## 3 Vertical Cracks with Small Eddy Current Penetration Depth  
A vertical crack lies perpendicular to the surface. It is assumed to be a barrier to the electrical current, which flows along the crack sides, generating additional heating in this region (see also Figure 1a). The crack also hampers heat diffusion, accumulating heat in the corners (see Figure 1b). For ferro-magnetic materials, the eddy current penetration depth is negligibly small for high-frequency excitation and the results of all three models can be well compared. This comparison was done for several different crack depths and heating pulse durations, where a very good agreement was found. Figure 2 demonstrates the results for the same case already shown in Figure 1. The temperature at the sound surface increases like a square root function, which is similar to a semi-infinite large body (see Equation (3)). The temperature at the crack position increases to a higher value. The analytical model and simu.1 provide almost the same results, and simu.2 yields a slightly lower temperature. This difference becomes stronger if the penetration depth of the eddy current is larger, which will be explained in later sections. Additionally, the temperature increase at the position \(x=d_{th}/2\), i.e., in the distance of half the thermal diffusion length, is also plotted in Figure 2. For a short time, after switching on the heating (\(t<0.025\) s = \(t_{pulse}/4\)), the temperature increases in the same manner to that at the sound surface. As the additional heat flow from the crack reaches this position, the temperature here increases more than that at the sound surface. The temperature distribution at the surface around the crack is shown in Figure 3a. The axis of abscissa shows a normalized distance, allowing for a time- and material-independent comparison. The temperature at the surface is perturbed around the crack up to a distance of \(2d_{th}\), but this distance increases with time. A very reliable detection result can be achieved if not only one single temperature image at the end of the heating pulse is considered, but if the whole time-dependent sequence during heating and cool-down is evaluated to one phase image using Fourier transformation [18]:  
\[F_{\tau}=\int\limits_{0}^{\tau}T(t)\,e^{-i2\pi t/\tau}dt\,,\quad\Phi=\arctan \biggl{(}\frac{\mathrm{Im}(F_{\tau})}{\mathrm{Re}(F_{\tau})}\biggr{)} \tag{7}\]  
where \(\tau=t_{pulse}+t_{colddown}\) and \(t_{colddown}\) is usually taken in the range of \(t_{pulse}/2\) and \(t_{pulse}\). In this study, \(t_{colddown}=t_{pulse}\) was chosen, but also a selection of \(t_{colddown}=t_{pulse}/2\) leads to very similar results to the presented ones. The phase image provides an excellent measure for the rate of the heat flow, where  
Figure 2: Temperature increase at the crack position (\(x=0\)), at the sound surface far away from the defect, and at the position \(x=d_{th}/2\), compared with the three models for the same case as shown in Figure 1.  
it increases more quickly and more slowly. It is much less sensitive to disturbing factors, such as inhomogeneous heating or inhomogeneous emissivity values, and has a much higher signal-to-noise ratio [18]. In Figure 3b, the phase difference at the crack compared to the sound surface is depicted. At the crack position, the phase is higher than that at the sound surface, as the temperature here increases more quickly and decreases more quickly after the heating is switched off. The phase has a minimum around \(x=d_{th}/2\), as at this position, the temperature increases at the beginning with the same rate as at the sound surface, but due to additional heat flowing from the crack to this position, the temperature later increases at a higher rate. After switching off the heating, the temperature at this point remains higher due to the heat flow from the crack position. With increasing time, the additional heat around the crack spreads through the whole sample and becomes evenly distributed. The longer the heating pulse, the broader the heated region is around the crack.