# Signal Analysis and Waveform Reconstruction of Shock Waves Generated by Underwater Electrical Wire Explosions with Piezoelectric Pressure Probes

^{*}

## Abstract

**:**

## 1. Introduction

- The SW peak pressure can reach several to tens of GPa near the blast source [8,19,20,21] and tens to hundreds of MPa in the rather far field [22], so a reliable mechanical strength and proper structure are required to avert possible damage to the pressure sensors. Besides, the pressure sensors often work totally immersed in water, which calls for an impeccable waterproofing treatment.
- This ultra-wideband SW signal puts very high demands on the frequency response characteristics of pressure sensors. In order to ensure a large enough sensitivity, the sensitive element of a pressure sensor cannot be designed too small, but a finite size will stretch the wavefront due to the integrating effect. Besides, the pressure probe is often placed in water at a certain depth and inevitably connected with signal conditioning and recording devices with a rather long cable, then the long cable and the signal conditioning device will greatly distort the waveform and decrease the bandwidth.

## 2. Measurement System

#### 2.1. Experimental Setup

_{sw}) from the internal wall of the tank. The pressure probes could be moved at the distance range of 200–560 mm (D

_{meas}) from the SW source. At these distances, the SW could be regarded as a plane wave. The distances between four pressure probes and the SW source were almost the same and precisely measured before the experiment. At each distance, at least five repeated tests were produced with the same discharging parameters. The SW source and the pressure probes were placed at the deep enough position (water depth of 350 mm) to avoid the possible influence of the “bulk cavitation phenomena [30]” effects.

#### 2.2. Measurement System Consistency

_{peak}with Equation (2):

_{peak}is the peak pressure error. Considering the PCB138 probe has a relatively large size (9.6 mm in diameter), the installation error may be within ±5 mm. When r

_{0}= 260 mm and ±Δr = 5 mm, the relative error of the P

_{peak}is −0.95/+0.98%. When r

_{0}= 560 mm and ±Δr = 5 mm, the relative error of the P

_{peak}is −0.44/+0.45%. It is obvious that a distance error of less than 5 mm will not produce a peak pressure error larger than 1% in the distance range of 260–560 mm. That is, the probe does not have to be installed precisely for measuring the peak pressure.

#### 2.3. Influence of Incident Angle and Sensitive Element Size

## 3. Analysis of Signals Obtained by PCB138 and Müller-Plate Probes

_{m}is the peak pressure, τ is the decay time constant, ε(t − t

_{0}) is the step function, and t

_{0}is the SW arrival time. Within the time duration of one τ, Equation (3) can provide a fair approximation for SWs to the time of one τ, and then the SW decay rate slows down in longer times [15]. The time constants τ can be defined as the time required for the pressure to decay from P

_{m}to the pressure value P

_{τ}= P

_{m}/e.

#### 3.1. Analysis of the Peak Pressure vs. SW Propagating Distance

_{0}and α are unknown constants to be determined. When the coefficient of determination (R-square) is the closest to 1, the two well-fitted results, which are P

_{peak}= 42434r

^{−1.09}and P

_{peak}= 2724r

^{−1.06}, can be obtained separately, as shown in Figure 8. In order to verify the accuracy of the fitting results, we placed an extra observation point at the radical distance of 200 mm. The measured peak pressure by the Müller-plate probe is ~12.59 Mpa and is marked with a green star in Figure 8. It is obvious that this measured peak pressure value has a very good accordance with the predicted value given by the fitting curve, so the fitting results are fairly accurate.

_{peak}= P

_{0}r

^{−1}. In this paper, the distances range between the observation points and the SW source are 260–560 mm. In this distance range, the SW has not been fully attenuated, so the decay rate of peak pressures should be faster than that in the far field, i.e., the corresponding decay constant α should be larger than 1. The fitting results illustrated in Figure 8 can prove our analysis. Besides, the decay constants α (1.06 and 1.09) for the Müller-plate probe and the PCB138 probe are quite close to each other, so the decay characteristics of SW peak pressure obtained by two kinds of probes are similar. Furthermore, the relative errors of peak pressures between the Müller-plate probe and the PCB138 probe at different radial distances vary in a very small range of 23%–27%, as shown in Figure 8. The approximately constant errors indicate that the two kinds of pressure probes both have good uniformities and linearities. In fact, it is very hard to obtain the absolutely accurate peak pressure value of SWs. Measuring the relative pressure precisely is more meaningful in practical application. Due to the good stability and linearity, both the PCB138 probe and the Müller-plate probe can be regarded as standards for relative peak pressure measurement.

