Two-Dimensional Fast Fourier Transform Techniques

In addition to the analytical techniques discussed previously, the dispersion curves for a propagating wave may be obtained from the results of FE modelling or experi- mental measurements. There are several methods to do this, in this work a technique which utilises two-dimensional fast Fourier transforms (2D-FFT) was used [105]. The basis of this technique is that the surface displacement of a wave, propagating

in the x direction, may be described as:

u(x, t) =A(ω)ei(ωt−kx−θ). (4.27)

HereA(ω) is a frequency dependent amplitude andθ is a phase angle. Thus it can be seen that the displacement can be considered to have a spatial and temporal dependence. By applying a 2D-FFT to equation 4.27, a new amplitude is obtained which is a function of frequency and wavenumber:

H(k, f) =

Z Z +∞

−∞

u(x, t)e−i(kx−ωt)dxdt. (4.28)

The peaks in this amplitude function correspond to the propagating wave modes. As such, this technique can be used to obtain dispersion curves with relative ease, even when there are multiple modes present in the signal.

This technique can be very useful for obtaining dispersion curves for waveg- uides with complex geometries, as the surface displacementu(x, t) may be measured experimentally, or obtained from FE modelling, measured at a series of closely spaced points along a line. The spacing of these measurement points must be kept much smaller than the minimum wavelength of interest, again to prevent any alias- ing. The time traces from each of the measurement positions can then be arranged into an array of amplitudes dependent on position and time, effectivelyu(x, t), to which the 2D-FFT may be applied in order to obtain the dispersion curves.

4.3

Conclusions

This chapter has introduced some of the fundamental concepts related to wave propagation. Analytical models of wave propagation in both solid media and fluids have been presented, allowing the more complex case of guided waves to be intro- duced. The concept of dispersive wave propagation has also been discussed for the

case of Lamb waves in thin plates, through analytical modelling. This chapter has also provided an introduction to FE modelling, which will be used throughout this work as well as the ways in which this modelling may be used to study dispersion in waveguides with complex geometries. In the next chapter, the two-dimensional Fourier transform techniques discussed here will be implemented to study dispersion of low frequency ultrasonic waves propagating in a range of rectangular cross-section waveguides.

Chapter 5

Guided Waves In Rectangular

Cross-Section Strips

As discussed in the previous chapter, guided waves with low dispersion, such as the S0 Lamb wave mode at low frequencies, would be ideal for many industrial appli- cations. Additionally the S0 Lamb mode has the majority of its displacement in the direction of propagation which allows more efficient generation of compressional waves in the target fluid. However, the geometric requirements of the waveguide (e.g. the large width) to support Lamb or Lamb-like waves cannot easily be sat- isfied within practical experimental constraints. Low frequencies are required in ultrasonic flow measurement to reduce the effect of attenuation. These frequencies are generally below 1 MHz for gas and 5 MHz for liquid applications, meaning that we have relatively large wavelengths. In this work we are primarily operating around 150 kHz, which typically corresponds to a guided wave wavelength in the order of centimetres for the mode that we wish to use. As a result of this, the width of the waveguide strips considered in this work cannot be neglected and will lead to additional complexity.

Many analytical models have been suggested in order to include the addi- tional boundary conditions. One of the first, and simplest of these models was

suggested by Morse in 1950 [106]. He suggested a modification to the Rayleigh- Lamb equations, including traction free boundary conditions on the two additional boundaries resulting from finite strip width, leading to the following equations:

tan(q∗a) q∗ + 4p∗tan(p∗a)(s2+k2) (s2_+k2_−q∗2₎2 = 0, (5.1) with p∗2 =p2−s2, (5.2) q∗2 =q2−s2, (5.3) s= n+1_2π b, n= 0,1,2..., (5.4)

where the strip width is 2b. It can be seen that the form of this equation is very similar to that of the equation for symmetric Lamb waves, and if the terms is set to 0 then equation 4.19 may be recovered, as the strip width effectively becomes infinite. The range of s values given by equation 5.4, approximately satisfy the zero stress boundary conditions on all of the surfaces. The inclusion of this constant allows the existence of additional modes, which converge around the positions of the Lamb solutions. From these equations it is also clear that increasing the width of the strip, reduces the influence of the width of the strip on wave propagation, as sis inversely proportional to b. This very simple modification was found to agree well with experimental data, with better agreement for wider strips, whena/b was small [106].

Since this early work of Morse, many other analytical models have been developed [107]. These models are, generally, much more complex, predicting even higher numbers of modes to be present, including longitudinal, torsional and bending modes in both the thickness and width dimensions [108]. Due to the complexity of these analytical models the FE techniques introduced in Chapter 4 will be used to study the effects of the waveguide geometry on wave propagation.

Table 5.1: Material parameters used in finite element modelling

Material Density Longitudinal Velocity Transverse Velocity (kgm−3) (ms−1) (ms−1)

Stainless Steel 7890 5790 3100

Air 1.24 343 0