# On the Influence of Scattered Errors over Full-Field Receptances in the Rayleigh Integral Approximation of Sound Radiation from a Vibrating Plate

## Abstract

**:**

## 1. Introduction

- SLDV [39] introduced the concept of contactless measurement—adding no mass to the specimen—in time/frequency domains in the 1980s, expanding it to finer grids of scanned locations, thus extending the concept of the velocity sensors to a spatially detailed acquisition. Therefore SLDV is considered the reference in NVH when spatially detailed FRF measurements are needed: SLDV keeps the same peculiarities of lumped sensor technologies (and of established procedures, born to exploit few dofs only), but adds many dofs in the spatial domain at a reasonable cost, with a trade-off at high frequency in the sensitivity between displacements and accelerations. Due to the asynchronous scanning process, however, SLDV can not be properly called a native full-field instrumentation.
- Only the optical full-field technologies that acquire the motion-related information from photons synchronously recorded at every sensible site of an imaging sensor, normally in a much denser grid, can be instead called native. Earlier, in [23,24], the author proved the high quality of the datasets obtainable in the spatial domain from the native full-field measurement techniques, acknowledged in terms of consistency of the motion field among the neighbouring dofs. Since the 2000s, ESPI, among the native full-field technologies, has given extremely accurate displacement fields up to several kHz of frequency. Due to a lack of complete processing automation, up to now the main drawback of ESPI remains the time-consuming stepped sine excitation/acquisition, in order to have data at all the lines of a broad frequency band in stroboscopic light acquisitions [25,45].
- High-speed DIC is another native technology, with its first commercial prototypes starting around 2005. DIC can have good detail in the time-resolved displacement maps, but the processing of the correlated data can be time consuming [12,13]. Due to both a sensor/electronics bandwidth trade-off, between resolution and sampling rate, and the difficulty in properly exciting the higher frequency displacement components, DIC is generally more limited for complex structural dynamics in a broad frequency domain, although rapid electronics and processing improvements can be easily foreseen in the near future.
- Indeed, nowadays, scanning and native full-field optical technologies allow displacement and velocity measurements in a non-contact way with dense spatial mapping, without inertia-related distortions of the dynamics (due to added sensors, fixtures and cabling), and without any structural finite element (FE) or analytical models to be accurately tuned for the lumped sensor data expansion [42,46,47]. As it was previously shown in [13,34], ESPI technology actually permits the most precise estimation from non-contacting testing of superficial receptance FRF maps, which turn out to be optimal also for derived quantities like dense mapping of rotational dofs and strains.

## 2. Materials and Methods

#### 2.1. Direct Experimental Modelling for Full-Field FRFs

**Figure 1.**Full-field optical measurement instruments set-up in front of the specimen on the anti-vibration table with shakers on the backside: the instruments in ($\mathit{a}$), the restrained thin plate in ($\mathit{b}$) and the 2 shakers in ($\mathit{c}$).

#### 2.2. The Lab Activities for the TEFFMA Project in Brief

**Table 1.**Amplitude samplings of Green’s function with distance and frequency dependencies, ${c}_{0}$ = 300 m/s.

Distance | 20 Hz | 250 Hz | 512 Hz | 750 Hz | 1024 Hz |
---|---|---|---|---|---|

1 mm | 7.954e+01 | 7.916e+01 | 7.873e+01 | 7.834e+01 | 7.789e+01 |

2.5 mm | 3.180e+01 | 3.142e+01 | 3.099e+01 | 3.061e+01 | 3.017e+01 |

5 mm | 1.588e+01 | 1.550e+01 | 1.508e+01 | 1.471e+01 | 1.430e+01 |

15 mm | 5.272e+00 | 4.904e+00 | 4.517e+00 | 4.192e+00 | 3.846e+00 |

25 mm | 3.150e+00 | 2.793e+00 | 2.435e+00 | 2.149e+00 | 1.862e+00 |

50 mm | 1.559e+00 | 1.225e+00 | 9.310e-01 | 7.256e-01 | 5.446e-01 |

75 mm | 1.028e+00 | 7.164e-01 | 4.747e-01 | 3.267e-01 | 2.124e-01 |

150 mm | 4.982e-01 | 2.419e-01 | 1.062e-01 | 5.028e-02 | 2.126e-02 |

200 mm | 3.659e-01 | 1.396e-01 | 4.660e-02 | 1.719e-02 | 5.457e-03 |

300 mm | 2.339e-01 | 5.514e-02 | 1.063e-02 | 2.383e-03 | 4.260e-04 |

500 mm | 1.291e-01 | 1.161e-02 | 7.469e-04 | 6.178e-05 | 3.505e-06 |

750 mm | 7.750e-02 | 2.091e-03 | 3.411e-05 | 8.116e-07 | 1.097e-08 |

1 m | 5.234e-02 | 4.235e-04 | 1.753e-06 | 1.199e-08 | 3.861e-11 |

1.5 m | 2.830e-02 | 2.059e-05 | 5.484e-09 | 3.104e-12 | 5.669e-16 |

2 m | 1.722e-02 | 1.127e-06 | 1.930e-11 | 9.036e-16 | 9.364e-21 |

3 m | 7.550e-03 | 3.997e-09 | 2.834e-16 | 9.079e-23 | 3.029e-30 |

#### 2.3. Revisiting the Sound Pressure Radiation Formulation

#### 2.4. Comparative Tools

#### 2.4.1. Modal Assurance Criterion

#### 2.4.2. Frequency Response Assurance Criterion

#### 2.4.3. Signal-to-Noise Ratio

## 3. Results

#### 3.1. Sound Pressure Mapping from Full-Field Receptances

#### 3.1.1. Meshing the Acoustic Domain

^{®}Linux environment, and on a workstation with 192 GB of RAM, 12 physical cores in dual hexacore Intel

^{®}Xeon

^{®}X5690 CPUs running at 3.46–3.73 GHz) to take advantage of the common positioning vectors ${\mathit{r}}_{\mathit{aq}}$ in the Green’s functions and in Equations (12)–(15), between structural and acoustic domains; the ${\mathit{r}}_{\mathit{aq}}$ may be evaluated, in parallel among the threads, just once and kept allocated afterwards in memory for each frequency line, in the acoustic field approximations made at the same distance. Therefore, in this paper a rectangular mesh of size 500 mm × 500 mm was prepared, with 51 × 51 dofs (${N}_{a}$ = 2601, 10 mm among each dof), centred on the vibrating plate and positioned at four different distances [25 mm, 75 mm, 150 mm, 300 mm] above it as displayed in Figure 2; this grid size was chosen to speed up the calculations with the parallel computing routines as mentioned above, while keeping a very high meshing resolution. The parameters of the medium (air) were fixed in ${c}_{0}$ = 300.0 m/s and ${\rho}_{0}$ = 1.204 kg/m${}^{3}$.

