Ultrasonic attenuation and phase velocity of high-density polyethylene (HDPE) pipe material

Abstract Knowledge of acoustic properties is crucial for ultrasonic or sonic imaging and signal detection in nondestructive evaluation (NDE), medical imaging, and seismology. Accurately and reliably obtaining these is particularly challenging for the NDE of high density polyethylene (HDPE), such as is used in many water or gas pipes, because the properties vary greatly with frequency, temperature, direction and spatial location. Therefore the work reported here was undertaken in order to establish a basis for such a multiparameter description. The approach is general but the study specifically addresses HDPE and includes measured data values. Applicable to any such multiparameter acoustic properties dataset is a devised regression method that uses a neural network algorithm. This algorithm includes constraints to respect the Kramers-Kronig causality relationship between speed and attenuation of waves in a viscoelastic medium. These constrained acoustic properties are fully described in a multidimensional parameter space to vary with frequency, depth, temperature, and direction. The resulting uncertainties in acoustic properties dependence on the above variables are better than 4% and 2% respectively for attenuation and phase velocity and therefore can prevent major defect imaging errors.

method that discards none of the multiparameter dataset must inherently be multivariate. 50 Multivariate regression is necessarily nonlinear in frequency for viscoelastic media, and the 51 regression coefficients are not all independent for causal Kramers-Kronig constrained dis-52 persion relations. A suitable and efficient approach to tackling a multiparameter problem 53 such as this is to employ a neural network algorithm [19,20] to obtain such a numerical 54 description of such dispersions relations. 55 This article is set out as follows. Succeeding background theory in section II; the principles 56 of the method for obtaining and analysing the acoustic properties of HDPE and other such 57 viscoelastic media are provided in section III; the procedure for obtaining HDPE acoustic 58 properties, their dependence on frequency, propagation depth within the pipe, temperature, 59 and propagation direction within the pipe, and the method for fully determining this mul-

II. THEORY
Here we summarise a theoretical description of an ultrasonic bulk wave that propagates 66 in a viscoelastic medium such as HDPE. This description includes the theoretical constraints 67 to which such an ultrasonic wave in a viscoelastic medium must adhere. These constraints 68 are necessarily imposed in the frequency domain, and so this is how the wave description is 69 formulated.

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The spectrum of a 1-D harmonic plane wave can be expressed as where x is location, f is frequency, and k x (f ), or tersely k(f ), is the dispersion relation, where v p (f ) is phase velocity and α(f ) is attenuation. For this entire study, the following  Much time has been given to the study of how best to describe the dispersion relation 75 of viscoelastic media [11,14,17,[21][22][23][24][25][26][27]. From which, it is usual to conclude that the most 76 suitable description for attenuation of ultrasonic waves in a viscoelastic medium follows a 77 power law, where 0 < α 1 and 1 < y < 2 are coefficients to be determined. One such example of a 79 commonly used damping model is Kelvin-Voigt, but this loses accuracy via restriction to which, via use of equation (3) and selecting zero reference frequency, f 1 = 0, results in the following power law description of ultrasonic viscoelastic phase velocity, where positive phase velocity at zero frequency, 0 < v p1 , is to be determined. Importantly, frequencies close to f 0 , resulting in greater filtering of the frequency content of the main 144 spectrum.

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The immersion tank setup used to obtain waveforms that include reflections from the 146 front and back walls of a cuboid-shaped sample cut from HDPE pipe inspected with a 147 pulse-echo ultrasonic configuration at normal incidence to the sample surfaces is depicted in  Below are equations used to obtain attenuation and phase velocity and their dependences 153 on frequency, propagation depth, temperature, and propagation direction from these HDPE 154 pipe sample waveforms. These equations can also readily be applied to other highly atten-155 uative media.

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To yield a general description of damped bulk plane wave attenuation and phase velocity, equations (1) and (2) can be combined using ln(a b ) ≡ b ln(a) and c ≡ |c| exp(iφ c ) to yield, where φ S (f ) and φ S 0 (f ) are phases of the spectra. By equating real and imaginary com-159 ponents of equation (6) it follows that for a 1-D propagating wave the general forms of 160 attenuation and phase velocity are, and . surface that has finite radius, a, such as is used here to characterise HDPE pipe acoustic 165 properties, it is necessary to account for diffraction of the transmitted and reflected waves.

