# Experimental Investigation on Self-Excited Thermoacoustic Instability in a Rijke Tube
## 2 Theorical Background
### Tunable Diode Laser Absorption Spectroscopy  
The absorption spectroscopy measurement technique is schematically shown in Figure 3. When a beam of light, whose center frequency is \(\
u_{0}\) and intensity is \(I_{0}\), travels across a region of interest, its intensity will be attenuated to \(I_{v}\) due to the interaction between light and matter. The incident light, \(I_{0}\), and the transmitted light, \(I_{v}\) are correlated by Beer-Lambert law:  
\[I_{v}=I_{0,v}\exp(-k_{v}L) \tag{4}\]  
where \(k_{v}\) and \(L\) (cm) denote the absorption coefficient at frequency \(v\) (cm\({}^{-1}\)) and absorption length, respectively. The absorption coefficient is defined as  
\[k_{v}=P_{total}\cdot X_{i}\cdot S\cdot\phi(\
u) \tag{5}\]  
where \(P_{total}\) (atm) represents total pressure, \(X_{i}\) mole concentration of absorption species, \(i\), \(S\) (cm\({}^{-2}\) atm\({}^{-1}\)) the linestrength of a specific absorption line, \(\phi(\
u)\) (cm) lineshape function, which can be normalized to unity, i.e., \(\int\phi(\
u)dv\equiv 1\), for every specific absorption line. Additionally, in most cases, the lineshape function can be accurately described by the Voigt function, which is the convolution of a Lorentzian function and a Gaussian function.  
Figure 3: Schematic of absorption spectroscopy measurement technique.  
Figure 2: Interaction between unsteady heat source and acoustic waves in a Rijke tube.  
The linestrength of a specific absorption line at an arbitrary temperature is determined from partition function, low-state energy and the molecular reference temperature linesstrength [28]:  
\[\begin{split} S(T)=& S(T_{0})\frac{Q(T_{0})}{Q(T)}\left( \frac{T_{0}}{T}\right)\exp\left[-\frac{hcE^{\prime\prime}}{k}\left(\frac{1}{T}- \frac{1}{T_{0}}\right)\right]\\ &\times\left[1-\exp\left(-\frac{hcv_{0}}{kT}\right)\right]\left[ 1-\exp\left(-\frac{hcv_{0}}{kT_{0}}\right)\right]^{-1}\end{split} \tag{6}\]  
where \(S(T_{0})\) is the reference temperature linestrength, \(T_{0}\) = 296 K, \(Q\) the partition function, \(E^{\prime\prime}(\text{cm}^{-1})\) the lower-state energy and \(v_{0}\) (cm\({}^{-1}\)) the transition center frequency. The above mentioned parameters can be found in spectroscopic databases such as HITRAN [29]. _h, c_ and \(k\) are [PERSON]'s constant, speed of light and Boltzmann's constant.  
In TDLAS, the frequency of laser is tuned by current or temperature to scan across the overall or partial absorption profile of the targeted transitions, the scan rate ranging from a few of Hz to MHz. The absorption area, \(A=\int_{0}^{L}\int_{-\infty}^{+\infty}k_{\text{r}}dvdl\), is obtained when the transition is fully scanned across. Then, the gas temperature of interest is inferred from the dependence of linestrength on temperature by two-line thermometry [30].  
\[\begin{array}{ll}\frac{A_{1}}{A_{2}}&=\frac{\int_{0}^{L}\int_{-\infty}^{+ \infty}k_{\text{r},1}dvdl}{\int_{0}^{L}\int_{-\infty}^{+\infty}k_{\text{r},2} dvdl}=\frac{\int_{0}^{L}\int_{-\infty}^{+\infty}pXS_{1}(T)\phi_{\text{r},1}(v) dvdl}{\int_{0}^{L}\int_{-\infty}^{+\infty}pXS_{2}(T)\phi_{\text{r},2}(v) dvdl}\\ &=\frac{S_{1}(T)}{S_{2}(T)}=f(T)\end{array} \tag{7}\]  
The ratio of absorption areas of two transitions is solely dependent on temperature.