Sensors & Transucers, Vol. 184, Issue 1, January 15, pp. 53-59 Sensors & Transucers 15 by IFSA Publishing, S. L. http://www.sensorsportal.com Non-invasive an Locally Resolve Measurement of Soun Velocity by Ultrasoun Mario WOLF, Elfgar KÜHNICKE Soli State electronics Laboratory, Technical University of Dresen, 16, Germany Tel.: +49351 463-3167, fax: +49351 463-33 E-mail: Mario.Wolf@tu-resen.e Receive: 14 November 14 /Accepte: 15 December 14 /Publishe: 31 January 15 Abstract: A new metho to measure soun velocity an istance simultaneously an locally resolve by an ultrasoun annular array is evelope in meia with constant soun velocity an then applie in meia with continuously changing properties. Instea of using reflectors at known positions the echoes of moving scattering particles are analyze to etermine the focus position. Although the metho reaches a very high accuracy for constant soun velocity, there are systematic eviation between measurements an soun fiel simulations using Fermat s principle. The soun propagation has to be escribe with a moifie wave equation. A new approach etermining GREEN s functions for a half space with continuously changing properties is presente. It combines the high frequency approximation with an integral transform metho. Copyright 15 IFSA Publishing, S. L. Keywors: Ultrasoun annular arrays, Measurement of soun velocity, Locally resolve, High frequency approximation, Integral transform metho. 1. Introuction A locally resolve monitoring of soun velocity allows estimating locally physical quantities like concentration or temperature or material properties like ensity or elasticity. This facilitates investigating an optimizing many inustrial processes, like mixing or chemical reactions, as well as meical therapy like hyperthermia for cancer treatment. In this contribution, a measurement technique is applie, which allows measuring soun velocity locally resolve using an ultrasoun annular array with concentric rings. In contrast to conventional tomographic techniques, it works without any reflectors or aitional receivers at known positions. Instea of evaluating various propagation paths, the focusing of the array is varie an the focus position an the soun velocity to the focus point are etermine simultaneously by analyzing the echoes of moving scattering particles. This is possible because the focus position epens on the soun velocity an the parameters of the use transucer. Therefore, the time of flight to the focus is use with calibration curves for the simultaneous etermination of soun velocity an focus position. The time of flight to the focus point is etermine from the average amplitue of the echo signals: The emitte wave is reflecte at each particle while the amplitue of the reflecte signal is proportional to the amplitue of the incient wave (Fig. 1a). Therefore the echoes from particles within the focus area are strongest. Although a single echo nearly appears like noise averaging over a sufficient number of signals generates a clear maximum (see Fig. 1b). As particles are in movement it is possible to consier a uniform istribution of particles in average time. So the http://www.sensorsportal.com/html/digest/p_585.htm 53
Sensors & Transucers, Vol. 184, Issue 1, January 15, pp. 53-59 average echo signal amplitue is a measure for the pressure along the acoustic axis of the transucer.. Metho to Measure soun Velocity an Distance Simultaneously.1. Soun Fiel Calculation by GREEN S Functions GREEN s functions are a beneficial tool for moelling soun propagation in homogeneous meia, which escribe the impulse response of a point source in a space with a specific geometry. Exact solutions ha been foun for half spaces, plates an layere meia by means of integral transform [3], [4]. Also approximate harmonic GREEN s functions are calculate giving the transfer function in the Fourier transforme omain [5]. GREEN s functions for a half space are also use in this contribution 1 jkr G ω ( R, θ ) = S( θ ) e, (1) R Fig. 1. Normalize average echo signal amplitue for ifferent numbers of averages in water of 3 C. The basic set-up for the measurements is quiet simple. Only a focusing single probe an a evice for pulsing an ata acquisition are require. The probe can be locate in the examine meium without any ajustment. The meium has to contain scattering particles much smaller