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The effect of dispersion on long-range inspection using ultrasonic guided waves
NDT&E International 34 (2001) 1±9
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The effect of dispersion on long-range inspection using ultrasonic guided waves P. Wilcox*, M. Lowe, P. Cawley NDT Laboratory, Department of Mechanical Engineering, Imperial College, Exhibition Road, London SW7 2BX, UK Received 31 January 2000; received in revised form 31 March 2000; accepted 3 April 2000
Abstract The dispersion of ultrasonic guided waves causes wave-packets to spread out in space and time as they propagate through a structure. This limits the resolution that can be obtained in a long-range guided wave inspection system. A technique is presented for quickly predicting the rate of spreading of a dispersive wave-packet as it propagates. It is shown that the duration of a wave-packet increases linearly with propagation distance. It is also shown that the duration of a wave-packet after a given propagation distance can be minimised by optimising the input signal. A dimensionless parameter called minimum resolvable distance (MRD) is de®ned that enables a direct comparison to be made between the resolution attainable at different operating points. Some conclusions are made concerning the resolution of various operating points for the case of Lamb waves in an aluminium plate. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Dispersion; Guided waves; Lamb waves; Long-range testing
1. Background 1.1. Guided waves and non-destructive inspection Much work has been published on the use of Lamb waves and other guided waves for inspection purposes and a comprehensive review of applications may be found in Ref. [1]. Very broadly speaking, the use of guided waves for non-destructive inspection purposes falls into two categories depending on the distance of propagation. Firstly, there are short-range applications, where guided waves are used to obtain information about a specimen that cannot be readily obtained by more conventional means. These areas include the determination of the elastic properties of materials [2,3], the detection of defects near to interfaces such as in the inspection of adhesive joints [4] and air coupled ultrasonic inspection of thin specimens [5]. In these cases, sensitivity is of key importance and generally this is the main criterion for selecting a suitable guided wave mode. The effect of dispersion is relatively unimportant as the propagation distances are small. This paper is concerned with the second area of guided wave applications where the propagation distance is large. These include the detection of delaminations in rolled steel * Corresponding author. Tel.: 144-0171-594-7227; fax: 144-0171-5801560. E-mail address: [email protected] (P. Wilcox).
[6,7] and composites [8], pipeline [9±11] and plate [12] inspection. In long-range applications, the aim is to inspect large areas of a structure rapidly. 1.2. Long-range inspection using guided waves In long-range guided wave testing applications, the guided waves are excited by a short burst of energy (the input signal) applied by a suitable transducer at one location on a structure. The excitation causes a packet of guided waves (the wave-packet) to propagate away from the transducer into the surrounding structure. Then either the same transducer or a second transducer is used to detect signals caused by re¯ections of energy in the wave-packet from surrounding structural features or defects. The problems associated with the use guided waves for inspection purposes are well documented [13]. In summary, multiple modes of guided wave propagation are possible in most structures and these modes are generally dispersive (i.e. their velocities are frequency-dependent). In order to obtain useful data from a guided wave inspection system, it is necessary to selectively excite and detect a single guided wave mode while suppressing coherent noise due to other modes of guided wave propagation. For this reason, the design of the transducer and the input signal are tailored so that the excitation energy is targeted at a single point on a suitable guided wave mode at a suitable frequency. This point is called the operating point.
0963-8695/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0963-869 5(00)00024-4
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Fig. 1. (a) Numerical simulation of the space±time map illustrating the dispersive propagation of the S0 mode in a 1-mm thick aluminium plate when the input signal is a 5-cycle Hanning windowed toneburst with a centre frequency of 2 MHz. Below are numerical predictions from the same model that show the timetraces that would be received, (b) close to the source, (c) 50 mm from the source and (d) 100 mm from the source.
Examples of transducers that may be used to excite and detect guided waves include inter-digital or comb transducers that operate by either piezoelectric [14,15] or electromagnetic mechanisms [16,17]. Alternatively, conventional plane bulk wave transducers may be used in conjunction with a coupling wedge in the angle incidence con®guration [18,19]. Suitable input signals are windowed tonebursts with a precise centre frequency and a limited bandwidth. The reason for using a limited bandwidth input signal is twofold. Firstly, it helps to prevent the excitation of undesired modes at other frequencies and secondly it reduces the effect of dispersion on the propagation of the desired mode [13]. The studies presented here are concerned solely with the effect of dispersion. For this reason, it is implicitly assumed throughout that a suitable transducer can be designed so that single mode excitation can be attained at any operating point on any guided wave mode. The practicalities of actually achieving this are beyond the scope of this paper.
