# Adhesive Single-Lap Joint Evaluation Using Ultrasound Guided Waves
## 2 Theory  
### Ultrasound Guided Wave Propagation in a Single Plate  
Lamb waves are a type of guided ultrasound wave existing in plate structures placed in a vacuum that can propagate relatively long distances. They are very useful to detect disbonds, corrosion, and delaminations efficiently [53; 54; 55; 56]. If plates are immersed in water, these waves are called leaky Lamb waves, due to the energy leakage into the surrounding fluid. The well-known dispersion equations that govern symmetrical (Equation (1)) and antisymmetric (Equation (2)) leaky Lamb wave modes were introduced by [PERSON] [53],  
\[\frac{\tan(qh)}{\tan(qh)}+\frac{4k^{2}pq}{\left(q^{2}-k^{2}\right)^{2}}-i\frac {\rho_{0}k_{T}^{4}\ p\tan(qh)}{\rho\sqrt{k_{0}^{2}-k^{2}}{\left(q^{2}-k^{2} \right)^{2}}}=0, \tag{1}\]  
\[\frac{\tan(ph)}{\tan(qh)}+\frac{4k^{2}pq}{\left(q^{2}-k^{2}\right)^{2}}+i\frac {\rho_{0}k_{T}^{4}\ p}{\rho\sqrt{k_{0}^{2}-k^{2}}{\left(q^{2}-k^{2}\right)^{2} }\tan(qh)}=0, \tag{2}\]  
where \(i\) is the unit imaginary number, \(h\) is half the thickness of the plate, \(p=\sqrt{k_{L}^{2}-k^{2}}\), \(q=\sqrt{k_{T}^{2}-k^{2}}\), where \(k_{L}=\frac{\omega}{c_{L}}\), \(k_{T}=\frac{\omega}{c_{T}}\), \(k=\frac{\omega}{c_{P}}\), \(k_{0}=\frac{\omega}{c_{0}}\), \(\omega\) is the angular frequency, \(c_{L}\) is the longitudinal velocity in the plate, \(c_{T}\) the transversal velocity in the plate, \(c_{p}\) is the phase velocity of a certain propagation mode, and \(c_{0}\) is the velocity in the surrounding fluid. The plate and fluid densities are \(\rho\) and \(\rho_{0}\), respectively. The transcendental Equations (1) and (2) give rise to an infinite number of solutions that correspond to the propagation modes. The imaginary part of each equation represents the fluid influence. If this is not present, the equations characterize the plate in vacuum. The leakage of the fluid causes attenuation, which is highly dependent on the ratio of in-plane and out-of-plane displacements at the plate surfaces for a certain mode. For a particular propagation mode the phase velocity is complex, \(c_{p}=c_{pr}+ic_{pi}\), which gives rise to a complex wavenumber, \(k=\frac{\omega}{c_{p}}=\frac{\omega}{c_{pr}+ic_{pi}}=k_{r}+ik_{i}\)[57]. The real part \(k_{r}\) represents the propagation properties of the wave and \(k_{i}\) is related to leakage attenuation. Considering that usually \(c_{pr}\gg c_{pi}\), after some manipulation, the complex wavenumber can be represented as  
\[k\cong\frac{\omega}{c_{pr}}-i\frac{\omega c_{pi}}{c_{pr}r^{2}}. \tag{3}\]  
The absolute value of the imaginary part of Equation (3) represents an exponential decay, known as the attenuation coefficient, \(\alpha\) (Np/m). The solutions of the dispersion equations for a plate in vacuum are very similar to the real part of the solutions for an immersed plate, because it has been shown that the surrounding fluid gives rise to very small variations in the wave velocity [58]. The particle displacement variation as a function of the plate thickness for each propagation mode is very important. If the plate is immersed and long-range propagation is needed, then a low out-of-plane displacement should be used to maintain low attenuation values, due to leakage to the fluid. The displacements can be obtained using the method of potentials and the boundary conditions at the plate surface.