## 2 Mathematical Setup
### Basics of Thermal Wave Interference
#### 2.3.2 Two-Layered Coatings on a Substrate  
Next, consider the following system of three layers, \(M_{1}-M_{3}\), having thicknesses \(L_{1}-L_{3}\), where \(M_{3}\) is assumed to be thermally thick.  
Obviously, adding a second coating layer increases the complexity in which thermal wave trains can possibly propagate in the system, see, for example, the blue arrows in Figure 7. The first interface for the case \(n=1\) was a coating-to-substrate interface, while for \(n=2\) the first interface is a coating-to-coating interface. Transmitted thermal waves can no longer be ignored, because the second layer is not thermally thick anymore, i.e., the reflections still contribute to the surface temperature significantly, although they have a longer propagation path. A very elegant way of summarizing all possible wave trains is done by replacing a complex-valued effective reflection coefficient \(\Gamma_{1}\) for the real-valued reflection coefficient \(R_{1}\) from Equation (11), cf. [4; 19].  
Again, by introducing a new layer, the reflection process expands after passing the interface between \(M_{1}\) and \(M_{2}\) and appear at a lower level, too. Thus, interference effects occur in the new layer with material \(M_{2}\) as well. This implies that the reflection coefficient \(R_{1}\) is not sufficient to describe the whole process anymore, so it is necessary to change the expression in \(R_{1}\) analogously to the derivation in Equation (9) as  
\[\begin{split}\Gamma_{1}&\coloneqq R_{1}+T_{1}T_{1}^ {\prime}R_{2}\exp{(-2\sigma_{2}L_{2})}\sum_{n=0}^{\infty}\big{(}R_{1}^{\prime }R_{2}\exp{(-2\sigma_{2}L_{2})}\big{)}^{n}\\ &=R_{1}+(1-R_{1}^{2})R_{2}\exp{(-2\sigma_{2}L_{2})}\sum_{n=0}^{ \infty}\big{(}-R_{1}R_{2}\exp{(-2\sigma_{2}L_{2})}\big{)}^{n}\\ &=R_{1}+\Big{(}R_{2}\exp{(-2\sigma_{2}L_{2})}-R_{1}^{2}R_{2}\exp{ (-2\sigma_{2}L_{2})}\Big{)}\frac{1}{1+R_{1}R_{2}\exp{(-2\sigma_{2}L_{2})}}\\ &=\frac{R_{1}+R_{2}\exp{(-2\sigma_{2}L_{2})}}{1+R_{1}R_{2}\exp{( -2\sigma_{2}L_{2})}}.\end{split} \tag{15}\]  
Here, \(T_{1}\) (\(T_{1}^{\prime}\)) denotes the transmission coefficients in a downward (upward) direction at the \(M_{1}-M_{2}\) interface. Using \(R_{1}\) (\(R_{1}^{\prime}\)), one denotes the reflection coefficients at the same interface, where the thermal wave train stays in \(M_{1}\) (\(M_{2}\)). Analogously, \(R_{2}\) is the reflection coefficient for thermal waves in \(M_{2}\) that stay in \(M_{2}\).  
Therefore, because of \(R_{1}^{\prime}=-R_{1}\), the geometric series formula can be used again, since \(|-R_{1}R_{2}\exp\left(-2\sigma_{2}L_{2}\right)|<1\). Note that  
\[\lim_{L_{2}\to\infty}\Gamma_{1}=R_{1}, \tag{16}\]  
which effectively represents the case \(n=1\) again, if the second coating layer would be infinitely thick (or at least thermally thick).  
The quantity \(\Gamma_{1}\) is called the effective reflection coefficient. Since the thermal interference processes are the same in the first layer as discussed before, one ends up with the following surface temperature for \(n=2\):  
\[\begin{split} T(x=0,t)&=T_{0}A_{0}\bigg{[}\frac{1+ \Gamma_{1}\exp\left(-2\sigma_{1}L_{1}\right)}{1-R_{0}\Gamma_{1}\exp\left(-2 \sigma_{1}L_{1}\right)}\bigg{]}\exp\left[i\Big{(}\omega t-\frac{\pi}{4}\Big{)} \right]\\ &=:\hat{T}(x=0)\exp\left[i\Big{(}\omega t-\frac{\pi}{4}\Big{)} \right].\end{split} \tag{17}\]  
Once again, this expression provides a term for the amplitude,  
\[A_{\hat{T}}=\sqrt{\text{Re}\big{[}\hat{T}(x=0)\big{]}^{2}+\text{Im}\big{[}\hat {T}(x=0)\big{]}^{2}}, \tag{18}\]  
Figure 7: System of two layers of coatings \(M_{1}\)–\(M_{2}\) and one layer of substrate \(M_{3}\).  
as well as for the phase angle,  
\[\varphi_{T}=\tan^{-1}\left[\frac{\text{Im}\big{[}\hat{T}(x=0)\big{]}}{ \text{Re}\big{[}\hat{T}(x=0)\big{]}}\right], \tag{19}\]  
yielding  
\[\hat{T}(x=0)=A_{\hat{T}}\exp{(i\varphi_{T})}.\]