.m MAT f i l e s r e q u i r e d : N u m I n t e g r c o e f f i c i e n t s Author : G u i l l e r m o Moreno J u l y 2008; k i t x i n i t g m a i l. com

72 58 A. Matlab codes BEGIN CODE d e f i n i t i o n o f v a r i a b l e s k=(2 pi f r e q )/ ; a= ; r a d i u s o f t h e s p h e r e (m) ka=k a ; p h i=phi ; due t o t h a l a t e r m u l t i p l i c a t i o n o f v e c t o r s Order o f t h e t r u n c a t i o n N=5; Loading t h e w e i g h t i n g f a c t o r s load N u m I n t e g r c o e f f i c i e n t s ; i m p l e m e n t a t i o n o f t h e r e c o n s t r u c c t i o n P r s s r e c o n s=0; v a r i a b l e i n i c i a l i z a t i o n f o r n=0:n p r o d u c t=0; i n i c i a l i z a t i o n o f t h e v a r i a b l e f o r t h e n e x t loop f o r m= n : n end S p h e r i c a l harmonics c a l c u l a t i o n Ynm = s p h e r i c a l h a r m o n i c s ( n, m, t h e t a, p h i ) ; Gaussian n u m e r i c a l i n t e g r a t i o n Num inte=0; i n i c i a l i z a t i o n o f t h e v a r i a b l e f o r t h e n e x t loop t r y t o do t h i s w i t h m a t r i x m u l t i p l i c a t i o n and sum f u c t i o n..!!!! f o r i i =1: l e n g t h (A) temp1 = A( i i ) S p h e r e p r e s s u r e ( i i, 4 ) conj ( s p h e r i c a l h a r m o n i c s ( n,m,... S p h e r e p r e s s u r e ( i i, 2 ), S p h e r e p r e s s u r e ( i i, 3 ) ) ) ; Num inte = Num inte + temp1 ; end temp2 = Ynm Num inte ; t e m p o r a l s t o r a g e v a r i a b l e p r o d u c t= p r o d u c t + temp2 ; Product o f t h e e x p r e s i o n o f t h e r e c o n s t r u c t i o n C a l c u l a t i o n o f t h e B e s s e l f u n c t i o n s s p h e r i c a l b e s s e l f o r t h e r e c o n s t r u c t i o n r a d i u s j r = s p h b e s s e l ( k, n, Rrec ) ; s p h e r i c a l b e s s e l f o r t h e r i g i d s p h e r e r a d i u s j a = s p h b e s s e l ( k, n, a ) ; b e s s e l needed f o r t h e d e r i v a t i o n jm 1 = s p h b e s s e l ( k, n 1, a ) ; d e r i v a t i o n o f B e s s e l f u n c t i o n d j a = jm 1 (( n+1)/ka ) j a ; C a l c u l a t i o n o f t h e Hankel f u n c t i o n s s p h e r i c a l h a n k e l w i t h ka argument ha= s p h h a n k e l ( k, n, a ) ; s p h e r i c a l h a n k e l needed f o r t h e d e r i v a t i o n hm 1= s p h h a n k e l ( k, n 1, a ) ; d e r i v a t i o n o f Hankel f u n c t i o n dha=hm 1 (( n+1)/ka ) ha ; Coeff = j r/ ( ja ( d j a/ dha ) ha ) ; C o e e f i c i e n t d e p e n d e n t o f n o n l y. s e e e q u a t i o n end R e c o n s t r u c t e d i n c i d e n t p r e s s u r e P r s s r e c o n s = P r s s r e c o n s + Coeff p r o d u c t ; c o n d i t i o n i f Prss i s a m a t r i x i f s i z e ( P r s s r e c o n s )>1