1.3. Manifestation of dispersion effects The effect of dispersion is that the energy in a wavepacket propagates at different speeds depending on its frequency. This manifests itself as a spreading of the wave-packet in space and time as it propagates through a structure. This is illustrated in Fig. 1(a), which shows the propagation of the S0 Lamb wave mode in a 1 mm thick aluminium plate after excitation with a 5-cycle Hanning windowed toneburst with a centre frequency of 2 MHz. This type of graphical representation of dispersion will be referred to as a space±time map and the means by which it is calculated is summarised in the following section. The xaxis on the space±time map represents time measured from the moment the excitation signal starts and the y-axis is the distance of propagation measured from the excitation location. The greyscale level indicates the value of a suitable quantity that is affected by the passage of the guided wave. In this case the quantity used is the out-of-plane displacement of the surface of the plate. The propagation and
P. Wilcox et al. / NDT&E International 34 (2001) 1±9
spreading of the wave-packet in space and time can be clearly seen on the space±time map. The striped effect within the area of the wave-packet is due to the displacements from individual wave peaks and troughs. The points at which its envelope amplitude falls below a certain threshold level de®ne the boundaries of the wave-packet, and any points outside this are coloured white in the space±time map. The de®nition of this threshold level will be discussed later. A propagating wave-packet is detected by a transducer that is able to monitor a suitable quantity associated with the wave-packet, such as the out-of-plane surface displacement, as a function of time (a time-trace). A time-trace is a crosssection parallel to the time axis through the space±time map of that parameter. Example time-traces that would be measured after propagation distances of 0.1, 50 and 100 mm are shown in Fig. 1(b), (c) and (d). In these timetraces, the effect of dispersion appears as an increase in the duration of the wave-packet in time and a decrease in its amplitude. 1.4. Why dispersion is undesirable The effects of increasing wave-packet duration and decreasing amplitude due to dispersion are both undesirable in long range guided wave testing. The spreading of a wave-packet in space and time reduces the resolution that can be obtained. This problem is frequently encountered when attempting to detect defects in close proximity to structural features, such as welds. In such a case, the defect can only be reliably detected if its re¯ection can be resolved from that due to the feature. The reduction in amplitude of a dispersive wave-packet reduces the sensitivity of the testing system. Although, the studies presented here are primarily concerned with the increase in temporal duration of a wave-packet, the decrease in wave-packet amplitude can be estimated by using energy conservation. On this basis and neglecting other losses, it can be assumed to a ®rst approximation that the amplitude of a wave-packet will decrease in proportion to the square root of the increase in its duration. 1.5. Overview The ®rst part of this paper describes a simple method for quantitatively predicting the rate of wave-packet spreading due to dispersion at any operating point on the dispersion curves for a particular structure. The second part of the paper explains how the practical effect of dispersion at different operating points can be compared. The procedures described in this paper may be used for any structure to which long-range guided wave inspection techniques can be applied. The list includes metallic platelike structures such as pressure vessels, multi-layered structures such as adhesive joints and cylindrical systems such as pipes. The only requirement is that the dispersion curve data (phase and group velocity) is available for the mode or
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modes over the frequency range of interest. For the purposes of illustrating the techniques proposed in this paper, the simple case of Lamb waves in a 1-mm thick aluminium plate in vacuum will be used as an example. On occasions where it is necessary to consider a single operating point, the S0 Lamb wave mode at a frequency of 2 MHz will be used.
