Analysis of the Effects of the Viscous Thermal Losses in the Flute Musical Instruments
Round 1
Reviewer 1 Report
In the article under review, the authors investigate the analysis of the effects of the viscous thermal losses in the flute musical instruments. The research questions posed in this paper are really interesting. As far as I know, the results obtained are original. I think this work will be useful for specialists in this domain.
1. I am not able to find out from the manuscript clearly what is the results of the paper. Please give a clear introduction of the background of the work. What has been done? What is new in this work?
2. After the literature, state the purpose of the paper more specifically.
3. The authors should also include a discussion about the uncertainty of the calculation and the relative accuracy that can reach.
Author Response
Authors’ reply to Reviewer 1
Notation:
Q: Reviewer’s question or comment (in blue).
R: Author’s response (in black).
Q1. I am not able to find out from the manuscript clearly what is the results of the paper. Please give a clear introduction of the background of the work. What has been done? What is new in this work?
R: Actually in the previous work, the system was modelled physically using a servo valve and an artificial mouth that were also simulated and tested with both static and dynamic behaviour neglecting the viscous thermal losses that are the main topic of this article.
Concerning this work, it consists of modelling and simulating the viscous losses within the resonator of the flute musical instrument. This work is not unique as it is a continuity of three previous publications showing:
- the system consisting of the musician-flute that was implemented and modeled. It consisted of an air compressor, a servo-valve and an artificial mouth mounted to mimic the musician lungs and mouth. A control system was also developed to regulate the pressure and the flow delivered to the artificial mouth. Added to that, the flute exciter was directly coupled with the artificial mouth and some pressure and temperature sensors where placed within the resonator [17] [18];
- the knowledge model was developed to represent the transfer between the pressure source at the input of the tube at x = 0 and the flow at any point x of the tube of length L and of constant radius r. Partial differential equations aiming to causally decompose the global model into sub-models, and thus to facilitate analysis in the frequency domain were used in modelling [19].
Q2. After the literature, state the purpose of the paper more specifically.
R: The main target of this article is to analyze the effect of viscous thermal fractional order element on the sound delivered at the output of the flute. The novelty of the work resides in the numerical synthesis of the viscous thermal losses as well as in its simulation using the hardware-in-the-loop technique. This numerical simulation is important as it allows the sweeping of the fractional order viscous thermal variable, which is mainly linked to the geometry of the flute’s resonator as well as to the materials constituting it.
Q3. The authors should also include a discussion about the uncertainty of the calculation and the relative accuracy that can reach.
R: While modelling the resonator for the purpose of studying the viscos thermal losses, many parameters where included aiming to get out with a model that can define the real resonator with the maximum matching. As for the uncertainties of the model compared to the exact system, they were not treated in this work and they are part of our future works.
Author Response File: 
Author Response.docx
Reviewer 2 Report
The manuscript studies the viscous thermal losses in a circular tube with finite length. Particularly, the authors study the case of wide tubes and they obtain an analytical expression for the acoustic impedance, the acoustic admittance and the transfer function as a function of the position of in the tube. They also report the frequency response of the different magnitudes at different positions inside the tube.
Unfortunately, this reader cannot appreciate the interest of the reported results since one expects a clear discussion about the effects of viscous thermal losses on the performance of the flute. In other words, the manuscript is not well written and it findings are not clearly described. Many issues should be addressed. In what follows, I list some of them
1.- The abstract is too long. I recommend revising it to emphasize what are the relevant contributions of the work. The reader would like to know why the viscous thermal losses are important in understanding the performance of the flute.
2.- In line 88, the values of Table 1 are given without any support. I suggest citing the book of Leo Beranek where expression for the different magnitudes can be found.
3.- In line 101, the values for the characteristic lengths of viscous are thermal effect are given but there is no justification. I suggest adding some reference where their expressions can be found.
4.- In line 236, please, check the range.
5.- In line 400, please, revise the sentence. It seems that you just translate a chapter of someone's PhD thesis.
6.- The conclusion section should be revised. In my opinion, the main contributions of this work are not clear.
7.- In the list of references, there are many mistakes. For example, in Ref. 3 the volume and the page number are missing. Also, in Ref. 6 and 8. The names of the authors in Ref. 11 are probably wrong. References 29, 30 and 31 are non-sense.
