## 2 Theoretical Background of the Leaky Lamb Wave Propagation and Generation  
Lamb waves were first discovered by Sir [PERSON] in 1917 [4]. Lamb waves are comparable to resonances resulting from the superposition of longitudinal and shear wavesin thin plates. In free plates, Lamb waves can travel long distances that allow them to be a useful tool for long-range inspection. The Lamb modes can be separated into two categories: symmetrical (S) and antisymmetric (A). This denomination is based on the symmetrical nature of the displacement profile parallel to the surface of the plate. An infinite number of harmonics of the symmetrical and antisymmetric modes exist [6]. When a fluid surrounds the plate, a leaky attenuation appears as explained by [PERSON] [5]. The energy of the Lamb waves is converted into compressional waves in the liquid as shown in Figure 1. Each Lamb wave mode is associated with an angle of re-emission (and by reciprocity to an angle of incidence, as explained later). The dispersion equations that govern respectively symmetrical and antisymmetric leaky Lamb wave modes are recalled [6]:  
\[\frac{\tan\left(\frac{\frac{\sigma\sigma}{2}}{2}\right)}{\tan\left(\frac{\frac {\sigma\sigma}{2}}{2}\right)}+\frac{4k^{2}pq}{\left(q^{2}-k^{2}\right)^{2}}-i \frac{\rho_{0}k_{T}^{4}p\tan\left(\frac{\sigma\sigma}{2}\right)}{\rho\sqrt{k_{ 0}^{2}-k^{2}\left(q^{2}-k^{2}\right)^{2}}\tan\left(\frac{\sigma\sigma}{2} \right)}=0 \tag{1}\]  
\[\frac{\tan\left(\frac{\sigma\sigma}{2}\right)}{\tan\left(\frac{\sigma\sigma}{ 2}\right)}+\frac{4k^{2}pq}{\left(q^{2}-k^{2}\right)^{2}}+i\frac{\rho_{0}k_{T}^ {4}p}{\rho\sqrt{k_{0}^{2}-k^{2}\left(q^{2}-k^{2}\right)^{2}\tan\left(\frac{ \sigma\sigma}{2}\right)}}=0 \tag{2}\]  
where \(i\) is the imaginary unit. \(e\) is the thickness of the plate. \(p^{2}=k_{L}^{2}-k^{2}\), \(q^{2}=k_{T}^{2}-k^{2}\), \(k_{L}=\frac{2\pi f}{c_{L}},k_{T}=\frac{2\pi f}{c_{T}}\), \(k_{0}=\frac{2\pi f}{c_{0}}\), \(k\) (the wave number) and \(f\) (the frequency) are the solutions. \(k_{L}\) and \(k_{T}\) are respectively the longitudinal and transversal wave numbers in the plate. \(c_{L}\) and \(c_{T}\) are respectively the speed of the longitudinal and transversal wave in the plate. \(c_{0}\) is the speed of waves in the fluid. \(\rho\) is the density of the plate, and \(\rho_{0}\) is the density of the fluid. One can notice that the real part of the equations corresponds to the dispersion equations of a free plate [4]. The imaginary part corresponds to the fluid-structure interaction and is proportional to the ratio \(\rho_{0}/\rho\). The wave number solution of these equations is a complex number \(k=k^{\prime}+ik^{\prime\prime}\). Its real part \(k^{\prime}\) represents the propagative properties of the wave, and its imaginary part \(k^{\prime\prime}\) represents the leaky attenuation by re-emission in the fluid. The duets \((k,\ f)\) solutions of the dispersion equations are plotted in Figure 2a regarding the real part of \(k\) and in Figure 2b regarding the imaginary part of \(k\). In this work, the plate is 7.8 mm thick and made of stainless steel. The density of the plate is \(\rho=7950\) kg\(\cdot\)m\({}^{-3}\), and the celerities of the longitudinal and transversal waves are respectively \(c_{L}=5738\) m\(\cdot\)s\({}^{-1}\) and \(c_{T}=3143\) m\(\cdot\)s\({}^{-1}\). The density of the water is \(\rho_{f}=1000\) kg\(\cdot\)m\({}^{-3}\), and the velocity of the compressional wave in the water is \(c_{0}=1490\) m\(\cdot\)s\({}^{-1}\). On the one hand, a normalization on the frequency is performed because the dispersion equations are invariant regarding the product frequency \(\times\) thickness. On the other hand, the wave number and the attenuation are inversely proportional to the thickness, and a normalization is also performed. Figure 1: Generation of a compressional wave in the liquid by a leaky Lamb wave in the plate [30]. At a given frequency \(\times\) thickness, several modes may coexist and propagate at the same time.