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For such transducers, this can be achieved using Lommel diffraction correction [32] given c represents the longitudinal phase velocity in each medium -here water and HDPE.

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A smaller path in each medium, x, results in less beam spread but Lommel diffraction 170 correction is valid only in the far field of the wave, the distance of which is known to be, The spectra of the signals reflected off the front wall, F 1 (f ) = S(x w , f ), and back wall, and given k w,s are dispersion in both media, A 11 is the water-HDPE plane wave normal inci-175 dence reflection coefficient, A 22 = −A 11 is that of HDPE-water, A 12 is the water-HDPE 176 transmission coefficient, and A 21 is that of HDPE-water. The transmittance of a plane wave 177 at normal incidence to the interface is, where ρ w,s are density in both media, and v p,est is an estimate of phase velocity in HDPE.

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Estimating phase velocity has a small associated error given the following approximation,

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A 121 ∝ (v p,est ) −1 for ρ s v p,est > ρ w c w , varies with a low order power in v p,est and therefore is not 181 strongly dependent on it. For this reason, uncertainty can be neglected in estimated phase 182 velocity, and by the same reasoning, all other parameters in equation (14). It is assuming that 183 plane wave conditions are valid for the beam profile, given the geometry of the transducer diameter, 2a, the minimum propagation, x w , and the wavelength. Normal incidence is a 185 valid assumption because the solid angle subtended in wave reception by the transducer is 186 small. Lommel diffraction correction, equation (9), is introduced to equations (12) and (13) given resulting empirical descriptions of attenuation and phase velocity are, and for the reflected pulse that occurs for ρ 1 c 1 > ρ 2 c 2 . Subscripts '1' and '2' represent before 192 and beyond the interface, which for B 1 is HDPE-water. In contrast with equation (8), frequencies; it is alike to shifting by the difference in travel time of pulses moving at the 207 group velocity, ∆t g,emp = x s /v g,emp , but rather its duration is optimised to be that which 208 provides continuous phase over a specified bandwidth. In this way, it can be considered 209 to be the frequency independent offset to phase velocity, from which the phase terms are

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This procedure is optimised such that it is inherently readily applicable to other highly 223 attenuative media.

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As depicted in figure 2, the signal generator that transmits half-cycle square impulses 225 is an Olympus 5077PR (Tokyo, Japan), the single probe used to transmit and receive ul-  (3) and that obtained empirically via the method in section IV. The R 2 is not close to unity 317 for phase velocity, and a slight negative gradient of velocity with frequency in the data 318 for T = 42.5 • C yields a negative R 2 value. This is probably in part caused by constraint obtained for attenuation for all depths, temperatures, and directions via, and for phase velocity via, and shown in figure 6. The fractional uncertainties are low and approximately constant The linear thermal expansion coefficient of HDPE is of the order κ ∼ 10 −4 K −1 therefore the where σ T are the standard deviations of each set of three temperature measurements. Using 357 quadrature, the total measurement fractional uncertainty is where T is the dominant term. The upper limit on the mean total attenuation and phase The attenuation and phase velocity of HDPE pipe material has been determined to within an 428 uncertainty of better than 4% and 2% respectively for a large range of inspection frequencies, 429 depths, and temperatures, for the three orthogonal directions within the pipe.

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Example values generated, using the above acoustic properties parameterisation, for at-    difference -which itself is a few percent. Therefore, the corresponding variation in acoustic 589 properties at half throughwall height for different pipe thickness is for the majority of cases 590 less than 1% for attenuation and less than a fraction of a percent for phase velocity, and so, 591 for both acoustic properties, this is not statistically significant compared with the associated 592 dispersion fractional uncertainties. As a result, the dispersion values presented in tables III 593 and IV are, within the provided fractional uncertainty ranges, applicable to the majority of