73 A.9. Particle velocity reconstruction 59 r e t u r n s t h e diagonal, which c o r r e s p o n d s t o t h e d e s i r e d v a l u e s P Recons=diag ( P r s s r e c o n s ) ; e l s e P Recons= P r s s r e c o n s ; end END CODE A.9 Particle velocity reconstruction f u n c t i o n [ U r a d i a l, U t h e t a, U phi ]= SNAH velocity ( t h e t a, phi, Rrec, f r e q, S p h e r e p r e s s u r e ) SNAH recon T h i s f u n c t i o n i m p l e m e n t s t e h SNAH r e c o n s t r u c t i o n e q u a t i o n f o r a r i g i d s p h e r e w i t h 50 mic f l u s h e d i n t h e s u r f a c e and a f i x r a d i o u s a. REturns t h e v a l u e f o r t h e r e c o n s t r u c t e c p a r t i c l e v e l o c i t y. S y n t a x : [ P r s s r e c o n s ]= SNAH recon ( Rrec, f r e q, S p h e r e p r e s s u r e ) I n p u t s : Rrec R e c o n s t r u c t i o n r a d i u s f r e q C u r r e n t f r e q u e n c y o f s t u d y S p h e r e p r e s s u r e M a t rix w i t h t h e s p h e r i c a l c o o r d i n a t e s o f t h e microphones i n t h e f i r s t 3 columns ( r, e l e v a t i o n, a z i m u t h ), and t h e p r e s s u r e v a l u e i n t h e l a s t columm t h e t a v a l u e or v e c t o r w i h t t h e e l e v a t i o n a n g l e/s d e s i r e d f o r t h e c a l c u l a t i o n p h i v a l u e or v e c t o r w i h t t h e a z i m u t h a l a n g l e/s d e s i r e d f o r t h e c a l c u l a t i o n O u t p u t s : U r a d i a l v e c t o r w i t h t h e complex v a l u e f o r t h e r a d i a l component U t h e t a v e c t o r w i t h t h e complex v a l u e f o r t h e p o l a r component U phi v e c t o r w i t h t h e complex v a l u e f o r t h e a z i m u t h a l component Other m f i l e s r e q u i r e d : PressureOnThe50micSphere. m, S u b f u n c t i o n s : s p h b e s s e l. m, s p h e r i c a l h a r m o n i c s. m, s p h h a n k e l.m MAT f i l e s r e q u i r e d : N u m I n t e g r c o e f f i c i e n t s Author : G u i l l e r m o Moreno J u l y 2008; k i t x i n i t g m a i l. com BEGIN CODE d e f i n i t i o n o f v a r i a b l e s k=(2 pi f r e q )/ ; a= ; r a d i u s o f t h e s p h e r e (m) ka=k a ; p h i=phi ; due t o t h a l a t e r m u l t i p l i c a t i o n o f v e c t o r s Order o f t h e t r u n c a t i o n N=5; Loading t h e w e i g h t i n g f a c t o r s load N u m I n t e g r c o e f f i c i e n t s ; Sumation i m p l e m e n t a t i o n v a r i a b l e s i n i c i a l i z a t i o n s f o r t h e loop P r s s r e c o n s=0; u r a d i a l = 0 ; u p h i = 0 ; u t h e t a = 0 ;

