Study on the Parameters of Phase Conjugation Arrays for Sound Localization and Identification

LIU Song,LI Sheng,ZHAO De-you

(a.State Key Laboratory of Structural Analysis for Industrial Equipment;b.School of Naval Architecture,Dalian University of Technology,Dalian 116024,China)

1 Introduction

Structural vibration and noise have been always concerned by the scholars.Vibration and noise may cause great harm to human body,and have devastating effects on the structures.The identification and location of the sound source are also very important for the development of the naval army.Currently the powerful technique used for this identification and reconstruction is NAH(near-field acoustical holography)[1].NAH is the acoustic processing methods combining of theory and experiment,and overcomes the acoustic diffraction resolution limit which the traditional acoustic holography technology can not break through.According to the complex pressure measured on the holographic plane near the source,the NAH method applies particular holographic algorithm to reconstruct the pressure and normal velocity on the surface of structure and then computes the pressure,intensity vector,the radiation power in the three dimension radiation field.The key technology of NAH is that it can farthest record the evanescent wave which contains the details of the source to rebuild the acoustic field and get the resolution which is far less than the acoustic wave.Because the evanescent wave can be ignored in one wavelength,the holographic plane has to be near the source,furthermore it needs a lot of measurement points and small measurement spacing which lead to many limitations in engineering[2].In addition,the NAH method needs to use the regularization method to solve the instability for the inverse wave propagation problem.

In order to investigate and resolve the above-mentioned problems for NAH such as the large area of the measurement array,the small measurement spacing,and the regularization issues,the identification and reconstruction of the radiation sound field for complex source are investigated based on the phase conjugation(time reversal)methods.The time reversal method can be applied because the linear wave equation contains the second derivation of sound pressure with respect to time only so that both p（ r,t）and p （ r,- t）are the roots of the wave equation where p（ r,t）represents the outgoing wave and the p （ r,- t）represents the incoming wave.Although TRM(time reversal mirror)can realize the reversal propagation of the wave and self-focusing to locate the source,the ability of this method is restricted by the acoustic diffraction limit resolution and theoretically the highest resolution is one-half wavelength,so it can not provide enough characteristics of radiation and information of source.At present the methods breaking the diffraction limit in TRM are:(1)That proposed by Fink[3],in the near field by using subwavelength scattering body,the evanescent wave will be converted to propagation wave,then by using the far field TRM receives thirtieth wavelength resolution.Its mechanism is the evanescent wave converted to the propagation wave is linear and reciprocal.(2)Rosny[5]realized the fourteenth wavelength resolution for the ultrasonic through proposed the novel acoustic sink applied the TRM(single array element)by experimental method.They think the diffraction limit of TRM is due to that after time reversal the back propagation wave generates the radiate outward wave as the same form to the source when gathering at source,which destroys the reciprocal of time reversal process,it needs to set the sink at the original source to absorb or cancel the back propagation wave after time reversal.(3)Conti[4]realized the twentieth wave length resolution through combination the near field microphone array and four outward TRM.Its principle is that the evanescent wave in the wave-number domain can be decomposed into the amplitude and phase parts which can be measured respectively through the near field microphones and the far field TRM and realize the amplification and the restoration of the evanescent wave in the wave number domain.Rosny[6]confirms that the array made of monopole transceivers through the pressure measurement can not break the diffraction limit(0.5λ)for the point source even fully time reversal in near field using two infinite plane array based on analytical method,but the array made of dipole transceivers through the pressure gradient measurement can break the diffraction limit.Applying the time reversal in time-domain corresponds to the phase conjugate in frequency-domain.Namely p（r,t）and p （ r,- t）are equivalent to p（ r,ω）and p*（r,ω），and p（ r,ω）represents the outgoing wave,p*（r,ω）represents the incoming wave.The superscript*means the conjugation.Phaseconjugate method has the better practicability and will get the precise results even the source is moving or the arrays are deformed.In addition,it can significantly improve the resolution of time reversal with the help of the inhomogeneity of the marine medium and the multi-path acoustic channel caused by the marine interface.The identification and reconstruction can be realized based on the specific time reversal or phase conjugation method through measured the radiation sound field.The parameters such as element spacing,array area,source frequency,array forms of phase conjugation arrays consisting of discrete elements are studied numerically for sound source localization.The influences of these parameters on the field magnitude and the spatial size of the conjugated field are investigated.