**Figure 2.**Relative position of the radiating surface (in blue tones) and the acoustic domain of interest (in grey tones), at specific distances of 25 mm in ($\mathit{a}$), of 75 mm in ($\mathit{b}$), of 150 mm in ($\mathit{c}$) and of 300 mm in ($\mathit{d}$), excitation from shaker 2 (in the yellow dot) at 20 Hz.

**Figure 3.**Vibro-acoustic FRFs in acoustic dof 941, obtained in Equations (14) and (18) from raw (red) and smoothed (blue) receptances respectively, in the [20–1024] Hz frequency range at specific distances of 25 mm ($\mathit{a}$), of 75 mm ($\mathit{b}$), of 150 mm ($\mathit{c}$) and of 300 mm ($\mathit{d}$) from shaker 2 in structural dof 931.

#### 3.1.2. Notes about the Evaluation of the Acoustic Transfer Matrix

**Figure 4.**FRAC mapping between raw acoustic transfer matrix in Equation (14) and smoothed acoustic transfer matrix in Equation (18) at specific distances of 25 mm in ($\mathit{a}$), of 75 mm in ($\mathit{b}$), of 150 mm in ($\mathit{c}$) and of 300 mm in ($\mathit{d}$) from shaker 2, with the statistics in Table 2.

Distance | Max | Min | Mean | Std.Dev. | snr |
---|---|---|---|---|---|

25 mm | 0.998 | 0.962 | 0.992 | 0.005 | 216.338 |

75 mm | 0.998 | 0.981 | 0.992 | 0.004 | 272.550 |

150 mm | 0.998 | 0.986 | 0.992 | 0.004 | 276.596 |

300 mm | 0.995 | 0.989 | 0.991 | 0.002 | 545.552 |

#### 3.1.3. Revealing Error Effects by Means of the FRAC Matrices

#### 3.1.4. Revealing Error Effects by Means of the MACii Functions

Distance | Max | Min | Mean | Std.Dev. | snr |
---|---|---|---|---|---|

25 mm | 1.000 | 0.680 | 0.978 | 0.056 | 17.329 |

75 mm | 1.000 | 0.402 | 0.976 | 0.077 | 12.647 |

150 mm | 1.000 | 0.389 | 0.986 | 0.055 | 18.064 |

300 mm | 1.000 | 0.605 | 0.993 | 0.032 | 31.116 |

Distance | 127 Hz | 250 Hz | 284 Hz | 336 Hz | 755 Hz | 991 Hz |
---|---|---|---|---|---|---|

25 mm | 1.000 | 1.000 | 0.988 | 0.962 | 0.796 | 0.703 |

75 mm | 1.000 | 1.000 | 0.994 | 0.994 | 0.686 | 0.411 |

150 mm | 1.000 | 1.000 | 0.998 | 0.999 | 0.617 | 0.459 |

300 mm | 1.000 | 1.000 | 0.999 | 1.000 | 0.801 | 0.605 |

**Figure 6.**MACii graphs in the comparison of the acoustic transfer matrix as in Equations (14) and (18), at specific distances of 25 mm in ($\mathit{a}$), 75 mm in ($\mathit{b}$), 150 mm in ($\mathit{c}$), and 300 mm in ($\mathit{d}$) at the specific frequency of 127 Hz, with excitation from shaker 2.

**Figure 9.**Error patterns of Equation (20) in vibro-acoustic FRF mapping from experiment-based receptances at 127 Hz at specific distances of 25 mm in ($\mathit{a}$), of 75 mm in ($\mathit{b}$), of 150 mm in ($\mathit{c}$) and of 300 mm in ($\mathit{d}$) from shaker 2.

#### 3.2. Uniformly Distributed Amplitude-Modulated Noise Effects

Distance | Raw Max | Raw Min | Raw Ratio | Smooth Max | Smooth Min | Smooth Ratio | Err. Max | Err. Min | Err. Ratio |
---|---|---|---|---|---|---|---|---|---|

25 mm | 9.517e-01 | 7.053e-02 | 13.49 | 9.480e-01 | 6.960e-02 | 13.62 | 9.321e-03 | 7.698e-04 | 12.11 |

75 mm | 3.997e-01 | 6.951e-02 | 5.75 | 3.978e-01 | 6.860e-02 | 5.80 | 3.875e-03 | 7.611e-04 | 5.09 |

150 mm | 1.933e-01 | 6.636e-02 | 2.91 | 1.918e-01 | 6.551e-02 | 2.93 | 2.068e-03 | 7.182e-04 | 2.88 |

300 mm | 9.580e-02 | 5.705e-02 | 1.68 | 9.487e-02 | 5.636e-02 | 1.68 | 1.062e-03 | 6.298e-04 | 1.69 |

**Table 6.**Value of the complex amplitude [1/m${}^{2}$] of the vibro-acoustic FRF field in acoustic dof 941 from the raw receptances in Equation (14), from the smoothed receptances in Equation (18), from the error function in Equation (20), and the ratio between the same error and the raw-based values at the specific frequency of 127 Hz.

Distance | Raw Value | Smoothed Value | Error Value | Ratio Error/Raw |
---|---|---|---|---|

25 mm | 5.655e-01 | 5.621e-01 | 3.317e-03 | 5.866e-03 |

75 mm | 2.727e-01 | 2.705e-01 | 2.132e-03 | 7.818e-03 |

150 mm | 1.612e-01 | 1.597e-01 | 1.573e-03 | 9.758e-03 |

300 mm | 9.084e-02 | 8.991e-02 | 9.699e-04 | 1.068e-02 |

**Figure 10.**Examples of receptance shapes with their experiment-related noise (in ($\mathit{a}$), Equation (1)) against the smoothed version (in ($\mathit{b}$), Equation (2)) and related error pattern (in ($\mathit{c}$), Equation (3)) at the specific frequency of 250 Hz, with excitation from shaker 2.

