Modelling I. The Need for New Formulas Calculating Near Field, Lateral Resolution and Depth of Field D. Braconnier, E. Carcreff, KJTD, Japan

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1 Modelling I The Need for New Formulas Calculating Near Field, Lateral Resolution and Depth of Field D. Braconnier, E. Carcreff, KJTD, Japan ABSTRACT In Non-Destructive Testing (NDT), awareness of the ultrasonic beam characteristics is vital to making an accurate measurement. To do so, formulas from the harmonic theory are commonly applied on pulsed excitation, which is inherently wide-band.. The reason lies in the fact that those formulas come from the Rayleigh surface integral and have closed-form solutions only for a harmonic excitation. The Fresnel distance and depth of field are derived from these solutions but are generally approximated. This is particularly true for the depth of field, in the case of focused transducers, where the expression from literature proves to be imprecise. If one takes a pulsed excitation into account, the calculation of the sound field is much more difficult and requires simulations. In this paper, the spatial impulse response (SIR) method is used in order to process the pulsed radiations and to measure the axial characteristics. In the case of SIR the Fresnel distance tends to be shorter whilst the depth of field tends to increase according to the pulsed mode. Moreover, interferences in the nearfield decrease as the bandwidth ratio increases. INTRODUCTION In NDT, an incorrect estimate of the beam characteristics can lead to an inaccurate inspection or a shifted probe design. Most of the time, harmonic theoretical formulas are employed in order to estimate those characteristics, although pulsed signals are used in real-world industrial applications. It is therefore interesting to verify to what extent the harmonic formulas are accurate in a pulsed context. First of all, a historical review of sound field calculations is made. Then, the basic harmonic theory is reviewed in order to recover the axial characteristics formulas. Finally, simulations with a pulsed excitation (wide band) are carried out in order to analyze the influence of the spectrum distribution. I will use the bandwidth ratio parameter (BWR), which describes the spectral amount in the excitation signal. Using the harmonic theory, flat and focused pistons will be compared in terms of maximum pressure and depth of field. In this study, large sizes of transducers (0 and 40λ) are considered in order to fit in with typical NDT applications. HISTORICAL REVIEW OF SOUND FIELD CALCULATION At the end of the nineteenth century, Lord Rayleigh ( ) was the first to introduce the solution of the sound field radiated by a plane vibrating surface by pursuing the work of Helmholtz and Kirchhoff. He stated that every point of a radiator can be considered as the source of an outgoing spherical wavelet, and that the field at a point of the space can be constructed from the superposition of these wavelets. This is analytically represented by the well-known Rayleigh surface integral in [1]. Physically, the Rayleigh integral is a statement of Huygens s principle. According to Rayleigh s work, dating from the beginning of the twentieth century, sound field radiation is analytically studied for a large shape of radiators: circular piston, rectangular piston, curved piston, array of sources, etc. As an example, we can cite the work of Backhaus, King, Stenzel, Schoch, etc. Reviews are carried out in [] and [3]. Since the middle of the twentieth century, in light of the increase of the calculation power, academic research has adopted transient field and near-field calculations. Complex simulations are proving to be simpler than calculating closed-form solutions of the sound field. In 1970 [4], Zemanek proposed a numerical calculation of the sound pressure in the near field for a circular piston. Thereafter, the method of the Spatial Impulse Response (SIR) was introduced by Stephanishen in 1971 [5] and improved by Lockwood and Willette in 1973 [6].

2 The principle of this particular method is that the sound field is processed by convolving the velocity of the piston with the spatial impulse responses of the observation points. This allows the calculation of the sound field for a pulsed excitation and for a large range of radiator shapes. This method was numerically optimized by Piwakovski in 1989 [7]. In 004, McGough proposed a high speed method to compute sound radiations by using a sectorial Gauss quadrature (see [8] and [9]). Nowadays, computing power allows high complexity algorithms like Finished Element Method (FEM) or Boundary Element Method (BEM). Those methods resolve the partial differential equations which have been formulated as integral equations. They originate from mechanical engineering and are not exclusively dedicated to acoustics. THEORETICAL HARMONIC BACKGROUND Flat Piston Let s consider a baffled flat piston of radius a (Figure 1) which is radiating in a uniform and isotropic medium. The radiating surface moves uniformly with a speed U ( t) = U0 exp( jωt) normal to the baffle. The sound field can then be computed by using the Rayleigh surface integral [1]: iωρ 0 p( r, t) = U ( t π ) S e ds, R i kr where S is the radiating surface and R = r-r 0 describes the distances from each point source r 0 to the observation point r. k is the wave-number, ω is the angle frequency and ρ 0 is the density of the material. By assuming that r is on the z axis, it leads to solution [10]: i( kz ωt ) ( a + z z) e k p( z, t) = ρ0cu 0 sin, where c is the celerity of the longitudinal waves in the material. According to the sine function, the axial pressure exhibits interference effects. An example of pressure on the transducer axis is presented in Figure. Figure 1 - Scheme of the flat circular piston oscillator Figure - Sound pressure on the axis radiated by a circular flat piston. F = 1 MHz, c = 1500 m/s, a = 5λ

