## 2 Theory  
### Lock-In Frequency Modulated Thermal Wave Imaging  
Active thermography is a technique for recording a thermal response for a certain period of time by injecting external thermal energy into an object. Heat wave incident on the surface propagate to the material by diffusion. This wave will be reflected by defects placed on the object (surface cracks, subsurface defects, voids, and delamination) [17]. The presence of these defects affects the rate of diffusion of the damaged area to the surrounding area, which obeys the laws of thermodynamics. An IR camera can be used to  
Figure 1: Classification of infrared thermographic testing approaches.  
detect harmonic response patterns radiated on the surface of an object that is interfered by incident and reflected waves.  
When thermal energy is incident on the opaque surface of the object, the object absorbs a portion of the incident energy and reflects a portion. The absorbed energy generates a localized heat flow on the sample surface. Time-dependent heat flow over time is defined by [PERSON]'s one-dimensional thermal diffusion equation by Equation (1) [17; 33]:  
\[\frac{\partial^{2}\mathrm{T}(\mathrm{x},\mathrm{t})}{\partial\mathrm{x}^{2}}= \frac{1}{\alpha}\cdot\frac{\partial\mathrm{T}(\mathrm{x},\mathrm{t})}{\partial \mathrm{t}} \tag{1}\]  
where T is the temperature, k corresponds to the thermal conductivity, and \(\alpha\) is expressed as material's thermal diffusivity (\(\alpha=\mathrm{k}/\,\mathrm{\rho}c_{p}\)); \(\mathrm{\rho}\) resembles the density; \(c_{p}\) is specific heat capacity. If this heat source is modulated, a periodic heat flow is produced in the material. This sinusoidal heat flow in an object refers to a diffusion process that generates a periodic temperature distribution called a heat wave. Defects present in the object may be detected by analyzing the phase shift and amplitude damping on the surface with the infrared camera.  
The periodical heat transfer through the semi-infinite (planar) body resulting thermal wave can be expressed by Equation (2) [34]:  
\[\mathrm{T}_{\mathrm{z},\mathrm{t}}=\mathrm{A}(\mathrm{z})\cos[\omega\mathrm{t }-\varnothing(\mathrm{z})]=\mathrm{T}_{0}\mathrm{e}^{-\frac{\mathrm{z}}{\mu} \mathrm{c}}\cos\left[\omega\mathrm{t}-\frac{2\pi\mathrm{z}}{\lambda}\right] \tag{2}\]  
where A(z) resembles the thermal amplitude; \(\omega\) is expressed as an angular frequency (\(\omega=2\pi\mathrm{f}\)); \(\varnothing(\mathrm{z})\) is the phase shift; t is the time; T\({}_{0}\) is an initial temperature change by incident thermal energy; z is the defect depth; \(\lambda\) is expressed as the thermal wavelength (\(\lambda=2\pi\mathrm{\mu}\)); and \(\mathrm{\mu}\) is the thermal diffusion length. The modulation frequency (f) and the thermal diffusivity can be expressed by Equation (3) [34]:  
\[\mathrm{\mu}=\sqrt{\frac{\alpha}{\pi\mathrm{f}}} \tag{3}\]  
From Equation (2), the phase shift (\(\varnothing\)) can be expressed as Equation (4), and it can be seen that it is related to the defect depth (z). By contrast, the amplitude (A(z)) can be expressed by Equation (5), and it can be seen that it is related to the initial temperature (T\({}_{0}\)) [35; 36].  
\[\varnothing=\frac{\mathrm{z}}{\mathrm{\mu}} \tag{4}\]  
\[\mathrm{A}(\mathrm{z})=\mathrm{T}_{0}\mathrm{e}^{-\frac{\mathrm{z}}{\mathrm{ \mu}}} \tag{5}\]