# Rail Diagnostics Based on Ultrasonic Guided Waves: An Overview
## Appendix A Guided Waves  
Any disturbance affecting a portion of an elastic medium propagates through the medium itself in a finite time as a mechanical sound wave (_elastic wave_). An ultrasound or ultrasonic wave is characterized by a frequency greater than 20 kHz. There are two types of ultrasonic waves [81]: _bulk_ (fundamental) waves and _guided_ waves. Bulk propagation refers to waves propagating without any boundary, like in an infinite medium (or in media whose boundaries does not influence wave propagation). Guided waves propagate in bounded media, like plates, rods, or tubes.  
In general, an elastic wave consists of two components, which propagate independently from each other; these components are known as _longitudinal_ and _transverse (shear)_ waves. In longitudinal waves, the variation of the propagating quantity is observed parallel to the propagation direction. In transverse waves, instead, the variation of propagating quantity is orthogonal to the propagation direction. The existence of such waves depend on the elastic properties of the medium in which they propagate [82].  
In a waveguide, interference phenomena arise from the waves bouncing back and forth inside the waveguide itself when impinging on its boundaries. Bouncing produces mode conversion because of reflection and refraction of longitudinal and shear waves. Depending on the angle and frequency, constructive, destructive, or intermediate interference takes place: this leads to hundreds of solutions of constructive interference points and, consequently, to guided wave packets traveling in the waveguide and named _modes_. The interference points can be represented through a _wave velocity dispersion curve_, which relates phase velocity to frequency. Each waveguide has its own set of dispersion curves, and every curve in the set is related to a specific mode of propagation. In the technical literature, dispersion curves are commonly employed to display all the types of waves and modes that can propagate inside a given waveguide. The ability to select, for a given dispersion curve, specific phase velocity and frequency has a significant impact on the penetration power and sensitivity of an ultrasound inspection system based on guided waves [83].  
Another important feature to be taken into consideration is the wave structure of the selected mode, i.e., the _mode shape_. This shows the mode in-plane displacement, out-of-plane displacement, or actual stress distribution that varies across the thickness of the considered waveguide [70]. This knowledge is useful to establish the maximum penetration power of waves in a given structure, or to establish the maximum sensitivity to a defect located in a specific area of a test specimen [83].  
When an ultrasonic wave propagates through a medium, it undergoes attenuation. The waveform and the amplitude of an ultrasonic wave are influenced by a number of factors, including ultrasonic beam spreading, energy absorption, dispersion, nonlinearity, transmission at interfaces, scattering by inclusions and defects, Doppler effect, etc. [81]. Attenuation is a fundamental factor to be always kept in mind when designing ultrasonic guided waves systems, especially if they are required to operate on a long range.  
A finite body can support an infinite number of guided wave modes; these represent the solutions of a differential guided wave problem, defined by the boundary conditions. Specific solutions to guided wave problems are those related to _Rayleigh_, _Lamb_, and _Stonely_ waves. Rayleigh waves are waves on the surface of a semi-infinite solid, Lamb waves are waves of plain strain occurring in a free plate, and Stonely waves are waves that occur at the interface between two media [70].