than wavelength an in a low concentration, so that the properties of the propagation meium are not influence. The metho ha been introuce by Lenz [1] for meia with constant soun velocity using a focusing single probe. This contribution enhances the metho to annular arrays, which allow moving the focus along the acoustic axis an so measuring in ifferent epths. At first this is qualifie for meia with constant soun velocity an than employe for meia with nonconstant soun velocity. A new metho for moelling soun propagation in meia with continuously changing material parameters is introuce for the evaluation. The moifie wave equation cause by non-constant material parameters is solve for a point source with an integral transform metho an a high frequency approximation []. This paper is ivie in five sections. Section explains how to measure soun velocity an focus position simultaneously. It motivates the moelling of soun propagation in meia with non-constant material properties which is iscusse in Section 3. Section 4 shows some measurement results for meia with constant soun velocity an a linear temperature graient in water. Section 5 gives a summary an perspectives. with the wave number k, ie istance R between source an observation point an the angle θ between the vector between source an observation point an the normal vector of the surface where the source is locate. The irectivity pattern S(θ) concerns the bounary conitions at the interface. An example for an interface of water an PZT (lea- zirconatetitanate) is shown in Fig.. Fig.. Directivity pattern of a point source on a water-pzt-interface... Annular Arrays for Focusing Annular arrays nee far less elements for a well focusing as the wiely sprea linear array. The array use in this contribution consists of 6 elements (see Fig. 3 for structure an imensions) an reaches an extension of the soun beam in the range of the wavelength. Focusing works by a superposition of measure signals where each is elaye with a calculate time: The soun paths from each element to the esigne focus point are etermine (Fig. 4). With a known soun velocity of the propagation meium these paths allow to calculate the ifferences in time of flight, which are use as elay times for each element so that all waves arrive at the focus point at the same time. 54
Sensors & Transucers, Vol. 184, Issue 1, January 15, pp. 53-59 Element ri [mm] ra [mm] 1 3.57 5.5 6.18 3 7.14 7.98 4 8.74 9.44 5 1.1 1.7 6 11.8 11.83 c = 143 m/s c = 156 m/s Fig. 3. Structure an imensions of the curve annular array, centre frequency: fm = 9 MHz, Curvature: R = 5 mm; ri an ra are the insie an outsie raii of the rings. As there is positive interference of course the pressure at this point becomes maximal. Fig. 5. Soun fiels for ifferent soun velocities. Fig. 4. Focusing soun paths to focus points for ifferent soun velocities. Of course if e.g. the temperature in the meium changes this causes a change of soun velocity. The focus moves because the positive interference arises at another point (Fig. 4, re an blue lines). As calculate soun fiels are in the Fourier transforme omain also the elay times have to be transforme into phase shifts. A superposition of the phase shifte soun fiels generates the focus at the esigne point. Fig. 5 shows the soun fiels of the use array, riven with the same elay times for ifferent soun velocities. They were chosen so that the focus woul arise at a istance of 4 mm in a meium with a soun velocity of c = 15 m/s..3. Simultaneous Determination of Soun Velocity an Focus Position It has been shown, that a change of soun velocity isplaces the focus position resulting in a ifferent time of flight to the focus. If the parameters of the transucer are known the time of flight can be calculate as a function of soun velocity. Fig. 6 shows functions for several sets of elay times. The labels mean that the use elay times woul cause a focus in a calibration meium with c = 15 m/s at the corresponing epth. Fig. 6. Calibration curves. The measure time of flight to the focus point can be use to etermine the soun velocity by these curves. Of course then the real focus position emerges. If the elay times are varie the focus is move along the acoustic axis of the array. So the soun velocity can be measure at each point, which allows measuring the istribution of soun velocity. 3. Soun Propagation in Meia with Non-constant Properties As mentione in the previous section the parameters of the transucer have to be known to calculate the soun fiel of the transucer. The focus position also has to be preicte for meia with continuously changing properties where aitional consierations are require. 3.1. Simultaneous Determination of Soun Velocity an Focus Position The most obvious way to moel the soun propagation in meia with continuously changing properties is applying Fermat s principle. It says that a wave propagates along that path which results in a minimal propagation time. 55
