## 4 Beam Profile Measurements
### Measured Beam Profiles and Effects of the Mode Superposition  
The measured vertical and lateral beam profiles radiated from the waveguide plate are shown in Figure 10. Without any quantitative evaluations on radiation characteristics, all three measured beam profiles seem to be visually different from those of the A(0,0) mode shown in Figure 5a,b, despite the attempt to excite only the A(0,0) mode, and they show different radiation characteristics depending on the excitation source position. In particular, drastic changes of the lateral beam are noticeable compared with those of the vertical beam; the lateral beams at \(d=300\) mm and \(d=400\) mm have a dual main beam, but one at \(d=350\) mm has a single main beam within the measurement domain. These characteristics cannot be explained by only pure width mode and demonstrate that the leaky radiation beams radiated from the waveguide plate with finite width are affected by the mode superposition of the width modes, as predicted from the dispersion curves of Figure 2.  
Figure 10: _Cont._  
Figure 9: Experiment for the measurement of the 3D radiation beam profile of a leaky wave radiated from a SS304 plate with 1.5 mm thickness and 15 mm width in water, (**a**) experimental setup and (**b**) block diagram, and (**c**) A-scan signal of the leaky wave measured at the center of the radiation surface. Moreover, the different characteristics changed by the excitation source position are strongly estimated to be affected by the change of the wave phase by phase velocity differences between the superposed modes. Figure 11 shows the wave phase changes on the waveguide plate calculated from the following equation at time \(t=0\), weighting constants \(a=b=1\), and phase delay constants \(\tau_{0}=\tau_{2}=0\):  
\[\
u(x_{0},y_{0},t)=a\cdot V_{0}(y_{0})e^{i(k_{P0}x_{0}-\omega(t-\tau_{0}))}+b \cdot V_{2}(y_{0})e^{i(k_{P2}x_{0}-\omega(t-\tau_{2}))} \tag{5}\]  
where \(V_{0}(y_{0})\) and \(V_{2}(y_{0})\) are the normalized velocity distributions of the A(0,0) and A(0,2) modes identified in Figure 3, respectively, and \(k_{P0}\) and \(k_{P2}\) are the wavenumbers of the A(0,0) and A(0,2) modes, respectively. According to the obtained dispersion curves, the A(0,1) and A(0,2) modes are found to have a high possibility to be excited with the A(0,0) mode. However, only the mode superposition between the A(0,0) mode and the A(0,2) mode was investigated here; the reason why the A(0,1) mode is excluded from the mode superposition is that the anti-symmetric width modes (including the A(0,1) mode) cannot be easily generated by general Lamb wave excitation methods such as the angle-beam method because they have symmetric distribution of the input wave energy in the width direction. It is known that the profile of the input energy in the width direction should match that of the target width mode to be generated [45; 46]. The two width modes excited by the high modal density are continually superposed in-phase or out-of-phase with each other during propagation along the waveguide plate. Therefore, from Figure 11, it can be recognized that the superposed modes make the spatial beating due to their differences in phase velocity and the beating on the radiation surface, varied by the excitation source position, results in the change in radiation characteristics. Figure 11: Change of wave phase on the waveguide plate by the mode superposition between the A(0,0) and A(0,2) modes. Figure 10: Measured leaky radiation beam profiles radiated from a SS304 plate (1.5 mm thickness, 15 mm width) with variation of the excitation source position, (**a**) \(d=300\) mm, (**b**) \(d=350\) mm, and (**c**) \(d=400\) mm. To validate the effect of the spatial beating on the leaky radiation beam, using the constructed RSI model, the leaky radiation beam profiles were simulated with the phase variations of superposed modes on the radiation surface.