Low-Phase-Error Underwater Acoustic Spiral Wavefront Array and Phase Error Compensation
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe article "Low-phase-error Underwater Acoustic Spiral Wavefront Array and Phase Error Compensation" submitted by Guo et al. is well written. However, I have suggested some revisions to the article for better understanding. After all these revisions are made, I recommend that the article be accepted.
Comments for author File: 
Comments.pdf
Author Response
Dear Reviewer,
We would like to express our sincere gratitude for your invaluable feedback on my manuscript. Your expertise and keen insights have been instrumental in helping us refine and improve the quality of my work. We truly appreciate the time and effort you have dedicated to this process. Thank you once again for your support and guidance.
Best regards,
The authors
Reviewer 2 Report
Comments and Suggestions for AuthorsComments:
This manuscript presents the design, modeling, fabrication, and experimental validation of a seven-element underwater acoustic transmitting array using bender disk transducers to generate both spiral and reference acoustic wavefronts with low phase error. The authors develop a simple circular array model and finite element analysis to optimize array parameters, followed by the proposal of a phase error compensation technique based on a linear input–output system assumption. Water tank experiments demonstrate that the fabricated prototype achieves sub-degree RMS phase error after compensation, with a minimum value of 0.09° and stable amplitude uniformity across tested frequencies. The work addresses the important challenge of reducing phase errors in acoustic spiral wavefront (ASWF) sources for underwater navigation applications, and demonstrates promising experimental results.
- While the results are strong, the manuscript should more explicitly compare its contribution with prior ASWF source developments [Refs. 2–9]. What makes this design and compensation strategy fundamentally more effective than previous approaches?
- More details are needed about the water tank setup—such as background noise level, calibration of hydrophone sensitivity, and boundary effects—since these factors could influence phase error measurements.
- Results are presented at three frequencies (5.5, 7.3, 9.0 kHz), but the long-term applicability of the compensation method across a wider bandwidth should be discussed.
- The authors acknowledge that compensation cannot correct all defects. Could the authors quantify the tolerance to manufacturing errors (e.g., element misplacements, admittance variability) beyond which compensation becomes ineffective?
- A short discussion on how this prototype could be scaled for real-world deployment (e.g., UUV navigation beacons, open-water environments) would significantly strengthen the impact of the paper.
Author Response
Thank you so much for your time spent on reviewing this manuscript. You can find our detailed responses below, and the revisions are highlighted as tracked changes in the resubmitted files.
Comments 1: While the results are strong, the manuscript should more explicitly compare its contribution with prior ASWF source developments [Refs. 2–9]. What makes this design and compensation strategy fundamentally more effective than previous approaches?
Response 1: Thank you for your helpful remark. In terms of sound source design, this paper for the first time directly utilizes the simple circular array (SCA) model to analyze and design the amplitude and phase responses of the ASWF source. The range of the ka value is focused the low-frequency range of 0–3 rad. These features were neither present nor discussed enough in previous works.
In terms of the compensation method at the ASWF source end, this paper for the first time introduces compensation of the amplitude and phase of the input signal, whereas previous work only involved input phase compensation. Therefore, the method in this paper is more effective.
Comments 2: More details are needed about the water tank setup—such as background noise level, calibration of hydrophone sensitivity, and boundary effects—since these factors could influence phase error measurements.
Response 2: We agree and have, accordingly, revised the manuscript to emphasize this point. The background noise level was below 93dB/√Hz from 5-10kHz. The hydrophone sensitivity of the B&K 8104 was calibrated at -208.0dB. The distance between the sound source and the hydrophone is 1.8 m, with a depth of 5 m. The shortest reflected sound path is greater than 10 m, and the time delay between the first reflected wave and the direct wave is greater than 5.5 ms. Given that the pulse width is 2 ms, the boundaries of the water tank will not affect the accuracy of the measurements and, so, the test met the free-field conditions.
Comments 3: Results are presented at three frequencies (5.5, 7.3, 9.0 kHz), but the long-term applicability of the compensation method across a wider bandwidth should be discussed.
Response 3: Thank you for bringing up this valuable point. We have accordingly revised the manuscript to emphasize this point. ‘In addition, the validation was only conducted at three frequncies; hence there was a high signal-to-noise ratio (SNR) because the energy focused on a single frequency point in one pulse. However, in terms of the long-term applicability across a wider bandwidth, the compensated performance might deteriorate.’
