# Experimental Investigation on Self-Excited Thermoacoustic Instability in a Rijke Tube
## 4 Results and Discussions
### Temperature  
The temperature fluctuations above flame, 30 mm from the Bunsen burner exit, during the thermoacoustic instability were captured by the single-ended TDLAS, with a measurement rate of 5 kHz. Figure 6 demonstrates the typical absorption spectrum of two H\({}_{2}\)O transitions of interest centered at 7185.59 cm\({}^{-1}\) and 7444.35 cm\({}^{-1}\). The red dots represent measured absorption signals, while the blue lines are the Voigt profile best-fitting to the measured signals. Both fitting residuals of H\({}_{2}\)O transitions are within 0.4%. The absorption areas of these two transitions, A\({}_{1}\) and A\({}_{2}\), can be deduced from the best-fitting curves. Then, the temperature is determined by using Equation (7). Figure 7 illustrates the measured temperature time histories during thermoacoustic instability with single-ended TDLAS and a J-type thermocouple in 1 s. It is evident that the temperature measured by TDLAS fluctuates from 1000 K to 1300 K in Figure 7a. The measurement by the thermocouple is stable at 1153 K due to its low response, making it fail to capture the dynamics of temperature. Figure 7b presents the zoom-in view of the green square region in Figure 7a. It can be found that the temperature fluctuates periodically, which is one of the main features of entropy waves.  
Figure 5: Partial measurement signals with time series. (**a**) is comparison between raw data and filtered data. (**b**) is the increment of every equal interval sampling in the green dot square in (**a**). The blue and pink lines denote raw data and low-pass-filtered (LPF) data, respectively.  
Figure 6: The typical absorption spectrum for H\({}_{2}\)O transitions centered at 7185.59 cm\({}^{-1}\) (**a**) and 7444.35 cm\({}^{-1}\) (**b**). the red dots represent measured absorption signals while the blue lines are the Voigt profile best-fitting to the measured signals.  
The measured temperature uncertainty comes primarily from the uncertainty in the reference linestrength and Voigt fitting routine [33]. The uncertainty in the reference linestrength of 7185.59 cm\({}^{-1}\) and 7444.35 cm\({}^{-1}\) is less than 2% and 5%, respectively [29]. On the other hand, the residuals of the best fitting to absorption signals of these two transitions are no more than 0.4%. Therefore, the measurement uncertainty is about 5.41%.  
In this work, both amplitude and frequency of temperature fluctuations are given equal attention. To find out the domain frequency of temperature, the short-time Fourier transform was performed on the measured temperature. Figure 8 displays the spectrogram of the temperature illustrating the frequency as a function of time. There exist a strong frequency at 230 Hz during the whole measurement corresponding to the fundamental frequency of the Rijke tube. Second harmonics of the frequency also appear. Furthermore, some frequencies with low amplitudes occur occasionally. Additionally, the frequency distribution explains why the temperature time series deviates from sinuous wave severely, as show in Figure 7b. This indicates that the local gas temperature suffers intricate perturbations, such as convection, within the inner wall of the Rijke tube.