## 3 Results
### Microstructure
Hence it can be concluded that the wave velocity itself cannot be used effectively to estimate the grain size due to its small sensitivity to grain size and its much larger temperature dependency. For a pure austenitic sample, an increase in grain size of 1 \(\upmu\)m means a velocity decrease of 1.6 m/s (Figure 12a), and for a pure ferritic steel this value is 1.1 m/s per \(\upmu\)m (Figure 12b). However, the same velocity decrease can also be caused by several other parameters, e.g., a temperature increase in the order of 1 to 2 \({}^{\circ}\)C (Figure 13a,b) or a small change in sample thickness (in the order of micrometers). In order to be able to determine such small velocity differences, parameters such as temperature and also sample thickness must be measured very accurately, rendering this method unfeasible for the determination of grain size. Figure 11: (**a**) The Voigt-Reuss-Hill averaged elasticity matrix of the samples of (**a**) pure austenite; (**b**) pure ferrite; and (**c**) pure martensite. All values are obtained from samples at a temperature of 27 \({}^{\circ}\)C. The elasticity matrices are independent of the grain size. Simulations of other microstructure realizations or randomizations (pure austenite, grain size 15 \(\upmu\)m, at 300 \({}^{\circ}\)C) lead to velocity variations less than \(\pm\)0.4 m/s. This relates to a grain size variation up to 0.2 \(\upmu\)m, which does not have a significant impact on grain sizing. Results of the attenuation analysis are shown in Figures 14 and 15, where the results of the fit to the attenuation of the second echo has been plotted, in terms of \(b^{\prime}\), as defined in Equation (6). For each of the constant temperature series in Figure 14, the sample with the middle grain size has been used as the reference sample. Hence, the corresponding data points all coincide at \(b^{\prime}=0\) for martensite at \(d=15\)\(\upmu\)m and for ferrite at \(d=10\)\(\upmu\)m. Similarly, as a reference sample in Figure 15, the sample with temperature \(T=300\)\({}^{\circ}\)C has been used for each of the grain sizes. Figure 12: The relation between the grain diameter \(d\), and observed wave velocity \(c\) for two material types and four temperatures: (**a**) shows 100% austenitic steel; and (**b**) 100% ferritic steel. Figure 13: The relation between the temperature \(T\), and observed wave velocity \(c\) for two material types and three grain sizes: (**a**) shows 100% austenitic steel; and (**b**) 100% ferritic steel. The curves, both as function of grain size and temperature, appear to be approximately linear for pure austenite and pure ferrite, as far as can be determined with only three or four data points per curve, which is in accordance with Wan [24]. A generic quantitative relationship cannot be properly determined. Note that the attenuation parameter for ferrite is a bit higher than expected for the smallest grain size (grain size less than 10 \(\upmu\)m, Figure 14b). It is assumed that this may be due to the fact that the wave propagation in the small grains of these samples is prone to computational errors: since there are less than 2 grid points per average grain, the results are assumed to be less reliable. The curves show slopes which depend on both grain size and temperature. Hence, a change in attenuation can be attributed to a change in grain size, in temperature or  
Figure 14: The relation between the grain diameter \(d\), and parameter \(b^{\prime}\) for two material types and four temperatures: (**a**) shows 100% austenitic steel; and (**b**) 100% ferritic steel. Figure 15: The relation between the temperature \(T\), and parameter \(b^{\prime}\) for two material types and three diameters: (**a**) shows 100% austenitic steel; and (**b**) 100% ferritic steel. in a combination of both. Knowing the sample temperature sufficiently accurate and using a curve corresponding with this temperature is essential to avoid sizing errors.