Sensors & Transucers, Vol. 184, Issue 1, January 15, pp. 53-59 It can be use to erive Snell s law but if soun velocity changes continuously the propagation path becomes curve an has to be etermine by a calculus of variation [6]. The time of flight T to a point P(x,y) an the path length S to this point are: with a = 1 c( x) 1+ 1 a c T = ln[ ], () c 1+ 1 a c ( x) 1 S = [arcsin( c ( x ) a ) arcsin( c a )], (3) a ( y c x x ) y + 4 y ( c + x) for a linear soun velocity graient (c(x) = c+ x,) an a point source in the origin of coorinates. This ha been use to erive fiels of point sources for linear soun velocity graients. The impulse response is assume by 1 Gδ ( x, y, t) = δ ( t T ( x, y)) (4) S( x, y) with the Dirac function δ. In the Fourier transforme omain this is 1 jωt ( x, y) Gω ( x, y, ω ) = e (5) S( x, y) Fig. 7 shows the phase plots of two examples where soun velocity increases (left) an ecreases (right) in positive irection x. p = v, χ p= v, (6) with pressure p, particle velocity v, an the material parameters mass ensity an elasticity χ. If the material parameters are constant, ifferentiating these equations with respect to location an time leas to the well-known wave equation. If these parameters are functions of location an aitional term appears in the graient of (6). This aitional term also appears in the wave equation: 1 Δp + p χp = (7) Also the efinition of potential Φ is moifie: 1 v = graφ (8) The following examination shall be one for a one-imensional epenence of the material properties in irection z. 3.3. One-imensional Solution Consiering a plane wave propagating in the irection of the graient of the material properties the wave equation for the potential is obtaine Φ+ z z 1 z Φ χ Φ = t (9) If the mass ensity epens linearly on z this equation has a solution in the form of a generalize power series [7]: n+ K ( z z ) (1) Φ = cn n= 1 Fig. 7. Phase plot for positive an negative soun velocity graient. However, there are alreay numerical problems in evaluating the coefficients an the solution with respect to the convergence for this one-imensional case. So it seems not to be feasible fining an exact solution for a two- or three-imensional problem. Although Fermat s principle gives the correct time of flight of the wave there is no information about the amplitue of the wave. 3.. Derivation of the Moifie Wave Equation The soun propagation in liquis is base on two funamental equations, the equations of motion an elasticity: 3.4. High Frequency Approximation an Integral Transform High frequency approximation ha been evelope in geophysics an is applie in techniques like ray tracing [8]. In this contribution, harmonic GREEN's functions shall be erive with this approach in combination with an integral transform metho. This allows calculating a transfer function for a point source for a specific geometry. Just the 56
Sensors & Transucers, Vol. 184, Issue 1, January 15, pp. 53-59 axially symmetric problem is solve because it is much easier manageable than the general problem. The wave equation in cylinrical coorinates (r, θ, z) is use: 1 = r + Φ +... r r r z 1 Φ χ Φ z z t (11) The single erivative to r in the secon term vanishes ue to its scalar multiplication with the graient of the mass ensity, having only a component in z irection. Applying a Hankel transform as escribe in [3] with respect to r Φ H ( ξ,z,t) = Φ( r,z,t) J ( jωξr)rr (1) leas to a one-imensional wave equation in the transforme omain, which is enote by the inex H: = z ξ Φ H H ω Φ 1 + Φ z z H χ Φ t H... (13) The high frequency approximation assumes that equation (8) can be solve by the following ansatz Φ = A jω t T ( x,z ) ( x,z) e ( ) (14) Now, Φ is replace with this ansatz an the terms are arrange accoring to its powers of ω. = Aω