2 Theory of phase conjugation

The reason that time-reversed sound waves travel backwards is a direct consequence of the lossless linear wave equation for the acoustic pressure p r,（）t:

This equation is time-reversal invariant because it contains only second-order derivatives with respect to time.Equation(1)ensures that if p（ r,t）is a solution then p（ r,- t）is too.Thus,if p（ r,t）represents sound waves expanding away from a sound source,then p （ r,- t）represents sound waves converging toward the same source.In the frequency domain p（ r,t）and p（r,- t）could be replaced by p（ r,ω）and p*（r,ω）respectively,where p has a harmonic time factor of eiωt.

For a perfect PC array,both the original field p and its normal derivativeshould be recorded to serve as the weighting factors for the arrays of monopole and dipole sources,the phase-conjugated field at the field point r due to a point sourceis given by[7]:

The phase-conjugated field pPC/Pis based on both the pressure and pressure gradient measurement and made of both monopole transceivers and dipole transceivers to reverse sound backwards.The phase conjugation arrays could also be based on pressure or particle velocity measurement and be made of monopoles or dipoles to reverse.

The phase-conjugated field by the array made of monopole transceivers based on the pressure measurement is:

The phase-conjugated field by the array made of dipole transceivers based on the pressure gradient measurement is:

3 Calculation the amplitude of the phase conjugation sound field

The amplitude of the phase conjugation field using the discrete array is derived subsequently.Assuming the fluid medium is reciprocal which means the fluid or its boundary is not moving,then G（ r, rs）=G（ rs,r ）,when the field point takes r=rs,the amplitude of the field for the closed array is:

where Im represents the image part.

Take the sound power radiation formula

into formula(6),the focus amplitude of the phase conjugation sound field is:

where S″is the area ratio.The field amplitude is proportional to the sound radiation power.

For the planar conjugate array having circular aperture,the solid angle is Ω=2π（1- cos（θ）），where θ is the angle between the normal direction of the conjugate array and the vector of the aperture to the source.The amplitude of the phase conjugation field received by the planar array is:

The amplitude of the phase conjugation field can be obtained via formula(8)and(9)for complex sound source.

4 Numerical simulations

4.1 The array element spacing and the area of the planar array

The sound field generated by the point source was calculated and the amplitude is taken to be 1 for simplicity.The calculated frequency is f=100Hz unless otherwise specified，the sound speed of the air is C=350m/s and the wave length is λ=3.5m.The array form is square with side length a=2.86*λ=10m.

(1)The calculated phase conjugation amplitude on x-axis among 0～2λ is shown in Figs.1～4 for different array element spacing and number of array elements,and distance from the planar arrays to the sound source(the coordinate origin).(Unless otherwise stated,the results are calculated in air.In these figures,PCP represents the combination measurement of pressure and pressure gradient to calculate the sound field,PCD represents the results only made of dipole transceivers through the pressure gradient,PCM represents the results only made of monopole transceivers through the pressure.Z represents distance from the planar arrays to the sound source.In addition:all of the following results for large number of array elements are used only for numerical simulation.)

Fig.1 Phase conjugation sound field at Z=0.05λ when the array number is 36

Fig.2 Phase conjugation sound field at Z=0.05λ when the array number is 441

Fig.3 Phase conjugation sound field at Z=0.05λ when the array number is 2 601

Fig.4 Phase conjugation sound field at Z=2λ when the array number is 441

From Figs.1～4 it can be seen that only made of dipole transceivers through the pressure gradient in the near field(Z=0.05λ)can break the diffraction limit resolution 0.5λ (resolution refers to the sound source focused spatial dimensions size).In accordance with the Helmholtz integral equation,the combination measurement of pressure and pressure gradient(PCP)should obtain the best resolution,but through the numerical simulation,PCP does not break the 1/2λ resolution.This is mainly due to that the formula for calculating PCP contains the imaginary part of Green function,which equals to zero in the near-field for high wave number and does not contain evanescent wave components,so that no matter how big the array to take,and close to the sound source,it can not break through the diffraction resolution.PCD precisely gets the high resolution for receiving the evanescent wave components.Made of monopole transceivers through the pressure(PCM)also can not break 0.5λ resolution even it included evanescent wave components.This is because multiplied the point source sound field and the Green function whose formula are exactly the same,the sound field amplitude is still decay in accordance with 1/R.

