## 8 Leaky Characteristics of Lamb Waves  
Lamb waves travel through a traction-free plate. Leaky Lamb waves are the waves that propagate through the same waveguide when the traction-free boundaries are replaced with fluid half-spaces. The theoretical literature on leaky Lamb waves focuses on the calculation of dispersion curves and determining how these curves differ from the conventional Lamb wave dispersion curves [15; 16]. The focus of this section is not this, but rather the leaky characteristics of Lamb waves, assuming there is no change in the dispersion curves. That is, the dispersion curves of a plate surrounded by a vacuum will be used as an approximation for the dispersion curves of a plate surrounded by air. This is a universal standard practice. The goal will be to use the partial-wave method to determine the characteristics of the acoustic leakage into the fluid from the Lamb wave travelling in the plate waveguide. Before continuing, it should be stated that for fluids, the Eulerian form of the balance of linear momentum for acoustics is:  
\[\rho\left[\frac{\partial}{\partial t}v_{i}+v_{j}\left(\frac{\partial}{ \partial x_{j}}v_{i}\right)\right]=-\frac{\partial}{\partial x_{i}}P \tag{51}\]  
This complicates the velocity-pressure relationship of the acoustic fluid. To simplify this, it is assumed that the displacement is time-harmonic, which makes the particle velocity:  
\[v_{i}=i\omega u_{i}. \tag{52}\]  
Figure 9: \(q\) value which is indicative of the phase difference between the top and bottom partial waves for various points on the Lamb wave dispersion curves for a 1-mm thick aluminum plate. The color axis is constrained to \(+1\) and \(-1\). It is also assumed that the displacement is spatially harmonic and longitudinal in polarization such that:  
\[u_{i}(x,y,z,t)=U_{o}\hat{k}_{i}e^{-ik_{j}x_{j}+i\omega t}, \tag{53}\]  
where \(\hat{k}_{j}\) is the unit vector denoting the direction of wave propagation. Using Equations (52) and (53) in Equation (51) results in:  
\[i\rho|v_{i}|(\omega-|k_{i}||v_{i}|)=i|k_{i}|P. \tag{54}\]  
The Eulerian form can approximate the Lagrangian form with these assumptions if:  
\[\omega\gg|k_{i}|\omega U_{o}. \tag{55}\]  
Equation (55) can be rearranged to:  
\[\omega\ll\frac{c_{p}}{U_{o}}. \tag{56}\]  
If it is assumed that the phase velocity is around \(10^{3}\) m/s and that displacements are about \(10^{-6}\) m, then for frequencies much less than \(10^{9}\) rad/s, the Eulerian form should approximate the Lagrangian form. As a result, the simpler pressure-velocity relationship:  
\[\rho c_{L}i\omega U_{o}=P, \tag{57}\]  
which was derived from the Lagrangian form of the balance of linear momentum for fluids assuming a time-harmonic displacement, can be used for the following example. As an acoustic wave in a fluid, the slowness curve can be plotted as a circle with a radius equal to \(1/c_{L}^{air}\). Plotted with respect to aluminum's slowness curve, the order of magnitude difference in radii makes all but the very dispersive part of the A0 mode, at very low frequencies, emit acoustic bulk waves (see Figure 10). At these low frequencies, the A0 mode is highly dispersive, and has a phase velocity lower than the acoustic wave speed in air. The comparison of slowness curves from various media, which are shown in Figure 10, is similar to an example in [PERSON] [2], but that example discussed the electromagnetic emission from a piezoelectric waveguide. The approach relies on [PERSON]'s law, dictating that the wavenumber (i.e., \(k_{x}\)  
Figure 10: Depiction of the aluminum slowness curves compared with the slowness curve for air. Slowness curves with solid lines denote real-valued \(k_{z}\) solutions, and dotted lines denote imaginary-valued \(k_{z}\) solutions (figure not to scale). should be equal at the interface between media. However, if [PERSON]'s law were to be used directly, complex-valued angles (i.e., \(\theta\)) would need to be used.