3 Nearfield length From p(z,t), it is possible to calculate the last maximum of pressure. It is the well-known nearfield λ length or Fresnel distance [10]: z = a λ NF. Assuming that a>>λ, the nearfield length can be 4 a approximate by: z NF. In the Figure example, z NF = mm. λ Depth of field The depth of field z is the z range where the sound pressure is greater than a specific level (typically -3dB in emission or -6dB in reception). The depth of field is then given for a>>λ with: 4 z = z NF. 3 Focused Piston Figure 3 - Scheme of the curved piston oscillator The scheme of the focused radiator is shown in Figure 3. The parameter A is the radius of the equivalent sphere and is the geometrical focal distance. The sound pressure field on the axis is given by O'Neil concerning focusing radiators [11]: p k, 0 1 z / a, i( kz ωt ) ( z t) p = sin a + ( z h) z e where p 0 is the pressure on the surface of the transducer and h = A - A -a. This formulation is valid for A > a because the focal distance cannot be shorter than a. An example of sound pressure on the axis is represented in Figure 4. The focal distance is A = 0.5 z NF 18.6 mm but it can be observed that the maximum pressure is located before this distance, at around 15 mm. It is the reason why we always differentiate the geometrical - or optical - focal distance (A), which we will call F opt from the maximum pressure level, which is the acoustical focal distance F ac. When F opt goes towards infinity, F ac goes towards the Fresnel distance of the flat piston. However, the values are almost equal when F opt is inferior to z NF.

4 Figure 4 - Example of sound pressure of a curved piston oscillator. F = 1MHz, c = 1500 m/s, a = 5λ, A = 0.5 z NF Depth of field The usual theoretical depth of field is written as (see [1] and [13]): Fac z = Kλ, a where K is often equal to 7 in the literature. Figure 5 shows the evolution of the K coefficient according to the focal distances for 3 sizes of transducer. It can be observed that K is not constant. It varies from 4 to 7 according to the focal length. In the case of infinite focusing (flat piston), K is equal to The value 7 is actually an approximation for a large radius and a small specific range of focal distances. Figure 5 - K coefficient versus acoustical and optical focal distances. a = 10, 50, 100 λ

5 TRANSIENT FIELD CALCULATIONS AND COMPARISON WITH THE HARMONIC THEORY In this section, I propose to provide clues to help understand the pulsed radiators. The simulations are processed with the Matlab toolbox DREAM [14], which uses the Spatial Impulse Response (SIR) method. This toolbox has the great advantage to work with any excitation signal. In the following section, I will be using a sine modulated by a Gaussian window. The bandwidth radio at -6 db (BWR ) is given by: f 6dB BWR = f 0, where f 0 is the central frequency of the transducer. Flat piston In Figure 6, the sound pressure has been plotted on the axis radiated by a 40λ transducer according to five bandwidth ratios: 10%, 30%, 50%, 70% and 90%. One can see that the interferences in the nearfield are attenuated as the BWR increases. After 70%, there are no more oscillations in the nearfield. The pressure is smoothed and more constant according to z. As for the last maxima of pressure (Fresnel distance), it is the same as the harmonic case if the BWR is small (<10%). It decreases then as the BWR increases. When the values of BWR are high (> 70%), one cannot even know whether a near-field last maximum exists. Figure 6 - Velocity signals and sound pressure on the axis according to 5 bandwidth ratios: 10%, 30%, 50%, 70% and 90% respectively. F = 1 MHz, c = 1500 m/s, a = 40λ Analysis of the nearfield length according to a pulsed excitation The last maxima of pressure are measured according to the bandwidth ratio. For a = 0λ and 40λ, the results are shown in Figure 7. One can observe that the assumption made in the previous part has been confirmed: the Fresnel distance is decreasing as BWR is increasing. Moreover, for these sizes of transducers, the evolution of the nearfield length is highly similar. We can assume that it is generally the case for large transducers.