#### 3.2. Analysis of the Pulse Width and Decay Time Constant

## 4. SW Pressure Evaluation and Reconstruction

#### 4.1. Peak Pressure Evaluation with the TOF Method

#### 4.1.1. SW Pressure Measurement Principle Based on the TOF Method

_{0}, ρ

_{0}, and E

_{0}are density and internal energy of undisturbed water, respectively. When the pressure is less than 2.5 GPa, the EOS of water is shown in Equation (7) [19]:

#### 4.1.2. Feasibility of Peak Pressure Evaluation with the TOF Method in Practical Application

_{p}/t

_{p}, in which, s

_{p}is the distance between two pressure probes and t

_{p}is the time lag of the pressure signals obtained by two pressure probes. D is the average speed between the two probes. If s

_{p}is too large, the error of D will be rather large due to the nonlinear propagation of the SW.

_{p}= 50 mm, D ≈ 1600 m/s, t

_{p}= s

_{p}/D = 31.25 μs, the distance error caused by installation Δs

_{p}≈ 1.6 mm, and the time error Δt

_{p}= 0, which indicates, except for the distance error Δs

_{p}caused by installation, all parameters are precisely measured. Then the absolute error of pressure ΔP will be ~43.5 Mpa according to Equation (10):

_{p}) of probes will lead to a rather large peak pressure error. Figure 12 shows our experimental and calculated results. We adopted two Müller-plate probes to measure the arrival times of the SW, because their sensitive elements are rather small (<0.5 mm in diameter), and the introduced distance error are smaller than with the PCB138 probes. The Müller-plate probes are carefully placed at the distances of 200, 250, 300, 350, 400, 450 and 500 mm from the discharge channel. Results show that the calculated peak pressures with the TOF method are overall larger than the values directly obtained with the Müller-plate probe, and this support our error analysis above.

#### 4.2. Reconstruction of the Real Pressure Waveform with the Energy Conservation Method

#### 4.2.1. Waveform Reconstruction Criteria

_{0}in the lower frequency band, the magnitude frequency response of the reconstructed signal P

_{rec}should theoretically be equal to that of the measured signal P

_{m}. Another criterion is based on the energy conservation law. According to Parseval’s theorem, the signal energy in the time domain is equal to that in the frequency domain. If two signals have similar energy characteristics in the time domain, their energy distribution in the frequency domain should also be similar. The frequency response of the PCB138 probe in the low-frequency range (for example 1–10 kHz) is flat and unity, we can suppose the measured signal in this frequency range is real and reliable, so the energy in the frequency domain of the restored signal E

_{rec}should be equal to that of the measured signal E

_{m}in this specific frequency range. The two waveform reconstruction criteria can be described by Equation (11):

_{m}and p

_{rec}are the pressure values of measured signal and the reconstructed signal. f

_{l}and f

_{h}are the lower and upper limit frequencies of the integral. P

_{m}and P

_{rec}are the frequency response of p

_{m}and p

_{rec}.

_{rec}should be strictly equal to P

_{m}, and E

_{rec}should be strictly equal to E

_{m}. However, in the calculation, some errors are inevitable. For example, the sample length cannot be infinite, and some frequency spectrum leakage is also unavoidable. The SW measurement with the pressure probes cannot be exactly accurate, so satisfactory reconstruction results may not be achieved only with Equation (11). In order to obtain more accurate waveform reconstruction results, we propose the reconstruction criteria based on the minimum errors, which require that both the magnitude error ErrorP and the relative error ErrorP be rather small. The two new criteria are given by Equation (12):

_{0}= 0 Hz as the calculation condition is not reasonable. In this paper, we take f

_{0}= 1 kHz, f

_{l}= 1 kHz and f

_{h}= 10 kHz as the calculation conditions. The threshold values for both ErrorP and ErrorE are both set to 1.5% to get the right reconstructed signal parameters.

#### 4.2.2. Three Kinds of SW Waveform Models

_{peak}is the peak pressure value of a signal, Δt is pulse width of the triangular signal, t

_{0}is the arrival time of the SW, τ is the time constant of an exponential decay signal and B is a constant. We set B = 2 in this paper.

