Ultrasonic beam manipulation and steering can be achieved by carefully controlling the frequency content of the generation signal used to simultaneously excite all the elements within the periodic array. The interference between the waves produced by each individual elements causes preferential propagation of ultrasound with a certain frequency along a particular angular path. The relationship between the angle of propagation and the excitation frequency is given by the diffraction grating equation, and occurs when the spatial periodicity of the array and the wavenumber of the excitation signal overlap in k−space. This conjunction occurs when there is constructive interference of that signal at that particular angle, when the path length differences between elements of the same polarity are equal to an odd integer number of wavelengths. As this condition of constructive interference changes with frequency, ultrasonic beam steering and sector scans can be achieved by simply varying the frequency of the excitation signal. This eliminates the need to be able
(a)
(b)
(c)
(d)
Figure 3.6: The directivity of a six element, 5 mm pitch array that generates SH waves in aluminium (cs = 3111 ms−1), which is being driven by a five cycle tone-
burst signal with a frequency of (a) 300 kHz, (b) 400 kHz, (c) 500 kHz and (d) 600 kHz.
individually control and phase each array element, thereby avoiding the requirement for expensive hardware.
A frequency domain model, based upon the Ewald construction used in ele- mentary diffraction modelling, was used to fully describe the frequency dependent directivity characteristics of the array. This model, unlike the diffraction grating equation, allows for the full directivity of the array to be calculated, not just the peak position of the ultrasonic beam. The model illustrates how the array’s abil- ity to strongly focus the ultrasonic beam changes with angle, with the beam being tightly focused for higher frequency excitations. The model is also very flexible, as it can trivially extended to two dimensional arrays and can handle known anisotropy in the velocity profiles of the generated wavefront [159, 168].
However, there is an inherent compromise between the axial and lateral resol- ution of frequency steered arrays, due to the principle of time-frequency duality. In addition, whilst a wide angular range (20◦ ≤ θ ≤ 90◦) can be scanned through this method, it requires a number of measurements, with the excitation frequency being swept through the appropriate range. The method, like the phased array methods described in section 2.3.3, can be relatively time consuming in acquiring the relevant ultrasonic data. A new approach, described in Chapter 4, shows how the periodicity of the array can be used to generate a wavefront that extends over the same angular extent. Consequently, the pulsed approach can inspect the same area as the frequency steered arrays, but only requires a single measurement. The pulsed wavefront also retains the frequency dependent directivity, which can again be utilised to locate any defects that may scatter this wavefront.
Chapter 4
Analytic Model of the Pulsed
Array Wavefront
4.1
Overview
The physical principles of a new and alternative way to operate a linear array of wave emitters is described in this chapter. The method involves driving all of the elements simultaneously with a pulse that must have a carefully controlled frequency content. However, unlike the frequency steered arrays shown in Chapter 3, the ex- citation signal is designed to contain a large range of frequencies. If the bandwidth of the generation signal is correctly chosen, the subsequently generated wave has a number of interesting characteristics. Firstly, the wavefront extends in two dimen- sions over a large range of solid angles, meaning that it samples a large volume of the propagation media with a single excitation. However, perhaps more interestingly, the wavefront also exhibits a variation in frequency as the propagation direction is changed. This effective frequency change occurs monotonically over a large an- gular range, and is smooth and continuous in nature [102]. This smooth variation means that there is a unique transformation between wave frequency and angle of propagation. By measuring the local frequency of the wave, the angle of propaga-
tion, relative to the centre of the array, can be determined. As all of the elements are pulsed simultaneously with the same excitation signal, there is a reduction in the complexity of the hardware required to operate the array. This reduction in complexity is particularly noticeable when compared to the hardware required to control modern phased array systems.
In a two-dimensional measurement using a one-dimensional array, the frequency variation of the wavefront can be used to locate the position of a defect with a single measurement, whereas with other methods, the defect position can only be localised to a circle (or an ellipse, depending upon whether or not the transducers are co- located), or located after many measurements (Phased array methods [1, 124, 141]). If the wave, as it travels through the medium, is scattered by a discontinuity, the information from this wave is sufficient to localise the discontinuity that caused this signal. The radial distance between the discontinuity and the array can be calculated from the time of flight measurement of the wave. By measuring the frequency of the wave, the angle at which the wave was scattered can be determined. So, with a single wavefront, a large volume of the medium can be inspected, with the location of any discontinuities being determined by combining of the time of flight measurements and the frequency of the scattered wave.
However, in order to utilise the pulsed array to locate any discontinuities, a quantitative knowledge of how exactly the frequency of the wavefront varies, and how it depends on angle, is required. The pulsed array, in a simplified form, can be modelled using an analytic approach.