2. Modelling dispersive propagation 2.1. Numerical prediction of dispersive propagation using Fourier decomposition In order to make quantitative measurements of dispersive wave propagation, it is necessary to be able to model how a guided wave-packet excited by an arbitrary input signal propagates when dispersion is present. In particular, it is desirable to be able to predict the rate at which the packet spreads out with propagation distance. An obvious way to do this is to make use of the phase velocity (or wavenumber) dispersion data available for the structure and then to use a Fourier decomposition technique to make an exact numerical prediction of the propagation. This technique was used to generate the space±time map and the time-traces shown in Fig. 1 and it is summarised below. Consider the case when a suitable guided wave transducer (e.g. an angle incidence device or an inter-digital transducer) is used to excite guided waves in a structure. For the purposes of this study, it is assumed that the transducer is ideal, in that it excites only one guided wave mode and in only one direction. Cartesian axes are de®ned using the same convention as Viktorov [18] for straight crested waves with wave propagation in the positive x-direction and the z-axis normal to the plane of the structure. The wave-crests are orientated parallel to the y-axis and these, as well as the structure and the transducer, are assumed to extend inde®nitely in the y-direction. The origin of the xaxis is de®ned so that the front edge of the transducer is located at x 0: The transducer is supplied with an electrical signal of ®nite duration V(t), which is converted into acoustic energy. This energy propagates away from the transducer in the positive x-direction as a single guided wave mode. It is assumed that at the transducer position, x 0; the variation of a parameter in the plate, such as out-of-plane surface displacement u with time t is directly proportional to V(t). Although this assumption is not strictly true, it is the only reasonable approximation that can be made without knowledge of the transducer characteristics and the excitability function [20] of the guided wave mode. The known function u(t) can be considered as being a `slice' taken at x 0 through the space±time map of the function u(x,t), which describes the propagation of the guided wave mode in space x and time t. If the phase velocity dispersion curve data for the system is available, then
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The boundaries of the wave-packet can then be computed using one of the de®nitions given below. 2.2. De®nition of the duration of a wave-packet
Fig. 2. The relative envelope amplitude of the wave-packet from the space± time map in Fig. 1(a) plotted using two different de®nitions for the reference level: (a) reference (0 dB level) is the peak amplitude of the signal at distance equal to zero; (b) reference is re-calculated at each distance as the peak amplitude of the wave-packet at that distance.
u(x, t) can be calculated at any other point in time and space in the manner described below. First u(t) is Fourier transformed to obtain its frequency spectrum: U v
1 Z1 u t e2iax dt 2p 2 1
1
where v is angular frequency. The value of u(x,t) associated with an individual spectral component of U(v )is given by the wave equation: U v ei k vx2vt
2
where k(v ) is the circular wavenumber that may be obtained from the phase velocity n ph(v ) dispersion curve data by: k v
v nph v
3
The overall value of u(x, t) due to the propagation of the input wave-packet u(t) is given by the integration of the contributions from all the spectral components of U(v ). To perform this integration, use is made of the fact that because u(t) was real, U 2v U p v: Hence the integration over the negative frequency range is equal to that over the positive frequency range and the expression for u(x, t) may be written: Z1 U v ei k vx2vt dv 4 u x; t 2 Re 0
Hence the time signal that would be obtained if u(x, t) was measured at any location can be computed. The Fourier decomposition method was used to predict the space±time map and the time-traces shown in Fig. 1. For the purposes of predicting the boundaries of the wave-packet, it is more convenient to work with its envelope and this can be readily computed using the Hilbert transform method, whereby Eq. (4) is replaced by: Z 1 5 Envelope u x; t 2 U v ei k vx2vt dv 0
To quantify dispersion, it is necessary to be able to measure the duration of a wave-packet and this requires a means of de®ning whereabouts in time a wave-packet begins and ends. The easiest way in which to do this is to de®ne the duration by the points in time at which the envelope of the wave-packet falls below a particular reference level. The problem that then arises is how to de®ne the reference level and the two space±time maps that are plotted in Fig. 2 illustrate this. These are both for the same example as that used in Fig. 1, except that in this ®gure the greyscale indicates the amplitude of the signal envelope rather than the individual wave peaks and troughs. In the space±time map shown in Fig. 2(a), the amplitude over the entire space±time map is referenced to a ®xed value, and this value (i.e. 0 dB) is taken as being the peak level of the wave-packet envelope at the source. If the duration of a wave-packet is de®ned in this manner then although it initially increases, it will, after a suf®cient propagation distance, begin to decrease and will ultimately reach zero. This can be seen in Fig. 2(a), where the contours of the greyscale form closed lobes. The reason for this is that the increase in wave-packet duration must be accompanied by a decrease in its amplitude as noted earlier. For the purposes of predicting the rate of wave-packet spreading due to dispersion, this de®nition is clearly not suitable since the rate will be dependent on the distance of propagation. A more suitable de®nition of the reference level yields the space±time map shown in Fig. 2(b). Here the amplitude at a particular propagation distance is referenced to the peak value of the wave-packet envelope at that distance. In this way, the measured duration of the dispersive wave-packet monotonically increases as it propagates. It can be seen from Fig. 2(b) that this increase is actually a linear function of the propagation distance. This is a useful fact and forms the basis of the technique described below for predicting dispersive propagation. 2.3. Prediction of dispersive propagation using group velocity dispersion curves The Fourier decomposition procedure described previously is an inef®cient and time-consuming method of obtaining the rate of spread of the wave-packet and it is not suitable for the type of iterative calculation that will be described later in this paper. This problem motivated the development of an alternative technique for predicting the duration of a received wave-packet after an arbitrary propagation distance. It was shown above that a dispersive wave-packet spreads out linearly in space and time as it propagates if a suitable de®nition for the boundary of the wave-packet is used. This means that the ends of the envelope of the dispersive wave-packet can be regarded as
P. Wilcox et al. / NDT&E International 34 (2001) 1±9
Fig. 3. Group velocity dispersion curves for a 1-mm thick aluminium plate in vacua. The vertical lines represent the bandwidth (based on the 220 dB points of the spectrum) of a 5-cycle Hanning windowed toneburst with a centre frequency of 2 MHz. The portion of the S0 mode that falls within this bandwidth is emboldened and the extrema of its group velocity are indicated.