74 60 A. Matlab codes f o r n=0:n i n i c i a l i z a t i o n o f t h e v a r i a b l e s f o r t h e n e x t loop p r o d u c t=0; p r o d u c t p h i=0; Y t h e t a=0; d Y t h e t a = 0 ; p r o d u c t t h e t a = 0 ; Pnm=0; Pn 1=0; f o r m= n : n S p h e r i c a l harmonics c a l c u l a t i o n Ynm = s p h e r i c a l h a r m o n i c s ( n, m, t h e t a, p h i ) ; c a l c u l a t i o n o f t h e s p h e r i c a l harmonics/ s i n, e x p r e s s i o n 13 i n P a r i s 08 paper s i n t h e t a = ones ( r e s o l u t i o n, 1 ) s i n ( t h e t a ) ; making a m a t r i x w i t h t h e r i g h v a l u e s Ynm phi = Ynm./ s i n t h e t a ; Gaussian n u m e r i c a l i n t e g r a t i o n Num inte=0; i n i c i a l i z a t i o n o f t h e v a r i a b l e f o r t h e n e x t loop t r y t o do t h i s w i t h m a t r i x m u l t i p l i c a t i o n and sum f u c t i o n..!!!! f o r i i =1: l e n g t h (A) temp1 = A( i i ) S p h e r e p r e s s u r e ( i i, 4 ) conj ( s p h e r i c a l h a r m o n i c s ( n,m,... S p h e r e p r e s s u r e ( i i, 2 ), S p h e r e p r e s s u r e ( i i, 3 ) ) ) ; Num inte = Num inte + temp1 ; end Product o f t h e e x p r e s i o n o f t h e r e c o n s t r u c t i o n f o r t h e r a d i a l componnet o f t h e p a r t i c l e v e l o c i t y p r o d u c t= p r o d u c t + Ynm Num inte ; Product o f t h e e x p r e s i o n o f t h e r e c o n s t r u c t i o n f o r t h e t a component o f t h e p a r t i c l e v e l o c i t y p r o d u c t p h i = p r o d u c t p h i + m Ynm phi Num inte ; c a l c u l a t i o n o f t h e t h e t a componnent o f p a r t i c l e v e l o c i t y Pnm=l e g e n d r e ( n, cos ( t h e t a ) ) ; g e n e r a t e s n+1 p o l i n o m i a l s Pn 1= l e g e n d r e ( n+1, cos ( t h e t a ) ) ; g e n e r a t e s n+2 p o l i n o m i a l s Y t h e t a=0; i f (m>=0) Only t h e p o s i t i v e ones t h e n e g a t i v e s are done i n t h e n e x t sigma i s t h e c o e f f i c i e n t o f t h e S p h e r i c a l harmonics sigma=( s i n ( t h e t a )./(1 ( cos ( t h e t a ) ). ˆ 2 ) ) s q r t ( ( ( 2 n+1) f a c t o r i a l ( n m) )/ ( 4 pi f a c t o r i a l ( n+m) ) ) ; gamma=( n+1) cos ( t h e t a ). Pnm (m+ 1, : ) (n m+1) Pn 1 (m+ 1, : ) ; Yp=exp ( i m p h i ) ( sigma. gamma) ; Y t h e t a=yp ; Computes t h e p o s i t i v e t e r m s end i n t h i s loop, o n l y t h e n e g a t i v e m i n d e x e s are c o n s i d e r e d... e x c l u d i n g m=0 which was done i n t h e p r e v i o u s loop i f (m<= 1) mm=abs (m) ; sigma=( s i n ( t h e t a )./(1 ( cos ( t h e t a ) ). ˆ 2 ) ) s q r t ( ( ( 2 n+1) f a c t o r i a l ( n mm) )/ ( 4 pi f a c t o r i a l ( n+mm) ) ) ; gamma=( n+1) cos ( t h e t a ). Pnm (mm+ 1, : ) (n mm+1) Pn 1 (mm+ 1, : ) ; Yn=(( 1) ˆmm) conj ( exp ( i m p h i ) ( sigma. gamma) ) ;