T 1 A T 1 T jω + +... z A z z z (15) z... + T χ z z ξ... 1 A A + z z A ( ξ,z) = A ( ( z ) χ( z ) ξ ) ( z) ( z) χ( z) ξ z ( ) ( ), (17) with the solely free parameter A. Note that this leas to the solution for a homogeneous meium if an χ are constants. All methos of generalize ray theory explaine in [3] like the erivation of source functions for point source acting on an interface consiering the bounary conitions can be applie to this solution. Finally, the inverse transformation has to be one: jω t T ( ξ,z ) ( ) = ω A( ξ,z) e Φ r,z ( ) J ( jωξr )ξξ (18) Current work is on an evaluation of this integral with a steepest escent approximation as it is escribe in [9]. The approximation facilitates the integral into a solvable form. This causes a neglect of surface waves being not of interest for the presente application. However, the metho is complicate because of the integral expression of T in the exponent. Though, the integral can be evaluate by a finite series expansion resulting in aitional terms containing higher powers of ξ. 4. Measurement Results 4.1. Measurements for Constant Soun Velocity Primarily, soun velocity was measure in ifferent meia with constant temperature, as escribe in Section II, to compare measurements an simulations. It is striking that the measure curve for Θ = 6 C fits very well to the simulate one, whereas the measure curve for Θ = 6 C eviates (Fig. 8). Therefore, aitional measurements with water-ethanol solutions were one. Different soun velocities can be ajuste with ifferent concentrations of ethanol, without a change in temperature [1]. Assuming large ω the first two lines are equate to zero inepenently an the frequency-inepenent thir line is neglecte. So T can be etermine irectly from the first line an with this solution A is etermine from the secon line T z ( ξ,z) = ± z) χ( z + z ( ξ z, (16) ) Fig. 8. Comparison of measure an calculate times of flight as a function of the use set of elay times for ifferent meia. 57
Sensors & Transucers, Vol. 184, Issue 1, January 15, pp. 53-59 It can be recognize that the measure curve of a water-ethanol solution with a soun velocity of 1565 m/s in Fig. 8 agrees to the simulate curve. This implicates that the working temperature of the probe has to be taken into account calculating the calibration curves. To analyze the accuracy measurements with ifferent numbers of averages have been one. Table 1 shows that the relative error of soun velocity is less than 1 %. Principle. Fig. 1 shows the comparison of calculate an measure times of flight to the focus as a function of the use set of elay times, corresponing with a focus (Fok) in the calibration meium water of C (soun velocity graient in water from 4 C at the transucer to 6 C at a istance of 5 mm). Table 1. Measurement capability. Number of average signals Stanar eviation ± 3σ of time of flight Absolute error of soun velocity 5 5 ns 13 m/s 5 ns 6 m/s 1 15 ns 4 m/s 5 1 ns.5 m/s 4.. Measurements for a Soun Velocity Graient An evient way to achieve a soun velocity graient is to generate a temperature graient with water, because the soun velocity as a function of temperature is well known for water [11] an it can be generate in a stable state. Fig. 9 shows the general set-up. Water in a basin locate at the bottom is kept at a temperature of 6 C. A secon smaller basin is place above. It contains a metal plate at its bottom for goo thermal conuction an a heat source at its top. This generates a vertical layere arrangement of warm water above cooler water, whereby a flui flow is avoie. The temperature is measure by an array of temperature sensors to etermine the soun velocity profile in the experimental set-up. Fig. 1. Comparison of measure an calculate times of flight as a function of the use set of elay times for a temperature graient. Although the notable ifference is just in the range of one microsecon this woul cause an error of more than 1 m/s in etermining the soun velocity. For aitional examinations, the soun velocity was measure conventionally via measuring the time of flight to a reflector at a known position. Moving the reflector along the acoustic axis allows a stepwise reconstruction of the soun velocity profile. Aitionally the soun velocity was etermine from temperature sensors again. Fig. 11 shows a comparison of the two conventional methos to etermine the mean