propagating with two different velocities, and it is this difference in velocities that causes the envelope to become longer. If these two velocities and the duration of the wavepacket at one point in space are known then the duration of the wave-packet at any other point in space can be calculated. The method described here takes the values for these two velocities from the group velocity dispersion curve for the guided wave mode. Consider a wave-packet as it passes a point in a structure.
Fig. 4. Comparison between group velocity and Fourier decomposition methods for predicting the boundary of the wave-packet when the S0 Lamb wave mode is excited in a 1-mm thick aluminium plate by a 5cycle Hanning windowed toneburst with a centre frequency of 2 MHz.
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At this point, a signal due to the wave-packet is recorded as a function of time. It is found that the wave-packet begins to pass the point at time t1 and ®nishes passing the point at the later time t2. The same wave-packet is then recorded as it passes a second point a distance l beyond the ®rst point. The wave-packet starts to pass the second point at time t3 and ®nishes passing at time t4. If the packet of guided waves contains waves with a range of group velocities from n min to n max, then the temporal limits of the wave-packet as it passes the second point may be expressed in terms of these velocities. The time t3, when the wave-packet ®rst reaches the second point cannot be earlier than the time taken for waves propagating at the maximum velocity n max to travel the distance l starting at time t1. Similarly, the time t4 at which the wave-packet ®nishes passing the second point cannot be later than the time taken for waves propagating with the minimum velocity n min to travel the distance l starting at time t2. Hence the start and end of the wavepacket after it has propagated a distance l are: t3 t1 1
l nmax
t4 t2 1
l nmin
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Again the example of the propagation of the S0 mode in a 1 mm thick aluminium plate after excitation with a 5-cycle Hanning windowed toneburst at a centre frequency of 2 MHz is considered. The bandwidth of such a signal is obtained by ®nding the frequencies at which the spectral amplitude falls a certain number of decibels below its maximum value at the centre frequency. The choice of this value is somewhat arbitrary but in this case a value of 20 dB is used which yields a frequency range from 1.32 to 2.68 MHz. The group velocity dispersion curves for a 1-mm thick aluminium plate are shown in Fig. 3. These were obtained using the software suite Disperse [21]. The bandwidth of the input signal is indicated and the dispersion curve for the S0 mode is emboldened in this region. The group velocity over this region ranges from a minimum of 1.791 mm/ms at the dip in the curve at 2.48 MHz to a maximum of 4.827 mm/ms at the lower limit of the bandwidth at 1.32 MHz and these are the values used for n min and n max. If t1 is set equal to zero, then t2 is the initial wave-packet duration of 2.5 ms (i.e. ®ve times the period of one cycle). With this information and the velocity limits just calculated, a space±time map showing just the boundary of the propagating wave-packet can be plotted as shown by the dotted lines in Fig. 4. For comparison, the signal envelope amplitude predicted using the Fourier decomposition method that was shown in Fig. 2(b) is also plotted in Fig. 4. It can be seen that boundaries of the wave-packet that are predicted by the two methods are in reasonably good agreement. It is important to stress that the bene®ts of the group velocity technique are its speed and simplicity and its use is for making comparative measurements of wave-packet spreading. It is not suitable for making absolute predictions because it is highly sensitive to the de®nition of the bandwidth of the input signal. This is a shortcoming that the
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Fig. 5. Example, in this case for the S0 mode at 2 MHz in a 1-mm thick aluminium plate, showing the variation of the duration of the received signal with the number of cycles in the input signal (a Hanning windowed toneburst) for various propagation distances.