6 Figure 7 - Normalized nearfield according to the bandwidth ratio. Left: a = 0λ, right : a = 40λ Analysis of the Depth of field length according to a pulsed excitation One can see in Figure 8 that the depth of field slightly increases when BWR increases. The value measured by simulations is very close to the theoretical value (coefficient = 4/3 1.33). After 60% of bandwidth ratio, the depth of field is not measurable. Nevertheless, we deduce that beyond this limit, the depth of field retains its meaning and tends to increase again. Figure 8 - Depth of field according to the bandwidth ratio. Left: a = 0λ, right: a = 40λ Focused piston In Figure 9 the velocity distributions and corresponding pressure profiles for several bandwidth ratios are plotted. As in the case of the flat piston, one may observe that the oscillations in the nearfield are decreasing as the bandwidth is increasing. Furthermore, we can see that the maximum pressure, situated on the acoustical focal length, remains the same and is not influenced by the BWR. Thereafter, the depth of field versus the bandwidth ratio for the optical focal length equal to 0.5 z NF is plotted on Figure 10. One may thus notice that the depth of field is smaller than the reference K = 7 and increases lightly whilst BWR increases. Therefore, the K coefficient varies according to the focal distance and the bandwidth. An analytical law is difficult to define. Furthermore, the smoothing of the profile due to the pulsed excitation lead to a non-measurable depth of field beyond BWR = 65%. Nevertheless, it is understandable that the depth of field should tend to increase beyond this value.

7 Figure 9 - Velocity signals and sound pressure on the axis according to 5 bandwidth ratios: respectively 10%, 30%, 50%, 70% and 90%. F = 1 MHz, c = 1500 m/s, a = 40λ, F opt = 0.5 z NF Figure 10 K coefficient according to the bandwidth ratio. F opt = 0.5 z NF. Left: a = 0λ, right: a = 40λ CONCLUSION I have compared the axial characteristics of the sound field using harmonic theory with a pulsed case (wide band). In order to do so, the nearfield length and depth of field formulas were brought back to mind. Then, thanks to the SIR method, simulations were carried out in order to measure these characteristics with a broadband excitation. In the case of the flat piston case, I have noted that the last maximum pressure, commonly called the nearfield length or Fresnel distance, decreases as the bandwidth increases. It can even be less than 80 % of the harmonic nearfield when the bandwidth is greater than 90%. For larger values of BWR, it is impossible to say whether a near-field last maximum exists because the field simply declines according to the depth. Moreover, the constructive interferences within the near-field disappear as the bandwidth increases. For large values of BWR, there are no more oscillations. Nevertheless, the depth of field slightly increases according to the bandwidth and remains true to the harmonic theory.

8 When it comes to the focused piston, the behavior of the oscillations along the axis in the nearfield is similar to that of the flat piston. However, the maximum pressure is correctly situated. General conclusions about the depth of field are difficult to draw since the K coefficient is dependent on the focal distance and the radius of the piston. It can nevertheless be said that the depth of field lightly increases as BWR increases. The common value of K usually assumed to be 7 is correct in most common applications since there is not so much meaning in focusing nearby or beyond the Fresnel distance. But this is not the case for applications implying an important focusing degree, in other words, focusing close to the transducer. In this case, the usual formula does not apply. I may add that this is typically the case with acoustic microscopy. This study shows that the pulsed excitation produces a decrease of the constructive interferences in the nearfield. This produces interferences at the natural focal place. Consequently, the sound pressure is maintained around the near-field length or the focal distance. It would be interesting to pursue this work out of the acoustic axis and onto other radiator shapes like rectangular or phased array. REFERENCES 1) Rayleigh J W S, Theory of sound, Dover, New-York, 1945, Volume II, p. 107, ) Freedman A, Transient fields of acoustic radiators, The Journal of Acoustical Society of America, (1) ) Harris G R, Review of transient field theory for a baffled planar piston, The Journal of Acoustical Society of America, (1) ) Zemanek J, Beam behavior within the nearfield of a vibrating piston, The Journal of Acoustical Society of America, (1) ) Stephanishen P R, Transient radiation from pistons in an infinite baffle, The Journal of Acoustical Society of America, (5) ) Lockwood J C and Willette J G, High-speed method for computing the exact solution for the pressure variations in the nearfield piston, The Journal of Acoustical Society of America, (3) ) Piwakowski B and Delannoy B, Method for computing spatial pulse response: Time-domain approach, The Journal of Acoustical Society of America, (6) ) McGough R J, The FOCUS toolbox, 010, 9) McGough R J and Samulski T V and Kelly J F, An efficient grid sectoring method for calculations of the near-field pressure generated by a circular piston, The Journal of Acoustical Society of America, (4) ) Kinsler L E and Frey A R and Coppens A B and Sanders J V, Fundamentals of acoustics, New York, John Wiley and sons, ) O'Neil H T, Theory of focusing radiators, The Journal of Acoustical Society of America, (5) ) Association francaise de normalisation, Faisceaux acoustiques generalites, Normalisation francaise, ) American Society of Nondestructive, Nondestructive Handbook : Ultrasonic Testing, Colombus, Patrick O. Moore, ) Lingvall F, The DREAM toolbox, Jan 004,