## 2 Lamb Wave Propagation in a Plate Strip with Finite Width
### Dispersion Curves and Wave Structures  
Figure 2 shows calculated dispersion curves of Lamb waves in a stainless steel (SS304) plate with 1.5 mm thickness and 15 mm width in a vacuum. Note that propagation characteristics in the plate coupled with liquid, such as the wave velocities and structures, are assumed to not be different from those in a vacuum [40]. Specifically, Figure 2a shows the phase velocity dispersion curves, whereas Figure 2b shows the group velocity dispersion curves. As shown in these dispersion curves, in contrast to those in an infinitely wide plate, Lamb waves in a finite width plate have numerous width modes. Therefore, a certain single mode of Lamb waves in the plate strip is named S(\(m\),\(n\)) or A(\(m\),\(n\)); the first index \(m\) is the order of the thickness mode; the second index \(n\) is that of the width mode. Furthermore, one can observe that the phase velocity increases and the group velocity decreases as the order of the width mode increases.  
Figure 1: Coordinate system of a plate strip with finite width.  
The most important observation in the dispersion curves of Figure 2 is that many Lamb wave modes are very close to each other at the operation point of the waveguide sensor, which indicates the difficulty of single mode excitation. In this high modal density area, neighbor width modes from the A(0,1) mode to the A(0,4) mode can have a high possibility of being excited with the lowest-order mode, the A(0,0) mode; the S(\(m\),\(n\)) modes are not in our interest because the flexural modes, A(0,\(n\)) modes are only used for the high radiation efficiency in the waveguide sensor. In addition, it can be reasonably inferred that the multiple width modes cannot be separated during the short propagation distance because they are not significantly different in group velocities at the operation point.  
Each higher-order width mode can be identified in the wave structure results, as shown in Figure 3. The out-of-plane velocity profiles in the width direction change appreciably according to the order of the width mode; they can be described as a combination of the trigonometric and hyperbolic functions [31; 32]. Here, the non-flat velocity profile in case of \(n=0\) is estimated to be caused by the free-free boundary condition and high frequency range. From the dispersion curves and the wave structures, it can be assumed that leaky Lamb wave propagation in the plate strip coupled with liquid might be affected by superposition among the higher-order width modes.  
Figure 3: **(a)** Wave structures and **(b)** particle velocity profiles (width direction) of the A(0,\(n\)) modes in a SS304 plate with 1.5 mm thickness and 15 mm width at 1.0 MHz; the velocity profiles are extracted on the neutral plane (\(z=0\)); solid line: out-of-plane direction (\(z\) direction); dashed line: in-plane horizontal direction (\(y\) direction); circle markers: in-plane extensional direction (\(x\) direction).  
Figure 2: Dispersion curves of Lamb waves in a SS304 plate with 1.5 mm thickness and 15 mm width: **(a)** phase velocity, **(b)** group velocity; SS304 plate properties: density \(\rho=7800\) kg/m\({}^{3}\), longitudinal wave velocity \(C_{L}=5800\) m/s and shear wave velocity \(C_{S}=3160\) m/s; solid line: even order of width mode (symmetric width mode); dashed line: odd order of width mode (anti-symmetric width mode).