Comments 4: The authors acknowledge that compensation cannot correct all defects. Could the authors quantify the tolerance to manufacturing errors (e.g., element misplacements, admittance variability) beyond which compensation becomes ineffective?
Response 4: We are thankful for your insight. Regarding the issue of tolerance to manufacturing errors, the authors believe that the effectiveness of compensation varies significantly depending on the type of ASWF source, making it difficult to generalize. For the source design presented in this manuscript, the authors consider it to be one with which the compensation method can readily achieve good results. This is characterized by the sufficiently small ka value of the element, such that manufacturing errors do not significantly alter the beam pattern of the element. A numerical simulation (not mentioned in the paper) showed that, even with large random position deviations of the array elements (as high as 1 mm (RMS)), compensation can still provide considerable effects. The phase directionality error after compensation is of the same order of magnitude as the case without position deviation. These related issues are worth further in-depth study.
Comments 5: A short discussion on how this prototype could be scaled for real-world deployment (e.g., UUV navigation beacons, open-water environments) would significantly strengthen the impact of the paper.
Response 5: Thank you for your helpful remark. The authors believe that the prototype in the manuscript can improve the positioning accuracy compared to large-phase-error sources, especially in high signal-to-noise ratio (SNR) situations. Here we provide two sets of our previous laboratory test results for your reference.
The first set of experiments was conducted in 2021, using four spherical sound sources to form an ASWF source, with a total phase directionality error of 1.87° (RMS), and a line trace positioning test with an azimuth span of 74° showed a 2.0° RMS error of azimuth estimation.
The other experiment was carried out in 2024, using the prototype in this study, and a line trace positioning test with an azimuth span of over 140° showed a 1.09° RMS error of azimuth estimation.
Due to the large time span between the two experiments, significant differences in experimental methods and conditions, and the fact that both experiments were conducted in laboratory settings rather than in real-world scenarios, and considering that the main focus of this paper is on the ASWF source rather than navigation experiments, the authors did not include these experimental results in the manuscript. If you think these data are necessary, we can organize these data and put them into the main text.
Thank you again.
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsThe manuscript represents a method for transmitting an array with low phase error. It also proposes a technique (techniques) to compensate the amplitude to minimise the error.
There are several items that should be fixed for the revision round.
1- Page 3, Line 94, "is define" should be changed to "is defined".
2- Page 4, please provide the reference for Equation (1a).
3- In the entire manuscript, "ka", defined as the product of wavenumber and array radius, is represented as an important item. Several figures were drawn based on the 'ka' parameter. What is the unit of 'ka'? Indeed, what is the unit of array radius?
4- English needs an update as some sentences miss "the". For example, on page line 175, "performance" should be changed to "the performance.
5- Figure 7 only shows a block of the transmitting array. This figure does not express (or explain) anything relevant to the sentences written in the last three sentences of page 6. How does that block show the link between electrical voltages (as the input) and the generated acoustic field (or pressure) (as the output of the block)? The figure may include several blocks showing how the input and output are linked through different blocks.
6- Page 7, Line 183, how are small-signal conditions defined?
7- Expression of Equation (4) (sentences relevant to Equation (4) ) and its link with the written Equation should be provided in a better format.
8- Table 2 is out of the margin. Please provide a better structure for it.
9- References need an update, as a very limited number of references were provided. Regardless of those papers published by the same author(s) of the manuscript, the newest reference was published in 2018 (more than 7 years ago). Providing newer references published in reputable journals (if there are any) will strengthen the quality of the manuscript.
Author Response
Thank you very much for taking the time to review this manuscript. Please find our detailed responses below and the corresponding revisions as tracked changes in the re-submitted files.
Comments 1: Page 3, Line 94, "is define" should be changed to "is defined".
Response 1: Thank you for pointing this out. We agree with this comment and have revised the manuscript accordingly.
Comments 2: Page 4, please provide the reference for Equation (1a).
Response 2: Thank you for pointing this issue out. We have marked the reference notation in the corresponding paragraph. Equation (1) is a variant of Equations (1)–(3) in Reference 8.
Comments 3: In the entire manuscript, "ka", defined as the product of wavenumber and array radius, is represented as an important item. Several figures were drawn based on the 'ka' parameter. What is the unit of 'ka'? Indeed, what is the unit of array radius?
