# Calculation of Guided Wave Dispersion Characteristics Using a Three-Transducer Measurement System
## 1 Introduction  
Guided waves have been extensively investigated in the recent years as a nondestructive testing (NDT) technique because of their advantages compared to ultrasonic testing (UT) inspection. The key benefit of this technology is the ability to interrogate the entire thickness of thin-walled structures over large areas from a single location. During guided wave propagation, waves experience dispersion that distorts the wave shape as the wave propagates, due to a dependence of velocity on frequency [1; 2]. Dispersion curves show the relation of phase and group velocities against frequency, for a particular geometry and material. Determination of dispersion curves is important for any guided wave application; accurate dispersion curves enable wave modes of received wavepackets to be distinguished, specific wave modes to be cancelled to clean the acquired signal, the propagation of wave modes in a particulardirection using phased arrays [3] and the spatial location of damage in the structure to be determined based on time-of-flight measurements [4]. Guided waves are being used in commercial products, for instance to detect wall thickness losses due to corrosion for pipeline inspection in the oil and gas industry. These commercial products generate a unique wave mode which propagates along the structure avoiding the creation of multimodal excitations, as this would increase the complexity of the analysis of the signals [5; 6]. Problems with the application of guided waves occur when the material properties or thickness of the structure to be inspected are unknown due to imprecise records, commercial confidentiality or manufacturing and maintenance uncertainty. This is a particular problem when evaluating composite structures, like wind turbine blades, where elastic constants are not often available. To inspect this kind of structure in an industrial environment, it becomes impracticable as currently there are no means of generating the dispersion curves for such situations. Therefore, there is a need to be able to calculate in situ the phase and group velocities at frequencies of interest in a quick and reliable way without requiring prior knowledge of material properties or thickness. There has been much work published on the determination of dispersion curves describing many analytical, numerical and experimental methods. For relatively simple structures like plates or pipes, dispersion curves can be predicted using the commercially available software Disperse(r), which uses the global matrix method [7]. Other methods, such as the transfer matrix method [8; 9] or pseudospectral collocation method [10], have been utilized to determine dispersion curves. Finite element (FE) [11] and semi-analytical finite element [12; 13] methods have been also used. All these numerical and analytical methods require knowledge of material properties and thickness of the structure to determine the dispersion curves. Experimental methods do not suffer this limitation, since guided wave data is directly acquired from the inspected structure. The most widely used experimental technique to measure dispersion curves is the 2D fast Fourier transform (FFT) [14]. This signal processing technique requires the acquisition of many signals along the wave propagation direction in order to carry out a double FFT in time and space. The result is a wavenumber-frequency matrix, which can be rearranged to give the phase velocity dispersion map. The acquisition of the signals from all required locations is manually prohibitive; therefore, a laser scanning vibrometer (LSV) is commonly used to automatically obtain the signals from preselected points. This device is highly sensitive and bulky, being restricted to controlled areas like laboratories [15] and difficult to use in situ. Other experimental techniques have been proposed recently. [PERSON] and [PERSON] [16; 17] presented a noncontact technique using an air-coupled transducer (ACT) to generate the wave mode on the plate and an LSV to acquire the propagating mode.