Fourier decomposition technique does not suffer from, as that technique uses contributions from all the spectral components of the input signal with their appropriate amplitudes. However, it should be stressed that neither of the techniques that have been described makes a perfectly accurate prediction of the wave-packet duration since they are both subject to the same initial approximation. This is the assumption that the spectrum of the excited wave-packet in the structure is the same as that of the input signal supplied to the transducer. This is not true due to the characteristics of the transducer itself and the excitability of a guided wave mode. Both of these are functions of frequency that cause the spectrum of the wave-packet to be distorted compared to the spectrum of the input signal. The justi®cation is that over a limited bandwidth these factors are suf®ciently slowly varying functions of frequency for their effects to be ignored. For the reasons mentioned earlier, the input signals used in long range guided wave testing are usually of limited bandwidth, so this approximation is reasonable.
3. Quanti®cation of dispersive propagation 3.1. Resolvable distance Having developed a tool for predicting the rate of wavepacket spreading and therefore the duration of a received wave-packet, the next stage is to obtain a useful quantity by which to make comparative measurements of the dispersion at different operating points on the dispersion curves for a particular system. An obvious quantity to compare is the rate of wave-packet spreading with propagation distance.
However, this can only be computed if the input signal is speci®ed, which leads to the problem of how to de®ne a standard input signal at different frequencies. One possibility is to de®ne the standard input signal as being a windowed toneburst containing a ®xed number of cycles at all frequencies. From this de®nition, it is straightforward to calculate the rate of wave-packet spreading at every point on the dispersion curves for a particular system using the technique described above. A second possibility is to de®ne the standard input signal as a windowed toneburst containing a number of cycles proportional to the centre frequency so that its duration is constant at all frequencies. From this de®nition, a second set of values for the rate of wave-packet spreading could be calculated for every point on the dispersion curves of a system. Unfortunately, the results obtained using these two de®nitions of input signal are signi®cantly different. Because of the ambiguity in how to specify the input signal, the rate of wave-packet spreading alone is not a suitable quantity for comparing different operating points. This motivated the development of the alternative procedure to compare different operating points described below. Instead of considering the rate of signal spreading at a particular operating point, the best resolution that can be obtained is examined. If the initial temporal duration of a wave-packet is Tin then after propagating a distance l the new temporal duration, Tout, will be: Tout Tdisp 1 Tin
7
where Tdisp is the increase in wave-packet duration due to dispersion. Using the technique for predicting dispersion based on group velocity described previously, this can be written as: Tdisp l 1=nmin 2 1=nmax
8
In order to obtain a measure of the spatial resolution associated with the wave-packet, its temporal duration Tout is multiplied by a nominal group velocity n 0. For the purposes of this study, n 0 is de®ned as the group velocity at the centre frequency of the wave-packet, since in practice this is velocity that is used when converting the arrival times of signals to propagation distances. In order to make the spatial resolution dimensionless, it is divided by a characteristic thickness dimension d of the system to give what will be de®ned as the resolvable distance: Resolvable distance
Tout n0 d
n0 l 1=nmin 2 1=nmax 1 Tin d
9
3.2. Minimum resolvable distance and optimised input signal It can be seen from Eqs. (7) and (9) that the duration of a wave-packet and hence the resolvable distance is governed by two terms, the ®rst term being due to the increase in
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Fig. 6. (a) MRD curves for Lamb waves in a 1-mm thick aluminium plate and a propagation distance of 1000 mm and (b) the associated curves illustrating the optimum number of cycles required in the input signals in order to attain the MRD at each point. The circles indicate points of maximum group velocity.
