# Computer Modeling of Barrel-Vaulted Sanctuary Exhibiting Flutter Echo with Comparison to Measurements

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Measurements

#### 2.1.1. Room Configuration

#### 2.1.2. Measurement Equipment

#### 2.1.3. Source and Receiver Locations

#### 2.2. Computer Modeling

#### 2.2.1. Geometrical Acoustics (GA) Methods

#### 2.2.2. GA Computer Modeling Methodology

#### 2.2.3. Wave-Based Methods

#### 2.2.4. Wave-Based Computer Modeling Methodology

#### 2.3. Signal Processing of Room Impulse Response

**Spectrogram Analysis**: Spectrograms were generated from 48 kHz sampled RIR data using a 10 ms Hann window with 120 sample overlap.**Reverberation Time**: Reverberation times (T(20) and T(30)) were calculated from the 48 kHz sampled RIR data for the 125–8000 Hz octave bands using the ITA toolbox [65].**Energy Decay Curve**: The energy decay curves were generated from the 48 kHz sampled RIR data using the ITA toolbox. The curves were normalized and plotted in dB using P${}_{ref}$ = 20 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ over a 40 dB reduction in SPL.

#### 2.4. Wavelet Multi-Resolution Analysis

#### Wavelet Selection

- The differences between the wavelet types which they applied were insignificant, therefore, any of the tested wavelet families could be applied with similar results.
- The length of the wavelet had a significant effect on the uniformity of the results, where wavelets with longer support gave rise to more consistent results.

`modwt`and

`modwtmra`commands with a Daubechies 6 (db6) wavelet (containing 6 vanishing moments and a corresponding wavelet length of 11). The result of using this wavelet family can be seen in the evaluation of the MRA of an impulse response sampled at 45,254 Hz in Figure 5.

## 3. Results and Discussion

**R01 and R04:**As both of these figures are the same distance from the sound source, with one located on the aisle and the other located off the aisle, comparison between these two points demonstrates the non-diffuse nature of the room.**Original and Panel (with R01 and R04):**In order to demonstrate the changes in the room acoustics as a result of the addition of acoustical panels, some figures feature side-by-side comparisons of the original room configuration and the room configured with panels as shown in Figure 4.**R03, R04 and R05:**These three receiver locations represent three different heights at the same location in the room as shown in Figure 3. This comparison demonstrates the unique behavior of the flutter echo at different heights relative to the source height.

#### 3.1. RIR Spectrogram

**Behavior of the room on and off the aisle:**Comparison of Figure 7 and Figure 8, demonstrates the non-diffuse nature of the room. As stated earlier, both receiver points R01 and R04 are the same distance from the sound source, however, because of their location with respect to the barrel-vaulted ceiling, their spectrograms display very different behaviors.In high frequencies, above 4000 Hz, the measured original R01 and R04 spectrograms exhibit similar behavior, with the majority of the energy having died out by 0.5 s. Below 4000 Hz, the behavior of the two locations are quite different. In the audience, the signal dies out almost completely by 1.5 s, while in the aisle, it remains well beyond 4 s. Additionally, the signal is characterized by a periodic pattern representative of the flutter echo present in the aisle. Furthermore, along the aisle, it is possible to identify the comb filter behavior expected from a flutter echo.**Behavior of room before and after the addition of panels:**In order to demonstrate the changes in the room acoustics as a result of the addition of acoustical panels, some figures feature side-by-side comparisons of the original room configuration and the room configured with panels as shown in Figure 1.Similar to what was observed in the comparison of R01 and R04, at higher frequencies (above 4000 Hz), the addition of the panels had very little effect on the spectrogram. Below 4000 Hz, in the audience area, the addition of the panels reduces the energy, such that the signal is essentially eliminated after 1.5 s. On the aisle, the effect of the panels is more pronounced, with the primary effect being to eliminate the majority of the late decay energy in the 250–2000 Hz octave bands after 2 s. The panels also have a secondary effect of reducing the amplitude of the peaks from 0.5 to 1.5 s.These results indicate that the panels had the desired effect of reducing the RT along the aisle without completely eliminating the spacious feel of the room promoted by the flutter echo.**Comparison of the GA and wave-based modeling with the measured data:**The spectrograms provide a means of comparison of the similarities and distinctions between the measured data and the two different modeling techniques, GA and wave-based. Comparison between measured and modeled response before 0.5 s highlights specific early behaviors (before 0.5 s) and later behaviors. Before 0.5 s, both models contain more energy above 8 kHz than the measured data, with Odeon predicting much more sustained high frequency energy. It is important to note that above 8 kHz, both Odeon and the wave-based model use ray-tracing techniques. Also, the measured data has a band of higher energy sustained past 0.5 s around 6500 Hz, which is not identified in the wave-based model.Most significantly, the very distinct pattern of the flutter echo detected in measurements on the aisle in Figure 8, are clearly identified in the wave-based model, but are not present in the spectrogram of the Odeon model RIR.

