## 2 Mathematical Setup
### Basics of Thermal Wave Generation  
Thermal waves can be mathematically characterized as solutions of the heat diffusion equation. The heat source at the surface represents the boundary conditions, influences the surface temperature distribution, and determines the generation of waves. The most common type of excitation is a periodic, planar energy input with high-performance laser beams of a single specific excitation frequency, i.e., lock-in excitation. For the sake of simplicity, an isotropic homogeneous semi-infinite medium \(M\) (that means an infinite extension of the medium in \(x\)-direction) is considered, which is illustrated in Figure 3.  
Here, \(k\) denotes the thermal conductivity, and \(c\) denotes the volumetric heat capacity of the material \(M\). The thermal diffusivity can then be calculated using \(\alpha:=\frac{k}{c}\) and the thermal effusivity by \(e:=\sqrt{kc}\). Figure 3: Thermal wave generation and propagation in a semi-infinite medium. It is assumed that the heated surface occupies the \(y-z\)-plane at \(x=0\). To obtain the temperature distribution at the surface of the medium, the Fourier equation must be solved,  
\[\frac{\partial^{2}T}{\partial x^{2}}=\frac{1}{\alpha}\frac{\partial T}{\partial t },\quad x,t>0. \tag{1}\]  
First, the boundary conditions need to be specified. The medium's surface is excited with a plane harmonic; thus, temporal heating of modulation frequency \(\omega:=2\pi f\) for some frequency \(f\) and source intensity \(Q_{0}\). This leads to an excitation of the form  
\[\frac{Q_{0}}{2}[1+\cos(\omega t)],\]  
yielding the generation of thermal waves inside the medium. Since the periodic thermal energy is subject to conduction into the solid, by using the appropriate rate equation, the boundary condition on the surface is given as  
\[ \begin{array}{rcl}-k\frac{\partial T}{\partial x}&=&\text{Re}\Big{\{}\frac{ Q_{0}}{2}[1+\exp{(i\omega t)}]\Big{\}}\\ &=&\frac{Q_{0}}{2}[1+\cos(\omega t)]\\ &=&\underbrace{\frac{Q_{0}}{2}}_{\text{dc component}}+\underbrace{\frac{Q_{0}}{2} \cos(\omega t)}_{\text{ac component}},\quad x=0,t>0.\end{array}  \tag{2}\]  
Here, dc means the _Direct Current_ and ac the _Alternating Current_. The dc component will not be relevant in later applications and can be neglected. Applying a time-harmonic approach then yields  
\[T(x,t)=\text{Re}[T(x)\exp(i\omega t)].\]  
Plugging this in Equation (1), one ends up with  
\[\exp(i\omega t)\bigg{(}\frac{\partial^{2}T(x)}{\partial x^{2}}-\frac{i\omega }{\alpha}T(x)\bigg{)}=0.\]  
Taking into account that \(T(x)\) must be finite for \(x\rightarrow+\infty\), one receives the solution of the boundary value problem (1) and (2) as  
\[T(x,t)=\frac{Q_{0}}{2k\sigma}\exp(-\sigma x+i\omega t),\quad\sigma\coloneqq(1+ i)\sqrt{\frac{\omega}{2\alpha}}. \tag{3}\]  
Using a multiplication with \(1=\frac{i+1}{\sqrt{2}}\exp{(-i\frac{\pi}{4})}\) and further simplifications, one obtains a more significant expression given as  
\[\mathbf{T}(x,t)=\frac{Q_{0}}{2e\sqrt{\omega}}\exp\left(-\frac{x}{\mu}\right)\exp \left[i\bigg{(}\omega t-\frac{x}{\mu}-\frac{\pi}{4}\bigg{)}\right], \tag{4}\]  
where  
\[\mu\coloneqq\sqrt{\frac{2\alpha}{\omega}} \tag{5}\]  
is the so-called thermal diffusion length. Hence, thermal waves are significantly dampened, where \(\mu\) controls the penetration depth into the material. For small thermal diffusivity \(\alpha\), the thermal waves do only slightly propagate into the interior of the material. In contrast, by decreasing the modulation frequency \(\omega\), a deeper penetration of the thermal waves into the material is obtained. This phenomenon is very useful in the photothermal measurement of layer thicknesses. Furthermore, note that there occurs a progressive phase shift using  
\[\varphi=-\frac{x}{\mu}-\frac{\pi}{4} \tag{6}\]  
between the temperature at the surface and a point \(x\) located at the propagating thermal wave in the material.