# The Dependence of Ultrasonic Velocity in Ultra-Low Expansion Glass on Temperature
## 6 Conclusions  
In this investigation, the aim was to theoretically analyze and validate the dependence of the ultrasonic velocity in ULE glass on temperature. Based on the simulation and experimental investigations, we can draw the following conclusions:  
1. The proposed pulse reflection immersion method provides reliable and stable measurements of the ultrasonic echo signal of ULE glass, and the calculation of the signal based on the correlation method can be used to conveniently and accurately extract the ultrasonic TOF of the tested sample and then obtain the ultrasonic velocity.
2. The application of the proposed method for six ULE samples with different CTE values indicates that ultrasonic velocity increases as the experimental temperature is increased. Furthermore, the \(c_{\rm L}\)-\(T\) exponential model was theoretically analyzed and experimentally fitted. The predicted \(c_{\rm L}\) using the exponential model at ten random temperature points shows good agreement with the actual measured ultrasonic velocities at the same temperature.  
These findings indicate the promising potential of the \(c_{\rm L}\)-\(T\) exponential model to determine the ultrasonic velocity at a specified temperature for ultrasonic nondestructive CTE measurement in large ULE glass, which will be critical for reliably evaluating the CTE homogeneity of large ULE glass at a specified temperature.  
There are some comments that can be made about the general applicability of our method. In this paper, the derivation and verification of the \(c_{\rm L}\)-\(T\) relationship are only for isotropic ULE glass, but the method for deriving exponential model can also be applied to anisotropic materials, such as anisotropic crystals, and for obtaining the longitudinal wave velocity, transverse wave velocity, and then the Poisson's ratio. The only distinction is \(\alpha_{\rm L}\). As for the experimental verification method for the exponential model, the correlation  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & **Temperature** & \multirow{2}{*}{\(c_{\rm m}\)**(m/s)**} & \multirow{2}{*}{\(c_{\rm p}\)**(m/s)**} & \multirow{2}{*}{\(c_{\rm m}\)**–\(c_{\rm p}\)(m/s)**} & \(\delta\)1 \\  & (\({}^{\circ}\)C) & & & & \% \\ \hline \multirow{8}{*}{5\#} & 10.6 & 5750.4 & 5749.7 & 0.7 & 0.012 \\  & 11.3 & 5750.2 & 5750.2 & 0.0 & 0.000 \\  & 13.5 & 5752.1 & 5751.8 & 0.3 & 0.005 \\  & 14.1 & 5753.1 & 5752.3 & 0.8 & 0.014 \\  & 15.1 & 5754.4 & 5753.0 & 1.4 & 0.024 \\  & 17.5 & 5754.2 & 5754.8 & \(-\)0.6 & 0.010 \\  & 20.3 & 5756.8 & 5756.8 & 0.0 & 0.000 \\  & 25.2 & 5760.3 & 5760.5 & \(-\)0.2 & 0.003 \\  & 25.9 & 5760.4 & 5761.0 & \(-\)0.6 & 0.010 \\  & 29.9 & 5764.8 & 5763.9 & 0.9 & 0.016 \\  & & & & \%\(c_{\rm c}=0.73\) & \\ \hline \hline \end{tabular} Note: \(c_{\rm m}\) is the actual measured ultrasonic velocity, \(c_{\rm p}\) is the ultrasonic velocity predicted by the fitting model, and \(\delta\) denotes the relative error. The deviation \(\sigma_{\rm c}\) is a measure of the inaccuracy.  
\end{table}
Table 3: Cont.  
method, it is also applicable for the measurement of the TOF, regardless of whether the material is isotropic.  
Conceptualization, W.W., Y.Z. (Yuanyuan Zhang) and H.L.; methodology, Y.Z. ([PERSON]) and H.L.; software, W.W. and Y.Z. ([PERSON]); validation, W.W. and H.L.; investigation, W.W. and M.S.; resources, Y.Z. ([PERSON]), M.S. and H.L.; data curation, W.W.; writing--original draft preparation, W.W.; writing--review and editing, Y.Z. ([PERSON]), Z.D. and H.L.; visualization, W.W.; project administration, Y.Z. ([PERSON]). All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.