In summary, I cannot recommend acceptation of this manuscript in its present version. I recommend a profound revision before resubmission.
Author Response
Authors’ reply to Reviewer 2
Notation:
Q: Reviewer’s question or comment (in blue).
R: Author’s response (in black).
Q1. The abstract is too long. I recommend revising it to emphasize what are the relevant contributions of the work. The reader would like to know why the viscous thermal losses are important in understanding the performance of the flute.
R: The abstract was corrected, reduced and restructured as follows:
This article presents the third part of a larger project whose final objective is to study and analyse the effects of viscous thermal losses in a flute wind musical instrument. After implementing the test bench in the first phase and modelling and validating the dynamic behaviour of the simulator, based on the previously implemented test bench (without considering the losses in the system) in the second phase, this third phase deals with the study of the viscous thermal losses that will be generated within the resonator of the flute. These losses are mainly due to the friction of the air inside the resonator with its boundaries and the changes of the temperature within this medium. They are mainly affected by the flute geometry and the materials used in the fabrication of this instrument. After modelling these losses in the frequency domain, they will be represented using a system approach where the fractional order part is separated from the system’s transfer function. Thus, this representation allows to study, in a precise way, the influence of the fractional order behaviour on the overall system. Effectively, the fractional behavior only appears much below the 20Hz audible frequencies, but it explains the influence of this order on the frequency response over the range [20; 20,000] Hz. Some simulations will be proposed to show the effects of the fractional order on the system response.
Q2. In line 88, the values of Table 1 are given without any support. I suggest citing the book of Leo Beranek where expression for the different magnitudes can be found.
R: The proposed reference was already added in the caption of table 1
Q3. In line 101, the values for the characteristic lengths of viscous are thermal effect are given but there is no justification. I suggest adding some reference where their expressions can be found.
R: The expressions for the values of the characteristic references can be found in the reference of table 1 [25].
Q4. In line 236, please, check the range
R: checked
Q5. In line 400, please, revise the sentence. It seems that you just translate a chapter of someone's PhD thesis.
R: checked
Q6. The conclusion section should be revised. In my opinion, the main contributions of this work are not clear.
R: The conclusion was reviewed and restructured as follows to clearly show our contribution:
The structure and progression of this article are organized in a didactic way so that readers with no idea about visco-thermal losses in wind instruments can "absorb" the dynamic behavior of an acoustic tube of constant radius. From the two partial differential equations which define the Webster-Lokshin model, a classical resolution in the operational domain leads to the analytical expression of the acoustic impedance and admittance of the function tube of position x, its length L and its radius r.
Moreover, a system vision is proposed aiming to causally decompose the global model into sub-models, thus facilitating analysis in the frequency domain. One of the conclusions of this frequency analysis is that the fractional model can be simplified over the range [20; 20,000] Hz of the audible frequencies. In Addition, the introduction of an uncertainty at the level of the fractional order (whose value considered as nominal is that of the initial Webster-Lokshin model, namely m0 = 0.5) allows to study the influence of the order m when this varies between 0 (conservative case) and 1.
As is often the case with fractional models, simulation in the time domain requires the establishment of rational forms. Thus, two rational forms composed of an integrator and N second-order cells, one in cascade and the other in parallel, where introduced. The parameters of the cascade form are then determined using the Frequency Domain System Identification (FDSI) module of the CRONE Toolbox. As for the parameters of the parallel form, they are obtained by a decomposition into simple elements of the cascade form.
More generally in the fractional model, this study of visco-thermal losses within the resonator of a wind instrument leads to a finding similar to that already made in other fields. Indeed, the main interest of the fractional form resides in the parametric parsimony, that is to say the capacity which the integro-differential operator of non-integer order has to model with a minimum of parameters the greatest number dynamic phenomena. Thus, the study of parametric sensitivity, in particular in the frequency domain, is simpler.
As a future work, a comparison between the simulated values and the exact outputs can be computed in order to compare the approximation effects from a practical point of view. Added to that, building resonators with the same fractional order as proposed in this article will be a major challenge as going from simulated systems to implemented ones will be an innovation in the musical instruments field.
Q7. In the list of references, there are many mistakes. For example, in Ref. 3 the volume and the page number are missing. Also, in Ref. 6 and 8. The names of the authors in Ref. 11 are probably wrong. References 29, 30 and 31 are non-sense.
