## 2 Lamb Waves  
Lamb waves are traction free surface waves that are induced in thin plates. There are two main types of modes of Lamb wave, the symmetrical (S) modes which can be represented by cosine functions and the asymmetric (A) modes which can be represented by sine functions. A diagrammatic presentation of these two wave modes are presented in Figure 1. The full numerical solutions are presented in [16]. Theoretically there are an infinite number of number of wave modes [17]. Though high order modes have shown potential for monitoring structures, particularly for small defects in their microstructure, their low amplitudes [18] and higher frequency content mean that they experience greater attenuation limiting their use in SHM systems [19]. It is for this reason that for monitoring larger structures the S\({}_{0}\) and A\({}_{0}\) modes are only typically excited in an SHM system to reduce the number of sensors required and limit the processing costs. Lamb waves exhibit a phenomenon known as dispersion where the velocity of the wave through a plate is a function of its frequency [20]. Typically, the symmetrical mode has a higher wave velocity than the antisymmetric mode of the same order. Due to velocity and wavelength being a function of the frequency of a Lamb wave, excitation frequency is an important consideration for pulse-receive type SHM systems. There have been extensive studies carried out on the behaviour of Lamb waves in plate like structures [21; 22]. A comprehensive experimental study supported by mathematical prediction of Lamb wave behaviour in metallic plate-like structures with adhesively bonded joints was performed by [PERSON] [23]. This work investigated the Lamb wave interaction and transmission through lap-shear joints. Mode conversion was studied as the primary phenomena to determine the condition of the joint. This work goes some distance towards determining appropriate modes and frequencies to be used for inspecting the condition of adhesively bonded lap shear joints. However, it only considered higher order wave modes for detecting the presence of disbonds [23]. The mechanics and theory behind the reflection and transmission of plate waves by vertical stiffeners was presented in [24]. This work highlights the complexity of Lamb wave interaction with vertical stiffeners. As Lamb waves interact with a stiffener some of the wave energy is transmitted through the stiffener while a percentage of the wave energy is reflected by the stiffener back towards the source and a small amount is attenuated by the stiffener. As a Lamb wave interacts with an adhesively bonded stiffener, it is possible for mode conversion to also occur due to changes in thickness and hence boundary conditions. This was numerically investigated using a boundary element method and further experimentally investigated by [PERSON] [25]. [PERSON] demonstrated that changes in thickness, particularly through a joint or a change in thickness due to a defect can result in different guided wave behaviours including reflection, transmission and mode conversion. [PERSON] and [PERSON] [26] also experimentally investigated the mode conversion phenomenon of Lamb waves interacting with a delamination in a composite panel based on a working hypothesis derived from the work of [PERSON] _et al._[27]. The work demonstrated that the diffraction effect exhibited by interaction with the defect did indeed induce mode conversion. It is also possible for mode conversion to occur at other  
Figure 1: A diagrammatic representation of symmetrical and antisymmetric wave modes. changes in acoustic impedance [28] such as a change in material (_i.e._, interaction with adhesive layer). Changes in wave mode can cause issues with the detection of Lamb waves, in particular for systems that primarily detect one wave component. [PERSON] _et al._[29] numerically investigated the interaction of the fundamental antisymmetric mode with a bonded stiffened section made from glass fibre reinforce plastic.