#### 3.2. RIR Reverberation Times

#### 3.3. Energy Decay Curves

#### 3.4. Wavelet Multi-Resolution Analysis

#### 3.4.1. Comparison of Flutter Echo On and Off the Aisle

`findpeaks`command with a minimum peak prominence of 10 $\times {10}^{-7}$, and a minimum peak distance of 1600.

#### 3.4.2. Comparison of Flutter Echo Before and After the Addition of Panels

#### 3.4.3. Structure of Triplets Based on Receiver Height

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**left**) photograph of Christ the King (CTK) church taken after remediation. The main aisle is in view, along with absorbing panels mounted under the barrel-vaulted ceiling. (

**right**) Simplified computer-aided design (CAD) model of CTK church after remediation with axis units in meters.

**Figure 2.**Brüel and Kjær TYPE 4292-L dodecahedral loudspeaker in the source location used for measurements.

**Figure 3.**Layout of source (S) and six receivers (R01–R06), from x–y (

**top**) and x–z (

**bottom**) cross-sectional views. Note: R03–R05 are coincident in the top view, and axes units are in meters.

**Figure 4.**(

**left**) CAD model for Odeon simulations (without panels); (

**right**) CAD model for wave-based simulations (with panels). Axes units are in meters.

**Figure 5.**Multi-resolution analysis (MRA) of impulse (45,254 Hz sample rate) using Daubechies (db6) wavelet. Levels 1–7, represent octave bands 250–16,000 Hz.

**Figure 7.**Spectrograms for receiver location R01 (in audience area, 3 m from source); before and after the addition of the acoustical panels.

**Figure 8.**Spectrograms for receiver location R04 (on aisle 3 m from source); before and after the addition of the acoustical panels. Note the presence of acoustical signals from 1.5–4.5 s not observed in Figure 7 above.

**Figure 9.**Reverberation times for receiver locations R01 (audience, 3 m from source) and R04 (aisle, 3 m from source); before and after the addition of the acoustical panels.

**Figure 10.**Energy decay curves for receiver location R01 (audience, 3 m from source) and R04 (aisle, 3 m from source); before and after the addition of the acoustical panels. For conciseness, only the 250 Hz through 4000 Hz octave bands are shown.

**Figure 11.**Energy decay curves for measured and modeled data before and after acoustical remediation. (

**a**) Receiver located off the aisle 3 m from source, (

**b**) receiver located on the aisle 3 m from source. Highlighted area reflects the data used for wavelet MRA in Section 3.4.1 and Section 3.4.2.

**Figure 12.**MRA levels 1–7, calculated using measured data sampled at 45.254 kHz from R04, representing octave bands 250–16,000 Hz, t = 1–1.3 s and 2–2.5 s. Based on these plots, the Level 4 MRA was identified for use in the presented wavelet analysis.

**Figure 13.**Level 4 MRA of original measured and modeled room impulse response (RIR), R01—in the audience, 3 m from source, R04—on the aisle 3 m from source. Dotted lines identify the middle peak of the triplets in the measured signal.

