## 5 Discussion and Design Direction for Performance Improvement of the Waveguide Sensor  
It seems difficult to completely physically avoid the spatial beating phenomenon and the mode superposition within a short propagation distance because of the high modal density by numerous width modes of the Lamb wave in a waveguide plate with finite width. Therefore, if a single narrow main beam is needed for the application purpose, the position tuning for the excitation source can be recommended for the wave energy concentration at the center in the width direction. Figure 16 shows the measured beam profiles of the leaky Lamb wave excited at \(d=330\) mm and a clear single main beam can be seen compared with the beam profiles measured for other excitation source positions shown in Figure 10.  
Figure 14: Simulation #3 of the leaky radiation beam profile based on the wave phase on the radiation surface. (**a**) Isotropic view of wave phase on radiation surface, (**b**) top view of wave phase on radiation surface, (**c**) vertical beam profile, and (**d**) lateral beam profile.  
Figure 15: Comparison results between the measured beam profile of a leaky Lamb wave radiated at \(d=350\) mm (Figure 10b) and the simulation result of Figure 13. (**a**) Response on acoustical axis and (**b**) lateral beam width at maximum intensity of the main beam.  
In addition, as the plate width increases, the phase velocity curve of the A(0,2) mode converges to that of the A(0,0) mode as shown in Figure 17. This is because the plate strip gets close to the infinitely wide plate with extension of the plate width. Accordingly, unless the plate width is not constrained in the waveguide sensor design, the plate width increment can extend the beating length by a difference in reduction between the A(0,0) and A(0,2) modes in wave velocities, and thereby the effect of the spatial beating on the leaky radiation beam can be reduced.  
Finally, a change in the frequency-thickness product (_fh_) can be also considered for performance improvement. Figure 18 shows the measured beam profile of the leaky Lamb wave excited at \(d=350\) mm with _fh_ = 1.5 MHz\(\cdot\)mm (1.5 MHz center frequency and 1.0 mm plate thickness), the same as the operational frequency-thickness product in the waveguide sensor (1.0 MHz center frequency and 1.5 mm plate thickness). As the excitation frequency increment makes the wavelength of the leaky Lamb wave short, the width-wavelength ratio is increased. Therefore, the effective plate width can be enlarged without an increment of the physical plate width, although the ultrasonic attenuation is increased and the radiation characteristics are also changed by the frequency increment of the leaky wave.  
Figure 16: Measured beam profiles of the leaky Lamb wave excited at \(d=330\) mm (1.5 mm thickness and 15 mm width, 1.0 MHz center frequency).  
Figure 17: Phase velocity dispersion curves of the A(0,2) mode in a 1.5 mm thick SS304 plate with variation of the plate width.  
As a result, leaky Lamb wave radiation from a waveguide plate with a finite width is affected by the mode superposition of the width modes. For this reason, radiation characteristics of the leaky radiation beam, in particular the lateral beam, are sensitive to the excitation conditions (excitation source position, excitation frequency, etc.). This can be an advantage or disadvantage for the plate-type waveguide sensor. Therefore, according to the application purposes and goals, it will be necessary to properly design and tune the excitation conditions in the plate-type waveguide sensor.