# Partial Energy Transfer Model of Lamb Waves Scattering in Materially Isotropic Waveguides
## 1 Introduction  
The scattering phenomena of Lamb waves [1, 2] on various types of discontinuities [3, 4] in elastic waveguides remains an attractive topic with a direct relationship to numerous applications in nondestructive testing or structured health monitoring. Due to the relatively high interest among the scientific community within this area, a number of experimental [5, 6, 7], numerical [8], and semi-analytical [9, 10] approaches, in addition to hybrid [11] methods that combine the semi-analytical and numerical approaches, have emerged in recent decades to comprehensively describe the scattering process of Lamb waves. Progressive development of computer technology has enabled the use of the finite element method (FEM) or, alternatively, the boundary element method (BEM) [12, 13, 14, 15]. The main disadvantage of these approaches, however, lies in their computational expense; in addition, these approaches do not provide more detailed insights into the scattering process. Subsequently, hybrid methods have been developed that combine FEM/BEM with suitable techniques, such as the complex mode expansion theory [16, 17] and partial wave technique [18], in addition to models that completely eliminate the use of FEM/BEM. A pioneering work within this area is the method published by [PERSON] [19], which addressed the reflection of the Lamb wave modes from the free plate end, and discussed the importance of modes withcomplex and pure imaginary wavenumbers in the computational process. [PERSON] et al. addressed a similar topic using the projection method [20], which was further improved by [PERSON] [21] and [PERSON] [22]. The modal expansion technique became a basis for many other works. [PERSON] et al. [23] used scattering analysis in the case of an isotropic plate containing horizontal delamination, in which special attention was paid to the conversion between Rayleigh and Lamb waves. In 2002, a novel approach was published by [PERSON] et al. [24; 25], which incorporated a mode matching technique in conjunction with the biorthogonality relation [26] and the stress and displacement discontinuities at the corresponding waveguide sections. Using this method, it is possible to explicitly derive the scattering matrix, which links the incident Lamb wave modes with the scattered wave modes. Such a multimodal approach was exploited and further developed by [PERSON], who adopted the mode matching (MM) method for a variety of problems [27; 28; 29; 30]. One of the important benefits of the MM method is its modular characteristic, which allows control of the amount of transmitted/reflected energy for a given waveguide section, and is thus able to simulate the partial contact between the crack faces. The importance of this phenomenon is evidenced by a number of publications in the field of fracture mechanics or ultrasonic testing. [PERSON] et al. [31] analyzed the effect of partial contact of an arbitrary location in the case of penny-shaped cracks, which often occur in metal casting, using the analytical and FEM approaches. These defects, including the given phenomena, may have an inconsiderable effect on the effective elastic and conductive material properties. In 2001, [PERSON] et al. [32] published a study of inclined edge cracks, which were subject to different friction conditions within the crack faces. The results show a considerable effect on the stress intensity factor versus different contact regimes within the crack faces. Another interesting study within this field is that of [PERSON] et al. [33], who presented a theoretical analysis of the crack-face contact effect on the stress intensity factor of various shell structures. Problems arising from partial contact of the crack face are also the subject of a number of publications focused on either conventional ultrasonic testing techniques [34; 35; 36], numerical and experimental methods incorporating the use of Lamb waves [37; 38; 39] or techniques coupled with power spectra subharmonic analysis [40].