**Figure 11.**MACii graphs in the comparison of the acoustic transfer matrix as in Equations (14) and (18), at specific distances of 25 mm in ($\mathit{a}$), 75 mm in ($\mathit{b}$), 150 mm in ($\mathit{c}$), and 300 mm in ($\mathit{d}$) at the specific frequency of 250 Hz, with excitation from shaker 2.

**Figure 14.**Error patterns of Equation (20) in vibro-acoustic FRF mapping from experiment-based receptances at 250 Hz at specific distances of 25 mm in ($\mathit{a}$), of 75 mm in ($\mathit{b}$), of 150 mm in ($\mathit{c}$) and of 300 mm in ($\mathit{d}$) from shaker 2.

Distance | Raw Max | Raw Min | Raw Ratio | Smooth Max | Smooth Min | Smooth Ratio | Err. Max | Err. Min | Err. Ratio |
---|---|---|---|---|---|---|---|---|---|

25 mm | 1.275e+00 | 1.236e-02 | 103.16 | 1.288e+00 | 1.465e-02 | 87.92 | 2.453e-02 | 1.685e-04 | 145.58 |

75 mm | 5.322e-01 | 1.920e-02 | 27.72 | 5.409e-01 | 2.825e-02 | 19.15 | 1.244e-02 | 2.066e-03 | 6.02 |

150 mm | 2.860e-01 | 1.071e-01 | 2.67 | 2.916e-01 | 1.099e-01 | 2.65 | 6.965e-03 | 2.060e-03 | 3.38 |

300 mm | 1.648e-01 | 9.755e-02 | 1.69 | 1.678e-01 | 9.978e-02 | 1.68 | 3.498e-03 | 1.905e-03 | 1.84 |

**Table 8.**Value of the complex amplitude [1/m${}^{2}$] of the vibro-acoustic FRF field in acoustic dof 941 from the raw receptances in Equation (14), from the smoothed receptances in Equation (18), from the error function in Equation (20), and the ratio between the same error and the raw-based values at the specific frequency of 250 Hz.

Distance | Raw Value | Smoothed Value | Error Value | Ratio Error/Raw |
---|---|---|---|---|

25 mm | 7.384e-01 | 7.411e-01 | 3.142e-03 | 4.255e-03 |

75 mm | 3.909e-01 | 3.964e-01 | 5.735e-03 | 1.467e-02 |

150 mm | 2.562e-01 | 2.608e-01 | 4.866e-03 | 1.899e-02 |

300 mm | 1.595e-01 | 1.624e-01 | 3.083e-03 | 1.933e-02 |

#### 3.3. Measurement-Related Noise Effects

**Figure 15.**Examples of receptance shapes with their experiment-related noise (in ($\mathit{a}$), Equation (1)) against the smoothed version (in ($\mathit{b}$), Equation (2)) and related error pattern (in ($\mathit{c}$), Equation (3)) at the specific frequency of 284 Hz, with excitation from shaker 2.

**Figure 16.**MACii graphs in the comparison of the acoustic transfer matrix as in Equations (14) and (18), at specific distances of 25 mm in ($\mathit{a}$), 75 mm in ($\mathit{b}$), 150 mm in ($\mathit{c}$), and 300 mm in ($\mathit{d}$) at the specific frequency of 284 Hz, with excitation from shaker 2.

**Figure 19.**Error patterns of Equation (20) in vibro-acoustic FRF mapping from experiment-based receptances at 284 Hz at specific distances of 25 mm in ($\mathit{a}$), of 75 mm in ($\mathit{b}$), of 150 mm in ($\mathit{c}$) and of 300 mm in ($\mathit{d}$) from shaker 2.

Distance | Raw Max | Raw Min | Raw Ratio | Smooth Max | Smooth Min | Smooth Ratio | Err. Max | Err. Min | Err. Ratio |
---|---|---|---|---|---|---|---|---|---|

25 mm | 8.410e-01 | 9.317e-02 | 9.03 | 8.347e-01 | 1.051e-01 | 7.94 | 1.883e-01 | 1.859e-03 | 101.29 |

75 mm | 4.320e-01 | 9.183e-02 | 4.70 | 4.346e-01 | 1.031e-01 | 4.22 | 5.740e-02 | 1.865e-03 | 30.78 |

150 mm | 2.347e-01 | 8.759e-02 | 2.68 | 2.402e-01 | 9.710e-02 | 2.47 | 2.143e-02 | 1.881e-03 | 11.39 |

300 mm | 1.198e-01 | 7.455e-02 | 1.61 | 1.240e-01 | 8.056e-02 | 1.54 | 7.968e-03 | 1.871e-03 | 4.26 |

**Table 10.**Value of the complex amplitude [1/m${}^{2}$] of the vibro-acoustic FRF field in acoustic dof 941 from the raw receptances in Equation (14), from the smoothed receptances in Equation (18), from the error function in Equation (20), and the ratio between the same error and the raw-based values at the specific frequency of 284 Hz.

Distance | Raw Value | Smoothed Value | Error Value | Ratio Error/Raw |
---|---|---|---|---|

25 mm | 4.901e-01 | 5.235e-01 | 3.907e-02 | 7.972e-02 |

75 mm | 3.021e-01 | 3.287e-01 | 2.835e-02 | 9.384e-02 |

150 mm | 1.978e-01 | 2.124e-01 | 1.550e-02 | 7.836e-02 |

300 mm | 1.133e-01 | 1.193e-01 | 6.680e-03 | 5.896e-02 |

**Figure 20.**Examples of receptance shapes with their experiment-related noise (in ($\mathit{a}$), Equation (1)) against the smoothed version (in ($\mathit{b}$), Equation (2)) and related error pattern (in ($\mathit{c}$), Equation (3)) at the specific frequency of 336 Hz, with excitation from shaker 2.

#### 3.4. Filtering-Related Noise Effects

**Figure 21.**MACii graphs in the comparison of the acoustic transfer matrix as in Equations (14) and (18), at specific distances of 25 mm in ($\mathit{a}$), 75 mm in ($\mathit{b}$), 150 mm in ($\mathit{c}$), and 300 mm in ($\mathit{d}$) at the specific frequency of 336 Hz, with excitation from shaker 2.

