## 3 Results and Discussion
### Stress Transfer Mechanism
#### 3.4.1 Long Flange (\(L_{\rm f}>L_{\rm s}\))
The vertical positions of the drawn lines are reasonably consistent with the plateau height of the first pulse (\(\bf B\) wave), verifying the hypothesis that the impact-generated stress wave is equally divided between the \(\bf B\) and \(\bf F\) waves. This hypothesis is further verified below. Consider the case where only the \(\bf F\) wave enters the reference plane after separating from the antecedent \(\bf B\) wave due to the long flange length. The ratios of reflection (\(R_{\rm r}\)) and transmission (\(R_{\rm t}\)) at a stress-free interface are given by [55]:  
\[R_{\rm r}=\frac{\sigma_{\rm r}}{\sigma_{\rm i}}=\frac{Z_{2}-Z_{1}}{Z_{1}+Z_{2}}, \tag{5}\]  
\[R_{\rm t}=\frac{\sigma_{\rm t}}{\sigma_{\rm i}}=\frac{2A_{1}\rho_{2}C_{2}}{Z_{1} +Z_{2}}, \tag{6}\]where the subscripts i, r, and t denote the incident, reflected, and transmitted waves, respectively; and the subscripts 1 and 2 indicate the incident and transmitted wave media (flange structure and incident bar), respectively. Starting from an **F** wave magnitude of 151.77 MPa (i.e., the hypothesised value) in Figure 4a (\(D_{\text{b}}=10\) mm), the magnitudes of the first, second, and third waves transmitted into the incident bar were calculated, using Equations (5) and (6), as 202.36, 67.45, and 22.48 MPa, respectively, as indicated by the dash-dot horizontal lines in Figure 4a. In Figure 4a (\(D_{\text{b}}=10\) mm), the vertical positions of the drawn lines are found to be consistent with the plateau heights of the first three \(\mathbf{R_{n}}\) waves on the green curve (\(L_{\text{f}}=30D_{\text{b}}\)). In Figure 4b (\(D_{\text{b}}=20\) mm), the plateau height of the first \(\mathbf{R_{n}}\) wave (\(\mathbf{R_{1}}\)) on the green curve (\(L_{\text{f}}=30D_{\text{b}}\)) is consistent with the drawn line at 202.36 MPa. Finally, in Figure 4c (\(D_{\text{b}}=20\) mm; \(A_{\text{s}}/A_{\text{b}}=0.5\)), starting from an **F** wave magnitude of 101.18 MPa (hypothesised value), the magnitude of the first \(\mathbf{R_{n}}\) wave (\(\mathbf{R_{1}}\)) was calculated to be 121.42 MPa. In Figure 4c, this value is also consistent with the plateau height of the \(\mathbf{R_{1}}\) wave for \(L_{\text{f}}=30D_{\text{b}}\). The consistency of the various calculation results for the \(\mathbf{R_{1}}\), \(\mathbf{R_{2}}\), and \(\mathbf{R_{3}}\) waves was verified in the previous paragraph under the hypothesis that the impact-generated stress (303.54 and 202.36 MPa for the cases where \(A_{\text{s}}/A_{\text{b}}=1.0\) and 0.5, respectively) is equally divided between the **B** and **F** waves. Based on verification of the hypothesis, the tensile **B** and compressive **F** waves that simultaneously form are interpreted to be the counterparts of each other across the impact plane; for instance, in the law of action and reaction, if the B wave is the action, then the F wave is the reaction. The interpretation that two separated pulses on the green curves (\(L_{\text{f}}=30D_{\text{b}}\)) in Figure 4b or Figure 4c (\(D_{\text{b}}=20\) mm) can be ascribed to the **B** and **F** waves in sequence can be verified from the view of the time lag to the second pulse from the first one. According to Figure 5, the travel distance of the \(\mathbf{F}+\mathbf{R_{1}}\) wave is longer than that of the **B** wave by \(2L_{\text{f}}\) (=60\(D_{\text{b}}\)). The time lag of the **F** wave is then \(2L_{\text{f}}/C\), which is 0.231 ms. This time interval is marked on the green curve (\(L_{\text{f}}=30D_{\text{b}}\)) in Figure 4c; the positioned mark is reasonably consistent with the time gap of the two separated pulses on the green curve. This observation also verifies the interpretation that two separated pulses on the green curves (\(L_{\text{f}}=30D_{\text{b}}\)) are attributed to the **B** and **F** waves in sequence.