**Figure 14.**MRA of measured and modeled RIR at R04 located on the aisle 3 m from source. Comparison of the original room and the room with the addition of the acoustical panels. Red lines represent exponential decay of peaks and are included to aid in visualization of the signal decay.

**Figure 15.**Structure of triplet pattern measured for the original room configuration on the aisle for three different receiver heights. R03 is located at a height of 1m, R04 at 1.5m and R05 at 2m. Highlighted area demonstrates the compression of the triplet for receiver R04 (height near source height).

**Table 1.**Receiver locations included in this study (locations specified in m in terms of the coordinates shown in Figure 1).

Receiver | X | Y | Z |
---|---|---|---|

R01 | 8.90 | 3.80 | 1.50 |

R02 | 9.60 | 1.90 | 1.50 |

R03 | 5.00 | 6.65 | 1.00 |

R04 | 5.00 | 6.65 | 1.50 |

R05 | 5.00 | 6.65 | 2.00 |

R06 | 1.66 | 6.65 | 1.50 |

**Table 2.**Absorption coefficients of materials used in GA and wave-based models. “Audience” material only used for Odeon, and “Seats” were used only in wave-based. Odeon only takes into account 63–8000 Hz data, whereas wave-based simulations took into account all octave band data displayed.

Material | 31 Hz | 63 Hz | 125 Hz | 250 Hz | 500 Hz | 1 kHz | 2 kHz | 4 kHz | 8 kHz | 16 kHz |
---|---|---|---|---|---|---|---|---|---|---|

Altar | 0.25 | 0.25 | 0.25 | 0.15 | 0.1 | 0.09 | 0.08 | 0.07 | 0.07 | 0.07 |

Audience | – | 0.1 | 0.1 | 0.07 | 0.08 | 0.1 | 0.1 | 0.11 | 0.11 | – |

Carpet | 0.08 | 0.08 | 0.08 | 0.24 | 0.57 | 0.69 | 0.71 | 0.73 | 0.73 | 0.73 |

Ceiling | 0.19 | 0.19 | 0.19 | 0.06 | 0.05 | 0.08 | 0.07 | 0.05 | 0.05 | 0.05 |

Panel | 0.2 | 0.42 | 0.89 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Seats | 0.44 | 0.44 | 0.44 | 0.56 | 0.67 | 0.74 | 0.83 | 0.87 | 0.87 | 0.87 |

Tile | 0.015 | 0.015 | 0.015 | 0.015 | 0.005 | 0.005 | 0.005 | 0.005 | 0.005 | 0.005 |

Walls | 0.19 | 0.19 | 0.19 | 0.06 | 0.05 | 0.08 | 0.07 | 0.05 | 0.05 | 0.05 |

Window | 0.35 | 0.35 | 0.35 | 0.25 | 0.18 | 0.12 | 0.07 | 0.04 | 0.04 | 0.04 |

Wavelet Level | Octave Band (Hz) |
---|---|

7 | 250 |

6 | 500 |

5 | 1000 |

4 | 2000 |

3 | 4000 |

2 | 8000 |

1 | 16,000 |

© 2020 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**

Lai, H.; Hamilton, B. Computer Modeling of Barrel-Vaulted Sanctuary Exhibiting Flutter Echo with Comparison to Measurements. *Acoustics* **2020**, *2*, 87-109.
https://doi.org/10.3390/acoustics2010007

**AMA Style**

Lai H, Hamilton B. Computer Modeling of Barrel-Vaulted Sanctuary Exhibiting Flutter Echo with Comparison to Measurements. *Acoustics*. 2020; 2(1):87-109.
https://doi.org/10.3390/acoustics2010007

**Chicago/Turabian Style**

Lai, Heather, and Brian Hamilton. 2020. "Computer Modeling of Barrel-Vaulted Sanctuary Exhibiting Flutter Echo with Comparison to Measurements" *Acoustics* 2, no. 1: 87-109.
https://doi.org/10.3390/acoustics2010007