**Figure 24.**Error patterns of Equation (20) in vibro-acoustic FRF mapping from experiment-based receptances at 336 Hz at specific distances of 25 mm in ($\mathit{a}$), of 75 mm in ($\mathit{b}$), of 150 mm in ($\mathit{c}$) and of 300 mm in ($\mathit{d}$) from shaker 2.

Distance | Raw Max | Raw Min | Raw Ratio | Smooth Max | Smooth Min | Smooth Ratio | Err. Max | Err. Min | Err. Ratio |
---|---|---|---|---|---|---|---|---|---|

25 mm | 4.238e-01 | 5.052e-02 | 8.39 | 5.092e-01 | 7.297e-02 | 6.98 | 2.685e-01 | 3.063e-02 | 8.77 |

75 mm | 1.819e-01 | 4.867e-02 | 3.74 | 2.295e-01 | 7.167e-02 | 3.20 | 1.254e-01 | 3.019e-02 | 4.15 |

150 mm | 1.020e-01 | 4.580e-02 | 2.23 | 1.427e-01 | 6.707e-02 | 2.13 | 7.198e-02 | 2.842e-02 | 2.53 |

300 mm | 5.805e-02 | 3.835e-02 | 1.51 | 8.404e-02 | 5.557e-02 | 1.51 | 3.852e-02 | 2.403e-02 | 1.60 |

**Table 12.**Value of the complex amplitude [1/m${}^{2}$] of the vibro-acoustic FRF field in acoustic dof 941 from the raw receptances in Equation (14), from the smoothed receptances in Equation (18), from the error function in Equation (20), and the ratio between the same error and the raw-based values at the specific frequency of 336 Hz.

Distance | Raw Value | Smoothed Value | Error Value | Ratio Error/Raw |
---|---|---|---|---|

25 mm | 1.166e-01 | 1.471e-01 | 8.006e-02 | 6.866e-01 |

75 mm | 1.153e-01 | 1.520e-01 | 8.142e-02 | 7.062e-01 |

150 mm | 8.934e-02 | 1.251e-01 | 6.160e-02 | 6.895e-01 |

300 mm | 5.584e-02 | 8.050e-02 | 3.679e-02 | 6.588e-01 |

**Figure 25.**Examples of receptance shapes with their experiment-related noise (in ($\mathit{a}$), Equation (1)) against the smoothed version (in ($\mathit{b}$), Equation (2)) and related error pattern (in ($\mathit{c}$), Equation (3)) at the specific frequency of 755 Hz, with excitation from shaker 2.

**Figure 26.**MACii graphs in the comparison of the acoustic transfer matrix as in Equations (14) and (18), at specific distances of 25 mm in ($\mathit{a}$), 75 mm in ($\mathit{b}$), 150 mm in ($\mathit{c}$), and 300 mm in ($\mathit{d}$) at the specific frequency of 755 Hz, with excitation from shaker 2.

**Figure 29.**Error patterns of Equation (20) in vibro-acoustic FRF mapping from experiment-based receptances at 755 Hz at specific distances of 25 mm in ($\mathit{a}$), of 75 mm in ($\mathit{b}$), of 150 mm in ($\mathit{c}$) and of 300 mm in ($\mathit{d}$) from shaker 2.

Distance | Raw Max | Raw Min | Raw Ratio | Smooth Max | Smooth Min | Smooth Ratio | Err. Max | Err. Min | Err. Ratio |
---|---|---|---|---|---|---|---|---|---|

25 mm | 9.757e-01 | 7.596e-03 | 128.45 | 1.004e+00 | 5.132e-03 | 195.64 | 5.358e-01 | 4.626e-02 | 11.58 |

75 mm | 2.388e-01 | 1.760e-02 | 13.57 | 3.299e-01 | 6.095e-03 | 54.13 | 2.563e-01 | 4.601e-02 | 5.57 |

150 mm | 8.683e-02 | 1.735e-02 | 5.00 | 1.589e-01 | 7.479e-03 | 21.25 | 1.410e-01 | 4.498e-02 | 3.13 |

300 mm | 3.343e-02 | 1.577e-02 | 2.12 | 7.036-02 | 1.333e-02 | 5.28 | 7.346e-02 | 4.045e-02 | 1.82 |

**Table 14.**Value of the complex amplitude [1/m${}^{2}$] of the vibro-acoustic FRF field in acoustic dof 941 from the raw receptances in Equation (14), from the smoothed receptances in Equation (18), from the error function in Equation (20), and the ratio between the same error and the raw-based values at the specific frequency of 755 Hz.

Distance | Raw Value | Smoothed Value | Error Value | Ratio Error/Raw |
---|---|---|---|---|

25 mm | 8.263e-01 | 3.508e-01 | 5.241e-01 | 6.343e-01 |

75 mm | 1.272e-01 | 1.310e-01 | 2.528e-01 | 1.987e+00 |

150 mm | 5.447e-02 | 1.199e-01 | 1.400e-01 | 2.559e+00 |

300 mm | 2.737e-02 | 6.187e-02 | 7.318e-02 | 2.674e+00 |

## 4. Discussion

**Figure 30.**Examples of receptance shapes with their experiment-related noise (in ($\mathit{a}$), Equation (1)) against the smoothed version (in ($\mathit{b}$), Equation (2)) and related error pattern (in ($\mathit{c}$), Equation (3)) at the specific frequency of 991 Hz, with excitation from shaker 2.

**Figure 31.**MACii graphs in the comparison of the acoustic transfer matrix as in Equations (14) and (18), at specific distances of 25 mm in ($\mathit{a}$), 75 mm in ($\mathit{b}$), 150 mm in ($\mathit{c}$), and 300 mm in ($\mathit{d}$) at the specific frequency of 991 Hz, with excitation from shaker 2.

## 5. Conclusions

**Figure 34.**Error patterns of Equation (20) in vibro-acoustic FRF mapping from experiment-based receptances at 991 Hz at specific distances of 25 mm in ($\mathit{a}$), of 75 mm in ($\mathit{b}$), of 150 mm in ($\mathit{c}$) and of 300 mm in ($\mathit{d}$) from shaker 2.

