## 3 Experiments  
A [0\({}^{\circ}\)/90\({}^{\circ}\)]\({}_{12}\) carbon fiber reinforced epoxy composite laminate made using a hot press process with unidirectional prepregs (Toray) is studied here; the experimental set-up is shown in Figure 1a. The plate is 510 mm \(\times\) 510 mm \(\times\) 3 mm in dimension. Since the thickness is small compared to the length and width dimensions, we will treat the plate as a two-dimensional model based on Equation (1). Due to the high attenuation of ultrasonic waves in the composite plate (4.3 dB/cm at 5 MHz [37]), the pitch-catch configuration is chosen and the gain is 17 dB. The high-pass filter is 1 MHz, while the low-pass filter is 10 MHz. The longitudinal transducer (A551S 5 MHz, Olympus, Inc., Tokyo, Japan) is excited using a pulsed source from a signal generator (Shantou Goworld Display Co., Ltd., Shantou, China). The waveform is recorded using a same type longitudinal transducer through an oscilloscope (TBS1202B, Tektronix, Inc., Beaverton, OR, USA) connected to a personal computer. The ultrasonic sensors are fixed to the plate by 3D printed fixtures with springs inside to apply constant forces on the sensors. A high viscosity couplant can provide good wave transmission and a traditional honey is applied. The defect is simulated using a circular piezoelectric medium attached to the composite plate surface [38]. Signals are amplified and averaged over 10 acquisitions for each source-receiver pair to minimize noises such as the rapid changes of capacitances between conductors due to flexing, twisting, or transient impacts on cables. Furthermore, since an important signal variation has been observed with a minor sensor movement, the defect is attached and detached repeatedly while keeping each source (S)-receiver (Rn, n = 1-4) pair fixed during signal recording (Figure 1b). A typical waveform from S-R\({}_{1}\) is presented in Figure 2. No first arrival time variation can be observed using an amplitude threshold picker [39]. The wave diffusivity and the dissipation parameter can be decided from the waveform envelope through the Hilbert transform based on Equation (1) that \(D\) = 100 m\({}^{2}\)/s and \(k\) = 3 \(\times\) 10\({}^{5}\)/s. The dissipation parameter \(k\) is an indication of viscoelasticity, which describes the exponential decay at late times. It is much larger than that of concrete with 5500/s [26]. These values fit well with waveforms in 0\({}^{\circ}\) and 90\({}^{\circ}\) directions. In contrast, the waveforms from off-axis directions (30\({}^{\circ}\) and 60\({}^{\circ}\)) have poor signal-to-noise ratios, as shown in Figure 3. It can be difficult for the diffusion property to be decided in those directions, which makes measurement with an arbitrary sensor location unreliable. Only the measurements along the X and Y axes are chosen here, and a constant diffusion property is assumed during the defect sensitivity kernel calculation under the measurement scheme in Figure 1b. The first arrival time is about 10 \(\upmu\)s and gives a group velocity of 8000 m/s along the X and Y axes. In fact, the directivity of the group velocity [40] does not interfere with the defect localization procedure according to Equation (6). The transport mean free  
Figure 1: (**a**) Experimental set-upand (**b**) measurement scheme. path \(L^{*}=2D/c\approx 0.025\) m is less than the transmitter-receiver pair distance 0.08 m, which ensures the ultrasonic wave is multiple scattered. The wave diffusivity is a characteristic of the microstructure which relates to the arrival time of the maximum energy density and decreases with frequency. A large variation of \(D\) has been found for concrete, from 10 m\({}^{2}\)/s [32] to 70 m\({}^{2}\)/s [26], because of its internal structure variation and frequency sensitivity to the diffuse wave. This can be expected as well for fiber reinforced composites. [PERSON] et al. found that \(D\) varies from 17 to 10 m\({}^{2}\)/s, while the crack size increases from 1 cm to 5.5 cm [32].