Transmission Phase Control of Annular Array Transducers for Efficient Second Harmonic Generation in the Presence of a Stress-Free Boundary
Round 1
Reviewer 1 Report
As always I'd like to see the next papers with laboratory verification and field verification. Maybe add a plan for this.
Author Response
Authors’ Reply to Reviewer Comments
The authors would like to deeply appreciate the valuable opinions of all reviewers. We believe that these comments and suggestions will definitely improve the quality of the paper. We have read all comments and requests carefully and have tried to revise the manuscript in the best possible way.
Reviewer 1
(C) As always I'd like to see the next papers with laboratory verification and field verification. Maybe add a plan for this.
(R) The following paragraph was added at the end of Section 5.1:
The current work covers ideal cases, but the actual phenomenon will not differ much from the theoretical investigation presented here. Future work should be able to include laboratory verification and field verification. Some of the challenges associated with experimental validation are making such a multiple element transducer. It should be able to carry high power input signals with minimal source harmonics. All elements of the transducer should be able to act as both a transmitter and a receiver and need to have a wide bandwidth to cover both the fundamental and second harmonic frequency components. In addition, at least 4 channels of function generator and high power amplifier may be required.
Author Response File: 
Author Response.docx
Reviewer 2 Report
Dear author,
Your work is devoted to an actual problem. The research results can be used to solve the problems of flaw detection and biomedical acoustics. The considered models and theoretical approaches to solving the problem, as well as the results obtained, are beyond doubt. The results of experimental studies and their comparison with the considered theoretical calculations could be of great interest. From the point of view of the reviewer, this problem should be given attention in subsequent works. The publication of the article in the presented form seems appropriate.
Some comments on the text of the work:
- Investigations of nonlinear acoustic phenomena in solids are usually associated with determining the deviations of the elastic force of an ideal lattice from Hooke's law. At small deformations, such deviations are insignificant, and therefore, in the overwhelming majority of works, they are limited to quadratic corrections in the equation of state of a solid, which are responsible for the generation of the second acoustic harmonic. However, bodies with microstructure or continuity defects (cracks, pores, delamination, etc.) are characterized by anomalously high elastic nonlinearity. Nonlinear effects manifested in such bodies can serve as a source of information about the structure of a solid. In this regard, the development of the theory of nonlinear interactions of higher orders and the determination of the nonlinear parameters of such interactions for their subsequent quantitative assessment is of particular relevance. Generation of the third harmonic was considered, for example, in the work: Van Den Abeele K., Breazeale M.A. // J. Acoust. Soc. At. 1996, Vol. 99, Is. 3, P. 1430. Perhaps, the work should reflect the limited application of the method based on the evaluation of the second harmonic nonlinearity. It is advisable to reflect in the conclusions on the work and the issues of the limitations of your proposed solution from a theoretical point of view.
2. More attention should be paid to the process of numerical calculations. How, in what software environment, were the calculations carried out, with what accuracy?
Author Response
Authors’ Reply to Reviewer Comments
The authors would like to deeply appreciate the valuable opinions of all reviewers. We believe that these comments and suggestions will definitely improve the quality of the paper. We have read all comments and requests carefully and have tried to revise the manuscript in the best possible way.
Reviewer 2
(C1) Nonlinear effects manifested in such bodies can serve as a source of information about the structure of a solid. In this regard, the development of the theory of nonlinear interactions of higher orders and the determination of the nonlinear parameters of such interactions for their subsequent quantitative assessment is of particular relevance. Generation of the third harmonic was considered, for example, in the work: Van Den Abeele K., Breazeale M.A. // J. Acoust. Soc. At. 1996, Vol. 99, Is. 3, P. 1430. Perhaps, the work should reflect the limited application of the method based on the evaluation of the second harmonic nonlinearity. It is advisable to reflect in the conclusions on the work and the issues of the limitations of your proposed solution from a theoretical point of view.
(R1) The following paragraph was added at the end of Section 5.2:
This study deals with the generation of second harmonic in relation to the evaluation of the properties of nonlinear solid materials. Just as the nonlinear image based on the second harmonic is superior to the linear image, the third harmonic image can have a better resolution in tissue characterization because the main lobe is narrower in the transverse beam pattern than the second harmonic. A measure of the third harmonic is the nonlinear coefficient C/A in fluids and biological media. Other studies have shown that the third harmonic is more sensitive to microstructure changes than the second harmonic by measuring the amplitude of the third harmonic or measuring the relative third-order nonlinear parameter for the damaged solid medium [32].
(C2) More attention should be paid to the process of numerical calculations. How, in what software environment, were the calculations carried out, with what accuracy?
(R2) The following paragraph was added in Section 5.1:
Eq. (11) for calculating the propagating sound beam fields is called the Rayleigh-Sommerfeld (RS) integral and serves as the exact solution to the linear wave equation, Eq. (9). Eq. (12) also serves as an exact solution to the second harmonic wave equation in the quasilinear theory since the exact linear solution is used in the right hand-side of Eq. (12). Therefore, all of the sound beam field equations derived in this study can be treated as exact solutions. Numerical calculations for the simulation were performed using MATLAB program. The amplitudes of the received displacements given in Table 1 can also be considered exact, although it may contain inherent errors that cannot be avoided in numerical calculations.
Author Response File: Author Response.docx
Reviewer 3 Report
The manuscript investigates a phase-shift technique for enhancing the second harmonic generation in the thin solid samples with stress-free boundaries using the pulse-echo method. The paper is well written and clear to understand for the reader. The author has reasonably presented the motivation behind the PE method, and why it is practical compared to the through-transmission method for nonlinear solid samples. I believe the proposed phase-shift technique with the PE method contributes well to the field of damage detection. The only minor concern of mine is about the last figure, where the nonlinear parameter is plotted. Why the case 4 has a higher value, and how can the user choose the optimum phase shift value between transducers to obtain the correct nonlinear value? Besides, this theoretical study is novel and is ready to be accepted.
Author Response
Authors’ Reply to Reviewer Comments
The authors would like to deeply appreciate the valuable opinions of all reviewers. We believe that these comments and suggestions will definitely improve the quality of the paper. We have read all comments and requests carefully and have tried to revise the manuscript in the best possible way.
Reviewer 3
(C) The only minor concern of mine is about the last figure, where the nonlinear parameter is plotted. Why the case 4 has a higher value, and how can the user choose the optimum phase shift value between transducers to obtain the correct nonlinear value?
(R) The following sentences were added in Section 5.2:
When a phase shift is accompanied for the measurement of only with R=-1, it seems that Cases 3 and 6 are suitable choices. The optimal phase shift criterion for the reliable determination of b using the four element transducer is to maximize SHG from the stress-free boundary and to minimize the dependence on the correction by making the correction value as close to one as possible.
It is noted that the largest value of occurs when in case 4 because the second harmonic amplitude is relatively large and the fundamental wave amplitude is the smallest.
Author Response File: Author Response.docx
