## 8 Leaky Characteristics of Lamb Waves
Based on this slowness curve representation, it can be concluded that the acoustic leakage will be an obliquely travelling bulk wave. The angle of incidence of the acoustic leakage would be between \(0^{\circ}\) and \(20^{\circ}\) for most Lamb waves. If these findings are applied to air-coupled transducers, it would suggest that there is an optimum angle of reception for the air-coupled transducers. The partial waves for air can be represented as an obliquely traveling longitudinal wave:  
\[u_{i}^{air}(x,z,t)=A_{air}\hat{k}_{i}e^{-ik_{j}x_{j}+i\omega t}, \tag{58}\]  
and the partial waves of a Lamb wave in an aluminum plate (see Equations (38) and (43)) can be used with five of the six available interface conditions:  
\[v_{i}=i\omega u_{i}. \tag{52}\]0  
\[v_{i}=i\omega u_{i}. \tag{52}\]1  
\[v_{i}=i\omega u_{i}. \tag{52}\]2  
\[v_{i}=i\omega u_{i}. \tag{52}\]3  
\[v_{i}=i\omega u_{i}. \tag{52}\]4  
\[v_{i}=i\omega u_{i}. \tag{52}\]5  
where an expression for \(P^{air}\) can be found by using Equations (57) and (58) to find that:  
\[v_{i}=i\omega u_{i}. \tag{52}\]6  
Although stress components are expressly used in the boundary conditions detailed in this paper, it is understood that they describe traction continuity. Equations (59)-(64) represent the traction and displacement continuity conditions indicative of a fluid-solid interface. We assume that \(B^{(+L)}=1\), \(B^{(-L)}\), \(B^{(+SV)}\), \(B^{(-SV)}\), \(B^{(+air)}\), and \(B^{(-air)}\) can be calculated as before by using the continuity conditions in Equations (59)\(-\)(64). \(B^{(+air)}\) and \(B^{(-air)}\) in this case would correspond with the amplitudes of the longitudinal waves at the top and bottom of the plate. All of the amplitudes were normalized by dividing each by:  
\[v_{i}=i\omega u_{i}. \tag{52}\]7  
Equations (38) and (58) can then be used to plot the wave structure, and verify that the interface conditions are satisfied. Since it is now possible to calculate the amplitude of the acoustic leakage, every point of the Lamb wave dispersion curve can be input to determine the ideal guided wave mode for maximum (or minimum) acoustic leakage. Figure 11 shows sample calculations for a 1-mm aluminum plate. The color axis of Figure 11 is the average amplitude of acoustic leakage from Lamb waves:  
\[v_{i}=i\omega u_{i}. \tag{52}\]8Assuming there is a direct correlation between the strength of acoustic leakage and the amplitude of the acoustic wave coupled to the Lamb wave, Figure 11 suggests several things:  
* the A0 mode can have a lower amplitude of acoustic leakage than the S0 mode (when normalized by the amplitude of the wave structure);
* the amount of acoustic leakage is not mode-dependent;
* comparisons to Figure 8 suggest a potential relationship between the dominant partial waves present and the amplitude of the acoustic leakage;
* when \(c_{p}=c_{L}\), the leakage of the symmetric modes shown in Figure 11 agrees with the previous analysis by [PERSON] [4](Section 8.7.1), which attributes the lack of leakage to the lack of out-of-plane displacement at that surface. Perhaps most interesting is that the maximum acoustic leakage is concentrated in Region 2 (Figure 1b) of the dispersion curves, while Region 1 (Figure 1a) has noticeably less acoustic leakage. A possible explanation for why this would be is that the guided wave modes in Region 1 (Figure 1a) are composed of surface partial waves, which are inherently coupled to the interface. In Region 2 (b), bulk shear vertical partial waves propagate through the thickness of the plate. Incident upon the aluminum-air interface, this can be modeled as an oblique incidence problem to determine the reflection and transmission characteristics, as shown in Figure 12 where \(\theta_{i}\) is the incident angle of a shear vertical bulk wave. \[v_{i}=i\omega u_{i}. \tag{52}\]9 Amplitude of the longitudinal partial waves in the air half-space after normalizing and averaging the magnitude for various points on the Lamb wave dispersion curves for a 1-mm thick aluminum plate.