75 A.10. FFT processing 61 Y t h e t a=yn ; Computes t h e n e g a t i v e t o t a l summation t e r m s end d Y t h e t a=d Y t h e t a+y t h e t a ; p r o d u c t t h e t a = p r o d u c t t h e t a + d Y t h e t a Num inte ; end o f c a l c u l a t i o n o f t h e t h e t a componnent end end C a l c u l a t i o n o f t h e B e s s e l f u n c t i o n s s p h e r i c a l b e s s e l f o r t h e r e c o n s t r u c t i o n r a d i u s j r = s p h b e s s e l ( k, n, Rrec ) ; s p h e r i c a l b e s s e l f o r t h e r i g i d s p h e r e r a d i u s j a = s p h b e s s e l ( k, n, a ) ; b e s s e l needed f o r t h e d e r i v a t i o n jm 1 = s p h b e s s e l ( k, n 1, a ) ; b e s s e l needed f o r t h e d e r i v a t i o n j m 1 r = s p h b e s s e l ( k, n 1, Rrec ) ; d e r i v a t i o n o f B e s s e l f u n c t i o n d j a = jm 1 (( n+1)/ka ) j a ; d e r i v a t i o n o f B e s s e l f u n c t i o n d j r = jm 1r (( n+1)/ k Rrec ) j r ; C a l c u l a t i o n o f t h e Hankel f u n c t i o n s s p h e r i c a l h a n k e l w i t h ka argument ha= s p h h a n k e l ( k, n, a ) ; s p h e r i c a l h a n k e l needed f o r t h e d e r i v a t i o n hm 1= s p h h a n k e l ( k, n 1, a ) ; d e r i v a t i o n o f Hankel f u n c t i o n dha=hm 1 (( n+1)/ka ) ha ; Coeff = 1/( ja ( d j a/ dha ) ha ) ; C o e e f i c i e n t d e p e n d e n t o f n o n l y. s e e e q u a t i o n R e c o n s t r u c t e d i n c i d e n t p r e s s u r e u r a d i a l = u r a d i a l + d j r Coeff p r o d u c t ; u p h i = u p h i + j r Coeff p r o d u c t p h i ; u t h e t a = u t h e t a + j r Coeff p r o d u c t t h e t a ; u r a d i a l= ( i/ ( rho c ) ) u r a d i a l ; u p h i= ( 1/( rho c k r ) ) u p h i ; u t h e t a=( i/ ( rho c k r ) ) u t h e t a ; c o n d i t i o n i f t h e v e l o c i t i e s are m a t r i x i f s i z e ( u r a d i a l )>1 r e t u r n s t h e diagonal, which c o r r e s p o n d s t o t h e d e s i r e d v a l u e s U r a d i a l=diag ( u r a d i a l ) ; U phi=diag ( u p h i ) ; U t h e t a=diag ( u t h e t a ) ; e l s e U r a d i a l= u r a d i a l ; U phi= u p h i ; U t h e t a= u t h e t a ; end END CODE A.10 FFT processing f u n c t i o n M e a s u r e d P r e s s u r e ( r e f, m a t f i l e n a m e ) MeasuredPressure P r o c e s s t h e t i m e domain s i g n a l s r e c o r d e d f o r t h e 50

76 62 A. Matlab codes c h a n n e l mic a r r a y o f B r u e l & K j e r. Saves a l l t h e f f t o f t h e c h a n n e l s i n a matrix, and Mat f i l e. S y n t a x : [ Mic, f r e q, ref num, Channel ]= MeasuredPressure ( r e f, m a t f i l e n a m e ) I n p u t s : r e f Channel d e s i r e d as a r e f e r e n c e, f o r t h e phase i n f o r m a t i o n. m a t f i l e n a m e f i l e name o f t h e t i m e o u t p u t mat f i l e from P u l s e a n a l y z e r O u t p u t s : Mic M a t rix w i t h t h e f f t and phase i n f o r m a t i o n f o r each c h a n n e l. Each columm c o r r e s p o n d t o t h e microphone f f t w i t h t h e same name as t h e number o f column f r e q f r e q u e n c y a x i s ref n u m number o f t h e c h a n n e l t o be d e s i g n e d as a r e f e r e n c e. The number w i l l c o r r e s p o n d t o t h e c h a n n e l name o f t h e r e f e r e n c e t h a t i s n o t gonna be t h e same as t h e number o f microphone. Channel i n a v e c t o r w i t h t h e i n f o r m a t i o n r e g a r d i n g t h e c h a n n e l a s i g n a t i o n t o each microphone. The columm number i s t h e c h a n n e l and t h e v a l u e o f t h e c e l l i s t h e mic a s o c i a t e d t o t h a t c h a n n e l. Other m f i l e s r e q u i r e d : none S u b f u n c t i o n s : none Author : G u i l l e r m o Moreno e m a i l : k i t x i n i t g m a i l. com J u l y 2008 BEGIN CODE l o a d i n g t h e mat f i l e c o i n t a i n i n g t h e 50 c h a n n e l s t i m e s i g n a l s from P u l s e q=[ l o a d, m a t f i l e n a m e ] ; e v a l ( q ) f u n t i o n t o make t h e f f t on t h e windowed s i g n a l L= F i l e H e a d e r. NumberOfSamplesPerChannel ; L= s t r 2 d o u b l e ( L ) ; sample f r e q u e n c y f s=f i l e H e a d e r. SampleFrequency ; f s=s t r 2 d o u b l e ( f s )/ 1 e6 ; window l e n g t h (800 l i n e s i n P u l s e i s a l e n g t h o f 2048) wl =2048; window w=hann ( wl ) ; R e f e r e n c e FFT C a l c u l a t i o n ( c h a n n e l 1) t i m e s i g n a l o f r e f c h a n n e l s s = [ Channel i n t 2 s t r ( r e f ) Data ] ; x r e f=e v a l ( s s ) ; g e t t i n g t h e mic number bb=[ Channel i n t 2 s t r ( r e f ) Header. SignalName ] ; ref num= s t r 2 d o u b l e ( s s c a n f ( e v a l ( bb ), ssh ) ) ; f r e q u e n c y v e c t o r f r e q = f s/ 2 l i n s p a c e ( 0, 1, wl/ 2 ) ; p r o c e s s i n g each c h a n n e l Mic=z e r o s ( wl, 5 0 ) ; Channel=z e r o s ( 0 ) ; f o r M=1:50 t i m e s i g n a l o f one c h a n n e l