soun velocity between the probe an a reflector at istance z. First, the time of flight for various reflector istances is measure (blue line). Secon, the temperatures are measure at various locations an converte to a soun velocity accoring to [11] an average over the propagation path (green line). The systematic eviation can be seen here, too. 15 1515 Measurement of mean soun velocity c via time of flight c via temperatur c mean [m/s] 151 155 15 Fig. 9. Experimental set-up for a soun velocity graient. The soun fiel for this graient an the time of flight to the focus were calculate applying Fermat's 1495 3 35 4 45 5 z [mm] Fig. 11. Comparison of two conventional methos to etermine the mean soun velocity between the probe an a reflector at istance z. 58
Sensors & Transucers, Vol. 184, Issue 1, January 15, pp. 53-59 Both eviations (see Fig. 1 an Fig. 11) result from the eficient assumption that wave propagation can be escribe with the wave equation for homogeneous meia. 5. Summary an Perspective This contribution iscusses a non-invasive metho to measure the soun velocity locally resolve along the acoustic axis of the use annular array. The three-imensional istribution of soun velocity can be obtaine by scanning. The metho has been qualifie for meia with constant soun velocity an then extene to meia with continuously changing properties. It has been shown that the continuous change of material properties has to be taken into account for the moelling of soun propagation. Fermat s principle gives the correct time of flight, but to obtain information about the amplitue, the moifie wave equation has to be solve. The potential of a point source has been calculate in the Hankel transforme omain. The inverse transform is actually realize an will allow calculating GREEN s functions for meia with continuously changing properties. Due to the assume change of material properties in only one imension a change of these properties in other imensions woul cause a lateral eviation of the focus position. This effect has to be examine in further works. Acknowlegements The authors woul like to thank Deutsche Forschungsgemeinschaft (DFG) for the financial support of the ongoing research project KU175/17-1. References [1]. M. Lenz, M. Bock, E. Kühnicke, J. Pal, A. Cramer, Measurement of the soun velocity in fluis using the echo signals from scattering particles, Ultrasonics, Vol. 5, Issue 1, January 1, pp. 117-14. []. M. Wolf, E. Kühnicke, Monitoring of Temperature Distribution in Liquis with Ultrasoun by Locally Resolve Measuring of Soun Velocity, in Proceeings of the 5 th International Conference on Sensor Device Technologies an Applications (SENSORDEVICES' 14) Lisbon, Portugal, 16- November 14, pp. 1-15. [3]. Y. H. Pao, R. R. Gajewski, The generalize ray theory an transient responses of layere elastic solis, in Physical Acoustics (W. P. Mason, e.), Acaemic Press, New York, Vol. 13, 1977. [4]. A. N. Ceranoglu, Y. H. Pao, Propagation of Elastic Pulses an Acoustic Emission in a Plate Part 1, Journal of Applie Mechanics, Vol. 48, Issue 1, 1981, pp. 15-13. [5]. E. Kuhnicke, Calculation of three-imensional harmonic waves in layere meia, in Proceeings of the IEEE International Ultrasonics Symposium (IUS), 7-1 Nov. 1995, pp. 811-816. [6]. M. Wolf, E. Kühnicke, M. Lenz, Moelling of soun propagation in meia with continuously changing properties towars a locally resolve measurement of soun velocity, in Proceeings of the IEEE International Ultrasonics Symposium (IUS), 1-5 July 13, pp. 145-148. [7]. L. Fuchs, Zur Theorie er linearen Differentialgleichungen mit veränerlichen Coefficienten, Journal für ie Reine un Angewante Mathematik, Vol. 1866, No. 66, 1995, pp. 11-16. [8]. V. Cerveny, Seismic Ray Theory, Cambrige University Press, 5. [9]. E. Kühnicke, Plane arrays Funamental investigations for correct steering by means of soun fiel calculations, Wave Motion, Vol. 44, Issue 4, March 7, pp. 48-61. [1]. C Burton, A Stuy of Ultrasonic Velocity an Absorption in liqui mixtures, Journal of the Acoustical Society of America, Vol., Issue, 1948. [11]. W. Marczak, Water as a stanar in the measurements of spee of soun in liquis, Journal of the Acoustical Society of America, 1997, pp. 776 779. 15 Copyright, International Frequency Sensor Association (IFSA) Publishing, S. L. All rights reserve. (http://www.sensorsportal.com) 59