# Rail Diagnostics Based on Ultrasonic Guided Waves: An Overview
## Appendix B Characteristics of Guided Waves in Rails  
Modeling the rail as a waveguide is not easy because of its complicated profile. For this reason, the accurate computation of the dispersion characteristics of guided waves traveling through rails represents a difficult problem, whose solutions are based on finite elements methods [7]. A frequently used method is the _semi-analytical finite element_ (SAFE) method.  
The dispersion curves of rails exhibit _mode repulsion_ and _mode crossing_: these phenomena cannot be easily distinguished. The meaning of these two terms can be understood by analyzing the dispersion curves of a waveguide. When two modes (curves) approach each other, they can cross (mode crossing) or suddenly diverge without crossing (mode repulsion). The mode shapes of rails, having a symmetric profile (e.g., see Figure 1), are either symmetric or anti-symmetric. A mode is symmetric when its energy is distributed symmetrically over the waveguide cross-section, i.e., it is equally distributed between field side and gauge side in the case of rails. A mode is antisymmetric when its energy is concentrated non-symmetrically in the waveguide, i.e., on one side only if rails are considered. It has been shown that: (a) symmetric and antisymmetric modes can cross each other; (b) the modes within symmetric and antisymmetric families do not cross each other [84]. Furthermore, the introduction of even a small asymmetry in the waveguide shape produces repulsion forces that prevent mode crossings. This information is useful in the selection of a propagation mode to be employed for rail integrity inspection. Examples of wavenumber vs. frequency curves and of the concentration of energy for different modes are shown in Figure 10.  
The energy of different (symmetric or antisymmetric) modes is concentrated in the head, web, or foot of the rail. This explains the possibility of locating defects in a specific portion of the rail cross-section by selecting the most appropriate mode [7].  
Some concerns can originate from the influence of ties, fasteners, and absorbing pads. It has been found that, at high frequencies (such as those of ultrasonic guided waves), their presence does not influence dispersion curves, but only decay rates [85]. Moreover, experimental studies have led to the conclusion that the most effective frequency interval for long range wave propagation along railway tracks is between 20 and 40 kHz [86].  
The three main technical problems observed in the use of guided ultrasonic waves are [7]:  
1. _Dispersion_--If different modes are excited, they travel at different (frequency-dependent) velocities in both directions. Consequently, each mode takes a different time to travel along the employed waveguide, so compromising spatial resolution (for instance, a given reflector can generate multiple echoes). This problem can be mitigated by resorting to dispersion compensation.
2. _Coherent noise_--This noise is observed in the same frequency band of the signal of interest. It is due to: (a) the excitation and reception of unwanted modes; (b) the transmission of waves in the wrong direction along the waveguide and the reception of echoes from that direction. Therefore, mitigating coherent noise requires exciting and sensing only the selected modes. This can be accomplished by using a proper transducer or excitation signal, and by suppressing unwanted modes.
3. _Changes in temperature or in material properties with age_--These changes, even if small, affect the above-mentioned dispersion phenomenon. This problem has to be taken into account when comparing signals received at different instants.