For the same array element spacing,the farther away from the sound source,the smaller the sound pressure amplitude and the lower the resolution.This is also consistent with the 1/R attenuation of the point source sound field.When the distance between the array and the sound source is Z=2*λ,the resolution is about 0.8λ as can be seen in Fig.4.

For the same area of the array(the side length a=10m=2.86λ),the more the number of array elements,the smaller the array element spacing,the higher the resolution,and the bigger phase conjugation sound pressure amplitude.When the parameters of the array area,the distance from the array to the sound source are constant,the phase conjugation amplitude is calculated with the array element spacing values ranging from Δ=0.01λ to 0.6λ,at a step size of 0.01(the paper only given several typical results such as Figs.1～4).Comprehensive consideration of various factors such as the resolution and the array element numbers,the optimal element spacing is 0.15λ(element 21*21=441,Fig.2)which according to the need for spatial sampling which the array elements spacing should be 1/（7-1）0 λ in NAH.When the array element spacing Δ≥0.5λ (36 array elements),even the array is much closed to the sound source,it still can not break the diffraction limit.

(2)Take the same array element spacing Δ=0.5m=0.15λ,the distances between the planar array and the sound source in normal direction are taken as 0.1λ,and 2λ respectively.For different side length of the array based on TRD method,the results are shown in Figs.5～6.

Fig.5 Phase conjugation sound field at Z=0.1λ for different array side length by PCD

Fig.6 Phase conjugation sound field at Z=2λ for different array side length by PCD

For the same array element spacing Δ=0.5m=0.15λ,the results are basically the same in near field which means that the area of the array has less influence on the sound field at this situation,as long as the side length of the array a≧5m or a≧1.4λ,it can break the diffraction limit.Even enlarging the area of the array(a=20m=6λ),the resolution is also 0.25λ;in far field,the area of the array should be large enough to get more acoustic information,namely,even a=30m=9λ,it could only get the resolution of 0.7λ.

(3)The phase conjugation sound field calculated in water at different frequency based on PCD method is shown in Figs.7～8 for the same array element spacing Δ=0.15λ.The distances between the array and the sound source are taken to be Z=0.05λ and Z=2λ which correspond to the side length a=1.4λ,and a=6λ (λ is the wavelength corresponding to the frequency.),respectively.

Fig.7 Phase conjugation sound field at Z=0.05λ for different frequency in water

Fig.8 Phase conjugation sound field at Z=2λ for different frequency in water

It can be seen from Figs.7～8 that in water,the results are basically the same as in the air,that is,the TRM method can break the diffraction limit based on PCD.The higher the frequency increased,the larger the phase conjugation sound field amplitude.On the contrary the higher the frequency increased,the lower the resolution.When f=4 000Hz,the wave length is λ=0.375m,the resolution is approximately 0.45λ.In far field the resolution is also merely 0.75λ.

4.2 Different array forms

The phase conjugation sound fields consisting of different array forms such as the curved array,the crisscross array of the diagonal of the planar array and the linear array are shown in Figs.9～15.The curved array is taken as the whole surface of the sphere and the spherical cap respectively for simplicity.

Fig.9 Phase conjugation sound field using sphere array when the radius is r=0.6λ

Fig.10 Phase conjugation sound field using sphere array when the radius is r=2λ

It can be seen from Figs.9～10 that for closed spherical array,no matter how decreased the spacing of the array,or increased the element number,it can not break the diffraction limit.The limit of formula(6)is k/4π,where k is the wave number.The calculated focused amplitude of the sound field is f/C,where f is the frequency and C is the sound speed.The focused amplitude is 0.286 while the same numerical results can be achieved by PCD and PCP.

For the closed spherical array,when sound power radiation of the point source is calculated by the surrounding sphere whose radius is r0=1,according to the formula(8),the focused sound field amplitude is 0.285 and the same numerical results can be achieved by PCD and PCP.