wave-packet length due to dispersion and the second term being the length of the input signal. At a particular frequency and with a small number of cycles in the input signal, Tin will be small, but its bandwidth will be large and dispersion effects will therefore be signi®cant. As the number of cycles in the input signal is increased, its bandwidth decreases and the size of the dispersion term in Eq. (9) also decreases. However, the Tin term will increase. At some point, an optimum input signal will be found that minimises the duration of the wave-packet and therefore the resolvable distance. To illustrate this, the example of the S0 Lamb wave mode in a 1 mm thick aluminium plate around a centre frequency of 2 MHz will again be considered. Fig. 5 shows a graph of the duration of wave-packet vs. number of cycles in the input signal for various distances of propagation. In all cases, the input signal is assumed to be a Hanning windowed toneburst. It can be seen that the curve for each propagation distance has a characteristic `tick' shape. The minimum of each curve represents the number of cycles in the optimum input signal for that propagation distance. It can be seen that as the propagation distance is increased, so does the number of cycles in the optimum input signal and the associated minimum duration of the wave-packet. The locus of this minimum is indicated by the dotted line. Although this line appears to be approximately straight, there is no reason why it should be since the group velocity curves are not linear functions of frequency. The minimum duration of wave-packet that can be achieved at an operating point for a given propagation distance de®nes the minimum
resolvable distance (MRD): MRD
n0 l 1=nmin 2 1=nmax 1 Tin umin d
10
The MRD curves for Lamb waves in the example structure can now be calculated and the results for a propagation distance of 1000 mm are shown in Fig. 6(a). The procedure used to calculate the value of MRD at each point on these curves is an iterative one whereby the number of cycles in the input signal is optimised to minimise the duration of the wave-packet after a propagation distance of 1000 mm. A side product of the calculation of MRD is the optimum number of cycles in the input signal. Hence a second set of curves showing the optimum number of cycles in the input signal may be obtained. These are plotted for the example structure in Fig. 6(b). 3.3. Discussion of results for Lamb waves in a 1-mm thick aluminium plate in vacuum For a long-range guided wave inspection system, it is desirable to operate at a point where the MRD is as low as possible. It can be seen from the curves shown in Fig. 6(a) that as the frequency is increased the MRD for each Lamb wave mode passes through an initial minimum followed by a number of further maxima and minima before eventually decreasing monotonically. This last portion occurs as the velocity of each Lamb wave mode tends to a constant value (equal to the Rayleigh wave velocity for the fundamental modes and the bulk shear wave velocity for
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P. Wilcox et al. / NDT&E International 34 (2001) 1±9
Table 1 Centre frequency, MRD and number of cycles required in optimum input signal at various points of low MRD on the ®rst six Lamb wave modes in a 1-mm thick aluminium plate. A propagation distance of 1000 mm is assumed Mode
A0
S0
A1
S1
A2
S2
Frequency (MHz) Optimum cycles in input signal MRD
1.47 8 26
0.15 1 51
2.67 27 56
4.05 41 65
6.27 69 53
6.78 52 48
higher order modes) at high frequencies. From the point of view of long-range guided wave testing, the region of interest on each mode is at lower frequencies. For this reason, it is desirable that such testing takes place at an operating point on or close to one of the minima on the MRD curve for a particular mode. It is also generally preferable to operate at a point where the group velocity is a maximum rather than a minimum. Reference to the group velocity dispersion curves for a 1-mm thick aluminium plate will indicate that the maximum in group velocity for a mode corresponds to the lowest frequency minimum in MRD for that mode. For the ®rst six Lamb wave modes, these points are indicated by circles in Fig. 6(a) and (b), and they are tabulated in Table 1. It is interesting to observe that the MRD at these points does not exhibit any signi®cant upward or downward trend with frequency. This is not the same as conventional bulk wave ultrasonic testing where high frequencies are associated with short wavelengths and high resolutions. In long-range guided wave testing, the resolution is determined by the length of the wave-packet and not by the wavelength of individual waves. The exception to this is at frequencies below the ®rst minimum in MRD on the S0 mode. Here the group velocity is almost constant with frequency and hence dispersion is negligible. For this reason, the optimum input signal is made as short as possible, which is toneburst containing a single cycle. Hence in this low frequency region on this mode, the MRD tends to a value equal to one wavelength. The conclusion from this study is that there is no bene®t (in terms of resolution) in using higher order Lamb wave modes for long-range testing. An operating point on one of the fundamental