Investigation of damping effects on low-frequency steady-state acoustical behaviour of coupled spaces

In the low-frequency range, the acoustical behaviour of enclosed spaces is strongly influenced by excited acoustic modes resulting in a spatial irregularity of a steady-state sound field. In the paper, this problem has been examined theoretically and numerically for a system of coupled spaces with complex-valued conditions on boundary surfaces. Using a modal expansion method, an analytic formula for Green’s function was derived allowing to predict the interior sound field for a pure-tone excitation. To quantify the spatial irregularity of steady-state sound field, the parameter referred to as the mean spatial deviation was introduced. A numerical simulation was carried out for the system consisting of two coupled rectangular subspaces. Eigenfunctions and eigenfrequencies for this system were determined using the high-accuracy eigenvalue solver. As was evidenced by computational data, for small sound damping on absorptive walls the mean spatial deviation peaks at frequencies corresponding to eigenfrequencies of strongly localized modes. However, if the sound damping is much higher, the main cause of spatial irregularity of the interior sound field is the appearance of sharp valleys in a spatial distribution of a sound pressure level.

Decision letter (RSOS-200514.R0) We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.
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Once again, thank you for submitting your manuscript to Royal Society Open Science and I look forward to receiving your revision. If you have any questions at all, please do not hesitate to get in touch. The authors have stated that "The way to reduce the value of the parameter D is to increase the sound attenuation inside the coupled spaces. This is due to the fact that with increased sound damping, the energy of the acoustic modes is physically attenuated, reducing the point-to-point variations in the a sound pressure level." This explanation is not correct. Because the authors' standard deviation is calculated using decibel values, changing the modal energy will not change D. The reduction in D with increasing sound absorption is due to the increase in modal bandwidth which increases the modal overlap. The increased modal overlap means that more modes are excited at a given frequency and this reduces the spatial standard deviation of the sound pressure level in the room (D). The increase of modal bandwidth with increasing sound absorption also explains why the variation of D with frequency becomes slower as the sound absorption is increased. The authors' use of a sound source position that is equidistant ( 1 m) from the floor and two of walls will create an unusual situation. The authors should repeat their calculations with the distances of the source chosen so that they are not in the ratio of small integers to each other.

Reviewer: 2 Comments to the Author(s)
This is a nicely presented paper. Well done! I have four minor comments: 1. I would suggest including some practical reasons why spatial variation of pressure fields may be an important metric in the design of acoustic enclosures.
2. The authors should check their use of "specific acoustic impedance" to describe their parameter zeta. Specific acoustic impedance is defined as pressure/velocity and is given by Z=rho*c*zeta. Zeta is simply a convenient non-dimensional ratio of specific acoustic impedance to the characteristic impedance of the fluid.

Decision letter (RSOS-200514.R1)
We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.

Dear Professor Meissner,
It is a pleasure to accept your manuscript entitled "Investigation of damping effects on lowfrequency steady-state acoustical behaviour of coupled spaces" in its current form for publication in Royal Society Open Science. The comments of the reviewer(s) who reviewed your manuscript are included at the foot of this letter.
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Comment 2
The authors' use of a sound source position that is equidistant (1 m) from the floor and two of walls will create an unusual situation. The authors should repeat their calculations with the distances of the source chosen so that they are not in the ratio of small integers to each other.

Response
Calculations of the mean spatial deviation D were repeated for a sound source located at the point: (8.9 m, 4.1 m, 1.2 m), so the distances from the source to the floor and room walls are not in the ratio of small integers to each other. Calculation results are shown in figures II-V and these data are equivalent to the results in figures 5 and 7-9 in the paper, obtained for the source located at the point: (9 m, 4 m, 1 m). Since there is a significant similarity of calculation results for both source positions, similar observations can be made and similar conclusions can be drawn for the newly selected source position, as in the case of the previously chosen source position.