R: References 3,6, and 8 are checked and corrected. References 29,30, and 31 are important as they refer to the CRONE toolbox explaining the procedure to rationalize a fractional order system which is the last part in our article.
Author Response File: Author Response.docx
Reviewer 3 Report
In this paper, the main objective is to analyze the effect of viscous thermal fractional order element on the sound delivered at the output of a flute. The novelty of the work resides in the numerical synthesis of the viscous thermal losses, as well as in its simulation using the Hardware-in-the-loop technique.
The paper is quite easy to read, and possibly promising for further generalizations; however, where possible, I suggest to try reducing the mathematics in Section 2, in order to improve its readability. I only stayed with a few detailed remarks, which are set out in the attached review.
Comments for author File: 
Comments.pdf
Author Response
Authors’ reply to Reviewer 3
Notation:
Q: Reviewer’s question or comment (in blue).
R: Author’s response (in black).
Q1. The paper is quite easy to read, and possibly promising for further generalizations; however, where possible, I suggest to try reducing the mathematics in Section 2, in order to improve its readability. I only stayed with a few detailed remarks, which are set out in the attached review.
R: The mathematics in section cannot be reduced due to the fact that it will be a good base for readers while trying to deeply understand and the concept especially if someone need to reproduce the work for the purpose trying to retest and modify our work
Author Response File: Author Response.docx
Round 2
Reviewer 2 Report
I’m sorry to say that still I cannot see the interest of this work, which studies the propagation in a cylindrical tube with finite length L and circular section with viscothermal losses. In addition, the study is constrained to the case wide tubes. The study of ducts and uniform tubes is a well-documented subject, starting with the classical book of M.L. Munjal “Acoustics of ducts and Mufflers” (Willey and Sons, 1987). The authors obtained analytical expressions that they solved for the single case of a tube with radius 5 mm and length L=0.3 m. The selection of this dimension is not justified nor explained. Perhaps they corresponds to an actual flute.
The readers would like to read a comprehensive discussion, for example, of the viscothermal effects as a function of the radius and length of the tube. There is any implications from the point of view of musical acoustics?
In the present version, this work has little interest and I do not recommend publication.
Author Response
Authors’ reply to Reviewer 2
Notation:
Q: Reviewer’s question or comment (in blue).
R: Author’s response (in black).
Q1. I’m sorry to say that still I cannot see the interest of this work, which studies the propagation in a cylindrical tube with finite length L and circular section with viscothermal losses.
R: As stated in the abstract and the introduction, the work presented in this paper is a part of a larger project. The objective is not to study the system from an acoustic musical point of view for which there are indeed many works, but from a control point of view (system approach, section 3), in particular within the framework of the dynamics of complex systems during the study of the coupling between the nonlinear exciter and the resonator (which is part of the continuity of this paper and which will be the subject of a future publication).
In addition, an extension of the fractional model to take into account the visco-elastic losses is proposed (relation 11), thus making it possible to vary the fractional order m from 0 (conservative system) to 1 (very dissipative system), and not to consider only m = 0.5 as is currently the case in the literature. This domain [0; 1] belonging to the order m makes it possible to better account for the influence of geometry (radius r and length L), the roughness and the nature of the material of the resonator.
Q2. In addition, the study is constrained to the case wide tubes. The study of ducts and uniform tubes is a well-documented subject, starting with the classical book of M.L. Munjal “Acoustics of ducts and Mufflers” (Willey and Sons, 1987). The authors obtained analytical expressions that they solved for the single case of a tube with radius 5 mm and length L=0.3 m. The selection of this dimension is not justified nor explained. Perhaps they corresponds to an actual flute.
R: The values of r=5 mm and L=0.3 m correspond to the dimensions of the experimental device developed in a first part of the overall project and used to validate a numerical simulator of the artificial mouth + nonlinear exciter + resonator assembly, simulator developed under MatLab / Simulink (experimentation / numerical simulation comparison, then resetting of the simulator).
Q3. The readers would like to read a comprehensive discussion, for example, of the viscothermal effects as a function of the radius and length of the tube. There is any implication from the point of view of musical acoustics?
From a system approach point of view, the extension of the fractional model (relation 11) makes it possible to easily vary, in numerical simulation, the fractional order m which is the image of visco-thermal losses, while, from an experimental point of view, it would be necessary to manufacture and test a large number of resonators with different dimensions, roughness and materials.