Response 3: Thank you for your helpful remark. In the manuscript, ka is a dimensionless quantity, and all the ka data are expressed in radians. We have revised the corresponding figures to emphasize this point.
The unit of array radius will not affect the final numerical value of the ka parameter, as the length in the radius and the length in the k parameter cancel each other out. In addition, the sole datum regarding the array radius is 50mm, which signifies the distance between the center of each element and the center of the array, as detailed in Chapter 2.3.
Comments 4: English needs an update as some sentences miss "the". For example, on page line 175, "performance" should be changed to "the performance.
Response 4: We agree and have added "the" in line 175. Furthermore, the manuscript has been revised using MDPI’s English editing service to improve its English language quality.
Comments 5: Figure 7 only shows a block of the transmitting array. This figure does not express (or explain) anything relevant to the sentences written in the last three sentences of page 6. How does that block show the link between electrical voltages (as the input) and the generated acoustic field (or pressure) (as the output of the block)? The figure may include several blocks showing how the input and output are linked through different blocks.
Response 5: Thank you for your attention to detail. The authors believe that the premise of the compensation method is the assumption of a linear system, rather than the specific internal composition of the system; therefore, it is not necessary to provide these details here. The block diagram in Figure 7 is given because the system's output is actually a continuously distributed sound field, which needs to be spacially sampled to facilitate operability for subsequent processing. The last three sentences in page 6 are provided to first declare this model, while the complete description of the model is given in the following two paragraphs.
Comments 6: Page 7, Line 183, how are small-signal conditions defined?
Response 6: Thank you for your comments. We appreciate your feedback and have carefully considered your concerns. The small signal condition is one of the fundamental assumptions of linear acoustics. It assumes that the amplitude of the input signal is sufficiently small, such that the propagation of sound waves can be approximated as a linear process. Under the small signal condition, the acoustic equations can be simplified to linear equations, which are easier to analyze and solve.
The 1 dB compression point (P1dB) or third-order intermodulation distortion (IMD3) can be used to determine whether the system meets the small signal condition.
The 1 dB compression point is an indicator for evaluating the linearity of an amplifier or active device, which is defined as the input signal power level when the gain of the amplifier's output signal decreases by 1 dB. By measuring the 1 dB compression point, the maximum input power of the amplifier under the small signal condition can be determined.
Third-order intermodulation distortion is an important indicator for evaluating the nonlinearity of a system. which assesses the linearity of a system by measuring the third-order intermodulation products generated by two closely spaced input signals in a nonlinear system.
If the input signal power is much lower than the 1 dB compression point, the system can be considered to meet the small signal condition. If the level of third-order intermodulation distortion is low (e.g., below -40 dB), the system can also be considered to meet the small signal condition.
Comments 7: Expression of Equation (4) (sentences relevant to Equation (4) ) and its link with the written Equation should be provided in a better format.
Response 7: We agree and have streamlined the subscripts of the compensated voltage vector and the target acoustic pressure vector, as well as adjusting the paragraph format to place Equation (4) immediately next to the sentence that references it.
Comments 8: Table 2 is out of the margin. Please provide a better structure for it.
Response 8: We are grateful that you pointed this out. Accordingly, we have adjusted the expression of the data in the table, representing all data with three digits and using consistent spacing to make the table neater. We have also added labels for CH1-CH7 to enhance the readability of the data, and have corrected its numbering to Table 1.
Comments 9: References need an update, as a very limited number of references were provided. Regardless of those papers published by the same author(s) of the manuscript, the newest reference was published in 2018 (more than 7 years ago). Providing newer references published in reputable journals (if there are any) will strengthen the quality of the manuscript.
Response 9: Thank you for bringing up this valuable point. We have, accordingly, reviewed the reference list. According to our review, the most recent non-self-cited papers referenced in our manuscript include three papers published in 2023. Our results show that there are currently few studies on ASWF sources related to underwater navigation, which have been mentioned in this paper. We acknowledge that there is another major category of underwater ASWF sources, which are more involved in communication and significantly differ from the those in the context of navigation; therefore, they were not cited. If you find any omissions, please do not hesitate to point them out to us.
Thank you very much.
Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
Comments and Suggestions for Authors
Author Response
Thank you very much for taking the time to review this manuscript. Please find our revisions as tracked changes highlighted in the re-submitted files.
We have revised some expressions in the Introduction (Line 15) and Conclusion sections (Line 335-337). Figure 13 has been redrawn in color for better presentation.
