# Pulsed Thermography Applied to the Study of Cultural Heritage
## 2 Method  
The studies reviewed in this paper have been performed by means of a pulsed IRT analysis. This basically consists in quickly heating the sample by the absorption of a light pulse, consisting of radiation in the ultra-violet (UV), visible (VIS), and near-infrared (NIR) range, and analyzing the subsequent temperature decay phase with an infrared camera which records the radiation emitted in the middle-wave infrared region (MWIR) or in the long-wave infrared region (LWIR) [16]. The presence of inhomogeneities of the thermal properties, possibly associated with subsurface defects, alters the local heat diffusion rate and, consequently, the temperature at the corresponding surface point where a thermal profile of the inhomogeneities can therefore be displayed in the infrared image, the so called thermogram, generated by the IR camera. The thermograms are recorded at regular time intervals during the thermal perturbation and the subsequent cooling phase associated with the heat diffusion into the sample volume. Pulsed IRT can be applied in two different modes: the front face heating mode, where the heating and the detection are carried out on the same sample surface, and the rear face heating mode, where they are carried out on opposite sides of the sample [16]. In both modes, the analysis of the time evolution of the temperature distribution at the sample surface also allows researchers to determine the value of the thermal diffusivity (\(D\)) of the constituting material [9; 24; 49; 56]. The determination of \(D\), a physical quantity which accounts for the rate at which a thermal perturbation diffuses throughout a material, is fundamental to perform quantitative studies of the geometrical characteristics of some subsurface features. The quantitative analysis of the thermographic results and, in particular, that of the time dependence of the thermographic signal, requires the processing of experimental results according to a theoretical model. Therefore, by solving the one dimensional heat diffusion equation when the pulsed light absorption occurs at a sample surface (\(x=0\)), one obtains the following expression for the time dependence of the temperature rise at a depth \(x\) in the sample [57; 58]:  
\[\Delta T(x,t)=\frac{Q}{\sqrt{\rho ck}}\frac{\sqrt{D}}{L}\left[1+2\sum_{n=1}^{ \infty}\cos\left(\frac{n\pi x}{L}\right)\exp\left(-\frac{n^{2}\pi^{2}}{L^{2}} Dt\right)\right] \tag{1}\]  
where \(\rho\) is the density, \(c\) is the specific heat, \(k\) is the thermal conductivity, and \(Q\) is the absorbed heat per unit surface. The time dependence in the exponential function is only characterized by the sample thickness \(L\) and the thermal diffusivity \(D\). Therefore, substituting in the expression \(x=0\) or \(x=L\), the solution at the sample front face or rear face heating modes, respectively, is obtained. For the front face heating mode, substituting \(x=0\), yields:  
\[\Delta T(0,t)=\frac{Q}{\sqrt{\rho ck}}\frac{\sqrt{D}}{L}\left[1+2\sum_{n=1}^{ \infty}\exp\left(-\frac{n^{2}\pi^{2}}{L^{2}}Dt\right)\right] \tag{2}\]  
Figure 1 shows the double logarithmic plot of the decay of the surface temperature with the time calculated for different \(L\) values and for \(D=0.19\) cm\({}^{2}\)/s, which is the typical value for bronze [24]. Two different behaviors are clearly shown. The first one, characterized by a \(-1/2\) slope line in the logarithmic plot, is observed at short times when the surface temperature change \(\Delta T(0,t)\) evolves as \(t^{-1/2}\), as expected from the theory of the heat diffusion in a semi-infinite homogeneous material according to which the temperature change can be expressed as [16]:  
\[\Delta T(0,t)_{L=\infty}=\frac{Q}{\sqrt{\rho ck}}\frac{1}{\sqrt{\pi t}} \tag{3}\]  
Deviations from this decay rate occur when the heat reaches the opposite surface \(x=L\) and further diffusion is inhibited.