Distance | Raw Max | Raw Min | Raw Ratio | Smooth Max | Smooth Min | Smooth Ratio | Err. Max | Err. Min | Err. Ratio |
---|---|---|---|---|---|---|---|---|---|

25 mm | 7.158e-01 | 3.704e-04 | 1932.51 | 4.729e-01 | 1.173e-02 | 40.32 | 2.787e-01 | 1.532e-03 | 181.91 |

75 mm | 1.524e-01 | 1.301e-04 | 1171.41 | 1.250e-01 | 1.201e-02 | 10.41 | 1.120e-01 | 2.095e-02 | 5.35 |

150 mm | 3.824e-02 | 1.430e-04 | 267.41 | 5.835e-02 | 1.285e-02 | 4.54 | 5.802e-02 | 2.033e-02 | 2.85 |

300 mm | 9.542e-03 | 5.899e-04 | 16.18 | 2.894e-02 | 1.310e-02 | 2.21 | 2.945e-02 | 1.780e-02 | 1.65 |

**Table 16.**Value of the complex amplitude [1/m${}^{2}$] of the vibro-acoustic FRF field in acoustic dof 941 from the raw receptances in Equation (14), from the smoothed receptances in Equation (18), from the error function in Equation (20), and the ratio between the same error and the raw-based values at the specific frequency of 991 Hz.

Distance | Raw Value | Smoothed Value | Error Value | Ratio Error/Raw |
---|---|---|---|---|

25 mm | 4.720e-01 | 2.092e-01 | 2.642e-01 | 5.597e-01 |

75 mm | 1.074e-01 | 1.927e-02 | 1.065e-01 | 9.916e-01 |

150 mm | 2.688e-02 | 3.140e-02 | 5.517e-02 | 2.052e+00 |

300 mm | 6.004e-03 | 2.420e-02 | 2.875e-02 | 4.788e+00 |

**Table 17.**Ratio between the evaluated error against the raw function value in acoustic dof 941, at specific frequencies of the comparison between vibro-acoustic maps, obtained from raw and smoothed optical full-field receptances, as in the last column of Table 6, Table 8, Table 10, Table 12, Table 14 and Table 16.

Distance | 127 Hz | 250 Hz | 284 Hz | 336 Hz | 755 Hz | 991 Hz |
---|---|---|---|---|---|---|

25 mm | 5.866e-03 | 4.255e-03 | 7.972e-02 | 6.866e-01 | 6.343e-01 | 5.597e-01 |

75 mm | 7.818e-03 | 1.467e-02 | 9.384e-02 | 7.062e-01 | 1.987e+00 | 9.916e-01 |

150 mm | 9.758e-03 | 1.899e-02 | 5.896e-02 | 6.895e-01 | 2.559e+00 | 2.052e+00 |

300 mm | 1.068e-02 | 1.933e-02 | 5.896e-02 | 6.588e-01 | 2.674e+00 | 4.788e+00 |

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DIC | Digital image correlation |

dof | Degree of freedom |

EFFMA | Experimental full-field modal analysis |

EMA | Experimental modal analysis |

ESPI | Electronic speckle pattern interferometry |

FRAC | Frequency response assurance criterion |

FRF | Frequency response function |

MAC | Modal assurance criterion |

NDT | Non-destructive testing |

NVH | Noise and vibration harshness |

ODS | Operative deflection shape |

SLDV | Scanning laser Doppler vibrometer |

$\left(\omega \right)$ | Frequency dependency |

$\mathbf{X}\left(\omega \right)$ | Displacement map |

$\mathbf{F}\left(\omega \right)$ | Excitation force |

${\mathbf{H}}_{\mathbf{d}}\left(\omega \right)$ | Receptance map |

men | $m\times {10}^{n}$ in C-language scientific/engineering notation, with m mantissa and n exponent |