#### 4.2.3. Waveform Reconstruction Process

- (1)
- A series of peak pressure P
_{peak}, pulse width Δt and time constant τ were assumed in a rather large value range, so SW waveforms with different parameters could be obtained with Equations (13)–(15). - (2)
- FFT calculations were performed, and the magnitude error ErrorP and the relative error ErrorP were calculated with Equation (12).
- (3)
- The proper peak pressure P
_{peak}, pulse width Δt and time constant τ could be picked up with the pre-set threshold values of both ErrorP and ErrorE, and the real SW signal can be obtained.

#### 4.2.4. Typical SW Reconstruction Results

_{0}for a triangular signal, exponential decay signal, and multi-exponential decay signal are 0.68%, 1.21% and 1.17%, respectively, and the energy errors ErrorE for the three signals are 0.46%, 1.23% and 0.97% separately.

#### 4.2.5. Verification of the Reconstruction Results

_{t = τ}when time equals to one τ can be obtained. Finally, a peak pressure can be calculated by P

_{m}= P

_{t = τ}× e. As shown in Figure 14, the pressure signal is measured by a Müller-plate pressure probe at the same distance as the PCB138 probe shown in Figure 13. At time τ, the fitted pressure P

_{t = τ}= 3.18 Mpa, and the calculated P

_{m}= 8.64 Mpa. This calculated peak pressure is quite close to the value given by the multi-exponential decay signal shown in Figure 13. This validates the feasibility and validity of the waveform reconstruction algorithm.

#### 4.2.6. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 4.**20 repeated temporal pressure waveforms at the distance of 260 mm from discharge channel. (

**a**) Pressure waveforms obtained by the PCB138 probe with the SW arrival times aligned; (

**b**) distribution of the SW arrival times obtained by the PCB138 probe; (

**c**) pressure waveforms obtained by the Müller-plate probe with the SW arrival times aligned; (

**d**) distribution of the SW arrival times obtained by the Müller-plate probe.

**Figure 6.**Pressure signals obtained by the PCB138 probe with different incident angles at the distance of 460 mm (

**a**) Consistency of the obtained pressure waveforms; (

**b**) rising edge of pressure signal with different incident angles.

**Figure 7.**Temporal waveforms obtained by pressure probes vs. distances. (

**a**) Müller-plate probe; (

**b**) PCB138 probe.

**Figure 8.**Dependence of the SW peak pressures and fitted values on the distance from the discharge channel.

**Figure 9.**Typical time-dependent pressure waveforms obtained with both pressure probes at the distance of 260 mm.

Parameters | PCB138A11 | Müller-Plate Needle Hydrophone |
---|---|---|

Sensitive element type | Tourmaline | PVDF |

Sensitive element size | ~Φ3.2 mm × 1 mm (measured) | <Φ0.5 mm |

Sensitivity uncertainty | ±15% | - |

Rise time | <1.5 μs | <50 ns |

Bandwidth | 2.5–1 MHz | 0.3–11 MHz |

Cable length | 20 m | 2 m |

Parameters | PCB138 Probe | Müller-Plate Probe | ||
---|---|---|---|---|

Peak Pressures | Arrival Times | Peak Pressures | Arrival Times | |

Mean value | 8.87 MPa | 179.91 μs | 9.70 MPa | 179.80 μs |

Standard deviation | 0.136 MPa | 0.460 μs | 0.196 MPa | 0.427 μs |

Coefficient of variance | 1.5% | 0.26% | 2.0% | 0.24% |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhou, H.; Zhang, Y.; Han, R.; Jing, Y.; Wu, J.; Liu, Q.; Ding, W.; Qiu, A. Signal Analysis and Waveform Reconstruction of Shock Waves Generated by Underwater Electrical Wire Explosions with Piezoelectric Pressure Probes. *Sensors* **2016**, *16*, 573.
https://doi.org/10.3390/s16040573

**AMA Style**

Zhou H, Zhang Y, Han R, Jing Y, Wu J, Liu Q, Ding W, Qiu A. Signal Analysis and Waveform Reconstruction of Shock Waves Generated by Underwater Electrical Wire Explosions with Piezoelectric Pressure Probes. *Sensors*. 2016; 16(4):573.
https://doi.org/10.3390/s16040573

**Chicago/Turabian Style**

Zhou, Haibin, Yongmin Zhang, Ruoyu Han, Yan Jing, Jiawei Wu, Qiaojue Liu, Weidong Ding, and Aici Qiu. 2016. "Signal Analysis and Waveform Reconstruction of Shock Waves Generated by Underwater Electrical Wire Explosions with Piezoelectric Pressure Probes" *Sensors* 16, no. 4: 573.
https://doi.org/10.3390/s16040573