77 A.10. FFT processing 63 s = [ Channel i n t 2 s t r (M) Data ] ; x=e v a l ( s ) ; g e t t i n g t h e mic number b=[ Channel i n t 2 s t r (M) Header. SignalName ] ; mic num=e v a l ( b ) ; Auto =0; C r o s s S p e c t r a=0; pos =1; n=1; while ( pos+wl <= L ) w h i l e enough s i g n a l l e f t f a c t o r o f 1. 5 f o r c o m p e n s a t i n g hanning window y = x ( pos : pos+wl 1). w; make window y y r e f = x r e f ( pos : pos+wl 1). w; make window y r e f C a l c u a l t i n g Spectrum f o r c r o s s s c p e c t r a c a l c u l a t i o n Y=2 f f t ( y )/ wl ; Y r e f=2 f f t ( y r e f )/ wl ; Cross=conj (Y). Y r e f ; C r o s s S p e c t r a=c r o s s S p e c t r a+cross ; end C a l c u l a t i n g A u t o s p e c t r u m o f t h e mic a u t o=( abs (2 f f t ( y )/ wl ) ). ˆ 2 ; Auto=Auto+a u t o ; pos = pos + wl/ 2 ; n e x t window, 50 o v e r l a p n=n +1; number i f windows averaged V e c t o r t h a t shows t h e c h a n n e l mic number a s s o c i a t i o n Channel (M) = s t r 2 d o u b l e ( s s c a n f ( mic num, ssh ) ) ; kk=channel (M) ; C r o s s S p e c t r a=c r o s s S p e c t r a/ n ; mic number on t h e s p h e r e i s t h e number o f row pha=atan2 ( imag ( C r o s s S p e c t r a ), r e a l ( C r o s s S p e c t r a ) ) ; temp=s q r t ( Auto/ n ). ( exp ( i. ( 1 unwrap ( pha ) ) ) ) ; i n v e r t phase t o t h e same as t h e measured i n p u l s e end Mic ( :, kk )=temp ; S e l e c t i n g t h e d e s i r e f r e q u e n c y p u t a v a r i a b l e t h a t i n d i c a t e s t h e r e a l f r e q u e n c y s e l e c t e d i n d e x=( d e s i r e f/f s ) wl+1; i n d e x=round ( i n d e x ) ; save ( f f t 5 0 M i c s, Mic, f r e q, ref num, Channel ) END CODE

78

79 Appendix B Baffled vibrating plate Level [db ref 20µPa] Baffled plate Reconstructed Y axis [m] Figure B.1 Simulated sound pressure level generated by a baffled vibrating steel plate at a 40 cm distance point. The plate is 20 by 20 cm and 5 mm thickness. Frequency ka= Level [db ref 20µPa] Baffled plate Reconstructed Y axis [m] Figure B.2 Simulated sound pressure level generated by a baffled vibrating steel plate at a 40 cm distance point. The plate is 20 by 20 cm and 5 mm thickness. Frequency ka=4 65

80

81 Appendix C Measurements images Figure C.1 Set up for the measurement of the monopole source. Figure C.2 Set up for the measurement of the monopole source. 67

82 68 C. Measurements images Figure C.3 Set up for the measurement vibrating plate. Figure C.4 Set up for the measurement vibrating plate.