## References

- Kirkup, S. Computational solution of the acoustic field surrounding a baffled panel by the Rayleigh integral method. Appl. Math. Model.
**1994**, 18, 403–407. [Google Scholar] [CrossRef] - Gérard, F.; Tournour, M.; Masri, N.; Cremers, L.; Felice, M.; Selmane, A. Acoustic transfer vectors for numerical modeling of engine noise. Sound Vib.
**2002**, 36, 20–25. [Google Scholar] - Citarella, R.; Federico, L.; Cicatiello, A. Modal acoustic transfer vector approach in a FEM–BEM vibro-acoustic analysis. Eng. Anal. Bound. Elem.
**2007**, 31, 248–258. [Google Scholar] [CrossRef] - Desmet, W. Boundary Element Method in Acoustics; Technical Report; Noise & Vibration Research Group, Mechanical Engineering Department, Katholieke Universiteit Leuven: Leuven, Belgium, 2004; In ISAAC 15, Course on Numerical and Applied Acoustics, Katholieke Universiteit Leuven, Belgium, Mechanical Engineering Department, Noise & Vibration Research Group; Available online: https://www.isma-isaac.be (accessed on 23 April 2023).
- Fahy, F. Foundations of Engineering Acoustics; Academic Press: London, UK, 2003; pp. 1–443. [Google Scholar] [CrossRef]
- Kirkup, S.; Thompson, A. Computing the Acoustic Field of a Radiating Cavity by the Boundary Element—Rayleigh Integral Method (BERIM). In Proceedings of the World Congress on Engineering, WCE 2007, London, UK, 2–4 July 2007; Ao, S.I., Gelman, L., Hukins, D.W.L., Hunter, A., Korsunsky, A.M., Eds.; Lecture Notes in Engineering and Computer Science. Newswood Limited: Hong Kong, 2007; Volume II, pp. 1401–1406. Available online: http://www.kirkup.info/papers/SKAT07.pdf (accessed on 23 April 2023).
- Arenas, J.P. Numerical computation of the sound radiation from a planar baffled vibrating surface. J. Comput. Acoust.
**2008**, 16, 321–341. [Google Scholar] [CrossRef] - Kirkup, S. The Boundary Element Method in Acoustics: A Survey. Appl. Sci.
**2019**, 9, 1642. [Google Scholar] [CrossRef] - Tenenbaum, E.; Bucher, I. Reconstruction of a nonlinear acoustic field based on acousto-optic effect computational tomography. In Proceedings of the ISMA2022 including USD2022—International Conference on Noise and Vibration Engineering, Leuven, Belgium, 12–14 September 2022; KU Leuven: Leuven, Belgium, 2022. Paper ID 201 in Vol. Optical Methods. pp. 2750–2762. Available online: https://past.isma-isaac.be/downloads/isma2022/proceedings/Contribution_201_proceeding_3.pdf (accessed on 23 April 2023).
- Gardonio, P.; Guernieri, G.; Turco, E.; Rinaldo, R.; Fusiello, A. Reconstruction of the sound radiation field from plate flexural vibration measurements taken with multiple cameras. In Proceedings of the ISMA2022 including USD2022—International Conference on Noise and Vibration Engineering, Leuven, Belgium, 12–14 September 2022; KU Leuven: Leuven, Belgium, 2022. Paper ID 196 in Vol. Optical Methods. pp. 2826–2837. Available online: https://past.isma-isaac.be/downloads/isma2022/proceedings/Contribution_196_proceeding_3.pdf (accessed on 23 April 2023).
- Zanarini, A. On the estimation of frequency response functions, dynamic rotational degrees of freedom and strain maps from different full field optical techniques. In Proceedings of the ISMA2014 including USD2014—International Conference on Noise and Vibration Engineering, Leuven, Belgium, 15–17 September 2014; KU Leuven: Leuven, Belgium, 2014; pp. 1155–1169, In Vol. Dynamic Testing: Methods and Instrumentation, Paper ID 676. Available online: https://past.isma-isaac.be/downloads/isma2014/papers/isma2014_0676.pdf (accessed on 23 April 2023).
- Zanarini, A. Comparative studies on full field FRFs estimation from competing optical instruments. In Proceedings of the ICoEV2015 International Conference on Engineering Vibration, Ljubljana, Slovenia, 7–10 September 2015; University Ljubljana & IFToMM: Ljubljana, Slovenia, 2015; pp. 1559–1568, Paper ID 191. Available online: https://www.researchgate.net/publication/280013709_Comparative_studies_on_Full_Field_FRFs_estimation_from_competing_optical_instruments (accessed on 23 April 2023).
- Zanarini, A. Broad frequency band full field measurements for advanced applications: Point-wise comparisons between optical technologies. Mech. Syst. Signal Process.
**2018**, 98, 968–999. [Google Scholar] [CrossRef] - Zanarini, A. Competing optical instruments for the estimation of Full Field FRFs. Measurement
**2019**, 140, 100–119. [Google Scholar] [CrossRef] - Van der Auweraer, H.; Steinbichler, H.; Haberstok, C.; Freymann, R.; Storer, D. Integration of pulsed-laser ESPI with spatial domain modal analysis: Results from the SALOME project. In Proceedings of the 4th International Conference on Vibration Measurements by Laser Techniques: Advances and Applications, Ancona, Italy, 21–23 June 2000; Volume 4072, pp. 313–322. [Google Scholar] [CrossRef]
- Zanarini, A. On the exploitation of multiple 3D full-field pulsed ESPI measurements in damage location assessment. Procedia Struct. Integr.
**2022**, 37, 517–524. [Google Scholar] [CrossRef] - Zanarini, A. On the defect tolerance by fatigue spectral methods based on full-field dynamic testing. Procedia Struct. Integr.
**2022**, 37, 525–532. [Google Scholar] [CrossRef] - Zanarini, A. On the approximation of sound radiation by means of experiment-based optical full-field receptances. In Proceedings of the ISMA2022 including USD2022—International Conference on Noise and Vibration Engineering, Leuven, Belgium, 12–14 September 2022; KU Leuven: Leuven, Belgium, 2022. Paper ID 207 in Vol. Optical Methods. pp. 2735–2749. Available online: https://past.isma-isaac.be/downloads/isma2022/proceedings/Contribution_207_proceeding_3.pdf (accessed on 23 April 2023).
- Zanarini, A. About the excitation dependency of risk tolerance mapping in dynamically loaded structures. In Proceedings of the ISMA2022 including USD2022—International Conference on Noise and Vibration Engineering, Leuven, Belgium, 12–14 September 2022; KU Leuven: Leuven, Belgium, 2022. Paper ID 208 in Vol. Structural Health Monitoring. pp. 3804–3818. Available online: https://past.isma-isaac.be/downloads/isma2022/proceedings/Contribution_208_proceeding_3.pdf (accessed on 23 April 2023).
- Zanarini, A. Introducing the concept of defect tolerance by fatigue spectral methods based on full-field frequency response function testing and dynamic excitation signature. Int. J. Fatigue
