## 2 Materials and Methods  
The experimental measurements on the vibrating plate were conducted in the Acoustics Laboratory of the Department of Engineering and Architecture at University of Parma, Italy. The setup was composed of a single-point laser doppler vibrometer by Polytec with processing unit OFV-5000 and sensor head OFV-505. The processing unit is equipped with an analog velocity decoder type VD-01, which was set to provide a measurement range of 25 mm/s/V, with a full-scale output peak of 0.25 m/s (at the maximum output voltage of 10 \(V_{peak}\)). In this configuration, the maximum frequency was limited to 50 kHz, and a built-in low-pass filter at 20 kHz was activated. The sensor head was moved by a two-axis handling system with stepping motors (Figure 1a) controlled via laptop computer through an ethernet connection and a dedicated electronic board. The mechanical crosstalk, which is the vibration transmitted to the laser sensor head through the solid path (ground, table, and handling system), was previously evaluated [29], showing a minimum attenuation of \(-\)55 dB below 1 kHz, therefore ensuring a satisfactory structural decoupling between the support and the measurement system. An aluminum plate of 300 mm by 200 mm and a thickness of 0.8 mm was positioned below the laser's sensor head, fixed on a plywood structure (baffle) to separate the front and the back radiation (Figure 1b), as [PERSON]'s integral hypothesis requires [30,31]. A Bruel&Kjaer (B&K) shaker type 4810 was used to excite the plate, and a load cell PCB 208C02 with 11,241 mV/kN sensitivity was screwed on it. The shaker was placed under the plate, rigidly connected to it by a stinger (Figure 2a). With the aim of replicating the experimental setup in the numerical model as closely as possible, the shaker was positioned at the center of the panel. A reference acoustic measurement of the plate was acquired too by positioning a B&K microphone type 4188 with preamplifier type 2186 above the plate, on-axis with respect to its center, at the standard distance of 1 m (Figure 2b). The Impulse Response (IR) was cut in the time domain for excluding the reflections of the environment (virtual anechoic room technique). The B&K microphone was calibrated with a B&K calibration system type 4231, which produces a pure tone at 1 kHz, having an RMS value of 1 Pa (a SPL of 94 dB re 20 uPa). Such measurement was used for comparing with the SPL reconstructed through [PERSON]'s integral. Since the OFV-5000 laser controller was not provided with an analog or digital output for delivering the test signal to the amplifier of the shaker, a ZOOM F8 audio interface was employed to manage all input and output signals synchronously (Figure 3). It was connected to the PC via USB and operating at a sampling frequency of 48 kHz. The test signal was an Exponential Sine Sweep (ESS) [32] of 10 s played through the first output channel. The signal was amplified before feeding the shaker at a voltage of 1 Volt, measured  
Figure 1: (**a**) Two-axis handling system of laser sensor head and (**b**) fully clamped aluminum plate mounted on a rigid baffle. Figure 2: (**a**) B&K shaker type 4810 connected to the plate via a stinger. The PCB load cell is visible at the top of the shaker. (**b**) Acoustic measurement of the plate with a B&K microphone at 1 m on-axis. with a true-RMS meter. While playing the ESS, three input signals were recorded by the soundcard: force from the load cell, sound pressure from a B&K measurement microphone and velocity from the LDV. Considering the target application, the frequency range of interest was comprised between 150 Hz and 1 kHz. Therefore, an ESS from 20 Hz to 2 kHz was employed for accounting a smooth fade-in and fade-out of 0.5 s, respectively. The acquisition system was fully calibrated before performing the measurements, so that the results could be directly compared with the simulations. The rated calibration value of 11.2 mV/N was employed for the load cell.