## 2 Materials and Methods
### Statistical Analysis  
In order to evaluate our hypothesis, we needed to analyze the differences between the acoustic responses from the defected sample and the acoustic response from the sample in pristine condition. Statistical technique was employed for the purpose, and analysis  
Figure 8: Actuation point and sensor location set in the simulation. (**A**) Left face of the geometry is configured as the uniform actuation application area visible as a blue line, (**B**) Detection point set at a coordinate value of (0, 5), visible as a red dot in the figure. Figure 7: Geometry details showing the dimension of the steel cylinder, the internal damage, and the approximate mesh size for the numerical analysis. of variance (ANOVA) tests were performed using software MATLAB 2017 version 9.3 (MathWorks). A confidence interval of 95% was defined for the ANOVA test input conditions. An ANOVA test determines whether data from several conditions of a parameter have a common mean. In other words, the ANOVA test enables us to find out whether different groups of an independent variable have different effects on the response variable. Mathematically, ANOVA can be expressed as a linear model with \(j\) number of different input groups, and there are \(i\) number of responses per group, given by Equation (10). \[y_{ij}=\alpha_{j}+\epsilon_{ij} \tag{10}\]  
Here, \(y_{ij}\) is the response for the i\({}^{\text{th}}\) observation of the \(j\)th group, \(\alpha_{j}\) is the mean for group number \(j\), and \(\epsilon_{ij}\) is assumed to be a random error with zero mean and a constant variance. ANOVA partitions the total variation in the data into following two components to test for the difference in the group means:  
* Difference of group means from the overall mean, i.e., \(\hat{y}_{j}-\hat{\Upsilon}\) (variation between groups), where \(\hat{y}_{j}\) is the sample mean of group \(j\), and \(\hat{\Upsilon}\) is the overall sample mean;
* Difference of observations in each group from their group mean estimates, \(y_{ij}-\hat{y}_{j}\) (variation within a group). In other words, ANOVA partitions the total sum of squares (SST) into sum of squares due to the between-groups effect (SSR) and sum of squared errors (SSE), as formulated in Equation (11). \[\sum_{i}\sum_{j}\left(y_{ij}-\hat{\Upsilon}\right)^{2}=\sum_{j}n_{j}\left(\hat {y}_{j}-\hat{\Upsilon}\right)^{2}+\sum_{i}\sum_{j}\left(y_{ij}-\hat{y}_{j} \right)^{2} \tag{11}\]  
Here, \(n_{j}\) is the sample size for the \(j\)th group, where \(j=1\), \(2\), \(\ldots\), \(k\). An ANOVA test is used to compare the variation between groups to the variation within groups. If the ratio of between-group variation to within-group variation is significantly high resulting in a low \(p\)-value, it statistically proves that the group means are significantly different from each other. This can be measured using a test statistic that has an \(F\)-distribution with \((k-1,N-k)\) degrees of freedom as presented in Equation (12). \[F=\frac{\sum_{i}\sum_{j}\left(y_{ij}-\hat{y}_{j}\right)^{2}/k-1}{\sum_{j}n_{j} \left(\hat{y}_{j}-\hat{\Upsilon}\right)^{2}/N-k} \tag{12}\]  
While comparing the acoustic response of the actuation between the damaged samples and the samples in pristine condition, two independent sample \(t\)-tests were performed, whereas paired \(t\)-tests were used for the comparison of the acoustic response of the damaged samples with different detector positions. Data are expressed as the mean \(\pm\) SD. A difference was considered to be statistically significant if the resulting \(p\)-value was less than 0.05. To verify the results based on detector location, independent sample and paired \(t\)-tests were performed. Table 5 presents the description of all the tests performed, along with the test purpose and the geometries utilized in each test.