**2022**, 165, 107184. [Google Scholar] [CrossRef] - Zanarini, A. Risk Tolerance Mapping in Dynamically Loaded Structures as Excitation Dependency by Means of Full-Field Receptances. In Computer Vision & Laser Vibrometry; Baqersad, J., Di Maio, D., Eds.; Conference Proceedings of the Society for Experimental Mechanics Series; Springer Nature Switzerland AG: Cham, Switzerland, 2023; Volume 6, Chapter 9; p. 14648. [Google Scholar] [CrossRef]
- Zanarini, A. Experiment-based Optical Full-field receptances in the Approximation of Sound Radiation from a Vibrating Plate. In Computer Vision & Laser Vibrometry; Baqersad, J., Di Maio, D., Eds.; Conference Proceedings of the Society for Experimental Mechanics Series; Springer Nature Switzerland AG: Cham, Switzerland, 2023; Volume 6, Chapter 4; p. 14650. [Google Scholar] [CrossRef]
- Zanarini, A. Dynamic behaviour characterization of a brake disc by means of electronic speckle pattern interferometry measurements. In Proceedings of the IDETC/CIE ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Long Beach, CA, USA, 24–28 September 2005; pp. 273–280. [Google Scholar] [CrossRef]
- Zanarini, A. Damage location assessment in a composite panel by means of electronic speckle pattern interferometry measurements. In Proceedings of the IDETC/CIE ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Long Beach, CA, USA, 24–28 September 2005; pp. 1–8. [Google Scholar] [CrossRef]
- Zanarini, A. Full field ESPI measurements on a plate: Challenging experimental modal analysis. In Proceedings of the XXV IMAC, Orlando, FL, USA, 19–22 February 2007; pp. 1–11, Paper s34p04. Available online: https://www.researchgate.net/publication/266896551_Full_field_ESPI_measurements_on_a_plate_Challenging_Experimental_Modal_Analysis (accessed on 23 April 2023).
- Zanarini, A. Fatigue life assessment by means of full field ESPI vibration measurements. In Proceedings of the ISMA2008 Conference, Leuven Belgium, 15–17 September 2008; pp. 817–832. [Google Scholar] [CrossRef]
- Zanarini, A. Full field ESPI vibration measurements to predict fatigue behaviour. In Proceedings of the IMECE2008 ASME International Mechanical Engineering Congress and Exposition, Boston, MA, USA, 31 October–6 November 2008; pp. 165–174. [Google Scholar] [CrossRef]
- Zanarini, A. On the role of spatial resolution in advanced vibration measurements for operational modal analysis and model updating. In Proceedings of the ISMA2014 including USD2014—International Conference on Noise and Vibration Engineering, Leuven, Belgium, 15–17 September 2014; KU Leuven: Leuven, Belgium, 2014; pp. 3323–3336. Available online: https://past.isma-isaac.be/downloads/isma2014/papers/isma2014_0678.pdf (accessed on 23 April 2023).
- Zanarini, A. Accurate FRFs estimation of derivative quantities from different full field measuring technologies. In Proceedings of the ICoEV2015 International Conference on Engineering Vibration, Ljubljana, Slovenia, 7–10 September 2015; University Ljubljana & IFToMM: Ljubljana, Slovenia, 2015; pp. 1569–1578, Paper ID 192. Available online: https://www.researchgate.net/publication/280013778_Accurate_FRF_estimation_of_derivative_quantities_from_different_full_field_measuring_technologies (accessed on 23 April 2023).
- Zanarini, A. Full field experimental modelling in spectral approaches to fatigue predictions. In Proceedings of the ICoEV2015 International Conference on Engineering Vibration, Ljubljana, Slovenia, 7–10 September 2015; University Ljubljana & IFToMM: Ljubljana, Slovenia, 2015; pp. 1579–1588, Paper ID 193. Available online: https://www.researchgate.net/publication/280013788_Full_field_experimental_modelling_in_spectral_approaches_to_fatigue_predictions (accessed on 23 April 2023).
- Zanarini, A. Model updating from full field optical experimental datasets. In Proceedings of the ICoEV2015 International Conference on Engineering Vibration, Ljubljana, Slovenia, 7–10 September 2015; University Ljubljana & IFToMM: Ljubljana, Slovenia, 2015; pp. 773–782, Paper ID 193. Available online: https://www.researchgate.net/publication/280013876_Model_updating_from_full_field_optical_experimental_datasets (accessed on 23 April 2023).
- Zanarini, A. Full field optical measurements in experimental modal analysis and model updating. J. Sound Vib.
**2019**, 442, 817–842. [Google Scholar] [CrossRef] - Zanarini, A. On the making of precise comparisons with optical full field technologies in NVH. In Proceedings of the ISMA2020 including USD2020—International Conference on Noise and Vibration Engineering, Leuven, Belgium, 7–9 September 2020; KU Leuven: Leuven, Belgium, 2020; pp. 2293–2308. Available online: https://past.isma-isaac.be/downloads/isma2020/proceedings/Contribution_695_proceeding_3.pdf (accessed on 23 April 2023).
- Zanarini, A. Chasing the high-resolution mapping of rotational and strain FRFs as receptance processing from different full-field optical measuring technologies. Mech. Syst. Signal Process.
**2022**, 166, 108428. [Google Scholar] [CrossRef] - Avitabile, P.; O’Callahan, J.; Chou, C.; Kalkunte, V. Expansion of rotational degrees of freedom for structural dynamic modification. In Proceedings of the 5th International Modal Analysis Conference, London, UK, 6–9 April 1987; pp. 950–955. [Google Scholar]
- Heylen, W.; Lammens, S.; Sas, P. Modal Analysis Theory and Testing, 2nd ed.; Katholieke Universiteit Leuven: Leuven, Belgium, 1998; ISBN 90-73802-61-X. [Google Scholar]
- Liu, W.; Ewins, D. The Importance Assessment of RDOF in FRF Coupling Analysis. In Proceedings of the IMAC 17th Conference, Kissimmee, FL, USA, 8–11 February 1999; pp. 1481–1487. [Google Scholar]
- Research Network QUATTRO. Final Report on RDOFs in QUATTRO Brite-Euram Project no: BE 97-4184; Technical Report; European Commission Research Framework Programs; European Commission: Bruxelles, Belgium, 1998; p. 83. [Google Scholar]
- Ewins, D.J. Modal Testing—Theory, Practice and Application, 2nd ed.; Research Studies Press Ltd.: Hertfordshire, UK, 2000; p. 400. [Google Scholar]
- Friswell, M.; Mottershead, J.E. Finite Element Model Updating in Structural Dynamics; Solid Mechanics and Its Applications, Kuwler Academic Publishers; Springer: Cham, The Netherlands, 1995; p. 292. [Google Scholar] [CrossRef]
- Haeussler, M.; Klaassen, S.; Rixen, D. Experimental twelve degree of freedom rubber isolator models for use in substructuring assemblies. J. Sound Vib.