Fig.11 Phase conjugation sound field using spherical crown array when the radius of sphere is r=0.6λ,the height of the crown is 0.5λ

Fig.12 Phase conjugation sound field using spherical crown array when the radius of sphere is r=2λ,the height of the crown is λ

When the array is curved surface,such as spherical cap,the resolution is roughly 1/2λ that is consistent with the resolution of planar array;merely the focused amplitude is different.So the phase conjugation method is not sensitive to the deformation of the array(from planar array to the curved array).

For the spherical cap array containing 19 165 array elements,the focused amplitude is 0.07 which is cosistent with the results calculated with formula(9).It indicates that formula(9)would be satisfied by enough array elements and small enough array element spacing.

Fig.13 Phase conjugation sound field using cross array at Z=0.1λ when the array number is 117 and the side length is a=2.86λ

Fig.14 Phase conjugation sound field using line array at Z=0.1λ when the array number is 41 and the side length is a=2.86λ

The crisscross array of the diagonal of the planar array is simple and utility with less number of array elements(117)and fully satisfying the engineering precision,but the planar array needs 441 elements(Fig.2)to get the same resolution.In addition,the distance between the crisscross array and the source Z should be less than 0.5λ to break the diffraction limit,and with Z increasing,the phase conjugation amplitude is getting smaller.When using linear array,the resolution can not break the diffraction limit because the linear array only obtained the acoustic information of one direction and lost the information of the perpendicular direction.

Fig.15 Phase conjugation sound field using cross array when the array number is 117 for different distance between the source and the array

4.3 The moving source

In the air,the position of sound source and the array is fixed,but in the ocean,the sound will be in motion and the array will be deformed due to various complex reasons.The deformation of the array has been discussed before.The effects on the phase conjugation method with the moving source are discussed below.

The Doppler formula is used to calculate the moving source.Fig.16 gives the phase conjugation field assuming that the sound source moves from the origin to 0.1λ at the X-axis,by using crisscross array(117 array elements).It can be seen that the moving source has little effect on the resolution.

5 Conclusions

Fig.16 The impact on the phase conjugation sound field with the moving source

The phase conjugation has the advantage of insensitive to the moving source and deformation of the array and could focus even the multi-path channel in the ocean.The parameters such as element spacing,array area,source frequency and array forms of phase conjugation arrays consisting of discrete elements are studied numerically for sound source localization.Conclusions can be derived from above numerical simulation as follows:

(1)The element spacing should be less than 0.5λ to break the diffraction limit with the planar array only made of dipole transceivers in the near field.The optimal element spacing is 0.15λ and the array side length should be greater than or equal to 1.4λ.With increasing frequency the field magnitude increases and the spatial resolution decreases.In far field the res-olution is also merely 0.75λ.

(2)If the closed array and the sound source are the conformal surfaces,the field amplitude is proportional to the sound radiation power.The relationship between the field amplitude and the radiation solid angle is also confirmed by spherical cap array.

(3)Regarding to the number of array elements and the performance the optimal array form is the cross array which leads to subwavelength focusing with fewer elements.

[1]Maynard J D,Williams E G,Lee.Near-field acoustic holography(NAH):I.Theory of generalized holography and the development of NAH[J].J Acoust.Soc.Am.,1985,78(4):1395.

[2]He Zuoyong,He Yuanan,Wang Man.The investigation on the physical acoustics problems of the NAH technique and its application[J].ACTA Acustica.2007,32(2):137-143.

[3]Lerosey G,de Rosny J,Tourin A,Fink M.Focusing beyond the diffraction limit with far-field time reversal[J].Science,2007,315:1120-1122.

[4]Conti S G,Roux P,Kuperman W A.Near-field time-reversal amplification[J].Journal of the Acoustical Society of America,2007,121(6):3602-3606.

[5]de Rosny J,Fink M.Overcoming the diffraction limit in wave physical using a time-reversal mirror and a novel acoustic sink[J].Physical Review Letters,2002,89(12),124301.

[6]de Rosny J,Fink M.Focusing properties of near-field time reversal[J].Physical Review A,2007,76,065801.

[7]Jackson D R,Dowling D R.Phase conjugation in underwater acoustics[J].Journal of the Acoustical Society of America,1991,89(1):171-181.