**2020**, 474, 115253. [Google Scholar] [CrossRef] - Pogacar, M.; Ocepek, D.; Trainotti, F.; Cepon, G.; Boltezar, M. System equivalent model mixing: A modal domain formulation. Mech. Syst. Signal Process.
**2022**, 177, 109239. [Google Scholar] [CrossRef] - Kreis, T. Handbook of Holographic Interferometry; Wiley-VCH: Berlin, Germany, 2004. [Google Scholar] [CrossRef]
- Rastogi, P.K. Optical Measurement Techniques and Applications; Artech House, Inc.: Nordwood, MA, USA, 1997. [Google Scholar]
- Van der Auweraer, H.; Steinbichler, H.; Haberstok, C.; Freymann, R.; Storer, D.; Linet, V. Industrial applications of pulsed-laser ESPI vibration analysis. In Proceedings of the XIX IMAC, Kissimmee, FL, USA, 5–8 February 2001; pp. 490–496. [Google Scholar]
- Craig, R.R. Structural Dynamics: An Introduction to Computer Methods; John Wiley & Sons Inc.: New York, NY, USA, 1981. [Google Scholar]
- O’Callahan, J.; Avitabile, P.; Riemer, R. System Equivalent reduction Expansion Process (SEREP). In Proceedings of the VII International Modal Analysis Conference, Las Vegas, NV, USA, 30 January–2 February 1989; Volume 1, pp. 29–37. [Google Scholar]
- Williams, E.G. Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography; Elsevier Ltd.: Amsterdam, The Netherlands, 1999. [Google Scholar] [CrossRef]
- Wind, J.; Wijnant, Y.; de Boer, A. Fast evaluation of the Rayleigh integral and applications to inverse acoustics. In Proceedings of the ICSV13, The Thirteenth International Congress on Sound and Vibration, Vienna, Austria, 2–6 July 2006; pp. 1–8. [Google Scholar]
- Richards, E.L.; Song, H.C.; Hodgkiss, W.S. Acoustic scattering comparison of Kirchhoff approximation to Rayleigh-Fourier method for sinusoidal surface waves at low grazing angles. J. Acoust. Soc. Am.
**2018**, 144, 1269–1278. [Google Scholar] [CrossRef] - Conta, S.; Santoni, A.; Homb, A. Benchmarking the vibration velocity-based measurement methods to determine the radiated sound power from floor elements under impact excitation. Appl. Acoust.
**2020**, 169, 107457. [Google Scholar] [CrossRef] - Chelliah, K.; Raman, G.; Muehleisen, R.T. An experimental comparison of various methods of nearfield acoustic holography. J. Sound Vib.
**2017**, 403, 21–37. [Google Scholar] [CrossRef] - Bendat, J.S.; Piersol, A.G. Random Data: Analysis and Measurement Procedures, 3rd ed.; John Wiley & Sons Inc.: Hoboken, NJ, USA, 2000. [Google Scholar]
- Wagner, P.; Huesmann, A.P.; van der Seijs, M.V. Application of dynamic substructuring in NVH design of electric drivetrains. In Proceedings of the ISMA2020 including USD2020—International Conference on Noise and Vibration Engineering, Leuven, Belgium, 7–9 September 2020; pp. 3365–3382. Available online: https://past.isma-isaac.be/downloads/isma2020/proceedings/Contribution_369_proceeding_3.pdf (accessed on 23 April 2023).
- Mueller, T.; Haeussler, M.; Sedlmair, S.; Rixen, D.J. Airborne transfer path analysis for an e-compressor. In Proceedings of the ISMA2020 including USD2020—International Conference on Noise and Vibration Engineering, Leuven, Belgium, 7–9 September 2020; pp. 3351–3364. Available online: https://past.isma-isaac.be/downloads/isma2020/proceedings/Contribution_287_proceeding_3.pdf (accessed on 23 April 2023).
- Allen, M.S.; Sracic, M.W. A new method for processing impact excited continuous-scan laser Doppler vibrometer measurements. Mech. Syst. Signal Process.
**2010**, 24, 721–735. [Google Scholar] [CrossRef] - Di Maio, D.; Ewins, D.J. Continuous Scan, a method for performing modal testing using meaningful measurement parameters; Part I. Mech. Syst. Signal Process.
**2011**, 25, 3027–3042. [Google Scholar] [CrossRef] - Baqersad, J.; Poozesh, P.; Niezrecki, C.; Avitabile, P. Photogrammetry and optical methods in structural dynamics—A review. In Mechanical Systems and Signal Processing; Elsevier: Amsterdam, The Netherlands, 2016. [Google Scholar] [CrossRef]
- Del Sal, R.; Dal Bo, L.; Turco, E.; Fusiello, A.; Zanarini, A.; Rinaldo, R.; Gardonio, P. Vibration measurements with multiple cameras. In Proceedings of the ISMA2020 including USD2020—International Conference on Noise and Vibration Engineering, Leuven, Belgium, 7–9 September 2020; pp. 2275–2292. Available online: https://past.isma-isaac.be/downloads/isma2020/proceedings/Contribution_481_proceeding_3.pdf (accessed on 23 April 2023).
- Del Sal, R.; Dal Bo, L.; Turco, E.; Fusiello, A.; Zanarini, A.; Rinaldo, R.; Gardonio, P. Structural vibration measurement with multiple synchronous cameras. Mech. Syst. Signal Process.
**2021**, 157, 107742. [Google Scholar] [CrossRef] - Mas, P.; Sas, P. Acoustic Source Identification Based on Microphone Array Processing; Technical Report; Noise & Vibration Research Group, Mechanical Engineering Department, Katholieke Universiteit Leuven: Leuven, Belgium, 2004; In ISAAC 15, Course on Numerical and Applied Acoustics, Katholieke Universiteit Leuven, Belgium, Mechanical Engineering Department, Noise & Vibration Research Group; Available online: https://www.isma-isaac.be (accessed on 23 April 2023).
- Allemang, R.J. The Modal Assurance Criterion—Twenty Years of Use and Abuse. Sound Vib.
**2003**, 37, 14–23. [Google Scholar] - Maynard, J.D.; Williams, E.G.; Lee, Y. Nearfield acoustic holography: I. Theory of generalized holography and the development of NAH. J. Acoust. Soc. Am.
**1985**, 78, 1395–1413. [Google Scholar] [CrossRef] - Veronesi, W.A.; Maynard, J.D. Nearfield acoustic holography (NAH) II. Holographic reconstruction algorithms and computer implementation. J. Acoust. Soc. Am.
**1987**, 81, 1307–1322. [Google Scholar] [CrossRef] - Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. Numerical Recipes in C: The Art of Scientific Computing, 2nd ed.; Cambridge University Press: Cambridge, UK, 1992. [Google Scholar]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zanarini, A.
On the Influence of Scattered Errors over Full-Field Receptances in the Rayleigh Integral Approximation of Sound Radiation from a Vibrating Plate. *Acoustics* **2023**, *5*, 948-986.
https://doi.org/10.3390/acoustics5040055

**AMA Style**

Zanarini A.
On the Influence of Scattered Errors over Full-Field Receptances in the Rayleigh Integral Approximation of Sound Radiation from a Vibrating Plate. *Acoustics*. 2023; 5(4):948-986.
https://doi.org/10.3390/acoustics5040055

**Chicago/Turabian Style**

Zanarini, Alessandro.
2023. "On the Influence of Scattered Errors over Full-Field Receptances in the Rayleigh Integral Approximation of Sound Radiation from a Vibrating Plate" *Acoustics* 5, no. 4: 948-986.
https://doi.org/10.3390/acoustics5040055