modes below the cut-off frequency of the A1 mode is attractive in practice, since only the two fundamental Lamb wave modes can propagate. Furthermore, the fundamental modes are well separated in phase velocity in this frequency region. Both of these factors make the design of modally selective transducers considerably easier. For a long-range testing application, the operating point at the minimum in MRD on the A0 mode at 1.47 MHz is especially attractive since the mode at that point is very easily excitable and detectable by any transduction method that couples to out-of-plane surface displacement [20]. The only problem with the A0 mode is that it is highly attenuated if the plate is immersed in a liquid. This is because the outof-plane component of the displacement at the surface of the plate couples to the surrounding liquid and causes energy to be radiated away from the plate in the form of bulk compres-
sion waves. For this reason, the A0 mode is not suitable for the inspection of structures such as liquid-®lled tanks and pressure vessels. In these situations, it is desirable to operate at a point where the attenuation of a guided wave mode due to leakage is small. Although several of the operating points in Table 1 can be shown to satisfy this requirement [20], there is no bene®t from the point of view of obtaining good resolution at operating at any point other than that on the S0 mode at 0.15 MHz. Here, the mode is again well separated from other modes in phase velocity, so the suppression of unwanted modes is straightforward. It should be stressed that all the results presented here relate purely to the propagation of guided waves and do not take any account of how a mode interacts with a particular feature or defect. For this reason, it should not be assumed that a point of good resolution on a mode is necessarily a point of high sensitivity to the defects it is desired to detect. 4. Conclusion A simple technique has been presented for predicting the spreading of a dispersive packet of guided waves as it propagates through a structure. If an appropriate de®nition for the duration of a wave-packet is made, then it has been shown that the spreading of a wave-packet is linear with propagation distance. A parameter called minimum resolvable distance (MRD) has been introduced that enables a comparison to be made between the effect of dispersion at different operating points. In the case of a 1-mm thick aluminium plate in vacuum, the MRD has been used to show that the best operating point in terms of resolution is at 1.47 MHz on the fundamental anti-symmetric mode A0. It has also been shown that there is no bene®t in terms of resolution from operating on a higher order mode at higher frequency. Resolution, attenuation and defect sensitivity are three of the criteria that make up a general rationalised strategy developed by the authors for selecting the operating point for a guided wave inspection system for a particular structure [20]. References [1] Chimenti DE. Guided waves in plates and their use in materials characterization. Appl Mech Rev 1997;50:247±84. [2] Chimenti DE, Nayfeh A. Ultrasonic re¯ection and guided wave
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[18] [19] [20] [21]
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for the inspection of large areas of thick plates. In: Thompson DO, Chimenti DE, editors. Annual review of progress in quantitative NDE, vol. 19. New York: Plenum Press, 2000 p. 1049±56. Alleyne DN, Cawley P. Optimization of Lamb wave inspection techniques. NDT & E Int 1992;25:11±22. Rose JL. Recent advances in guided wave NDE. IEEE Ultrason Symp 1995:761±70. Monkhouse RSC, Wilcox PD, Cawley P. Flexible inter-digital PVDF transducers for the generation of Lamb waves in structures. Ultrasonics 1997;35:489±98. Alers GA, Burns LR. EMAT designs for special applications. Mater Eval 1987;45:1184±94. Salzburger HJ. Guided shear horizontal (SH-) wavesÐexamples of the practical use of their bene®ts for new NDE applications. In: Thompson DO, Chimenti DE, editors. Annual review of progress in quantitative NDE, vol. 18A. New York: Plenum Press, 1999. p. 269± 76. Viktorov IA, Zubova OM, Kaekina TM. Investigation of Lamb wave excitation by the wedge method. Sov Phys Acoust 1965;10:354±9. Jia X. Modal analysis of Lamb wave generation in elastic plates by liquid wedge transducers. J Acoust Soc Am 1996;101:834±42. Wilcox PD, Dalton RP, Lowe MJS, Cawley P. Mode and transducer selection for long range Lamb wave inspection. Key Eng Mater 1999;16±68:152±61. Pavlakovic BN, Lowe MJS, Alleyne DN, Cawley P. Disperse: a general purpose program for creating dispersion curves. In: Thompson DO, Chimenti DE, editors. Annual review of progress in quantitative NDE, vol. 16A. New York: Plenum Press, 1997. p. 155±92.
www.elsevier.com/locate/ndteint
The effect of dispersion on long-range inspection using ultrasonic guided waves P. Wilcox*, M. Lowe, P. Cawley NDT Laboratory, Department of Mechanical Engineering, Imperial College, Exhibition Road, London SW7 2BX, UK Received 31 January 2000; received in revised form 31 March 2000; accepted 3 April 2000
Abstract The dispersion of ultrasonic guided waves causes wave-packets to spread out in space and time as they propagate through a structure. This limits the resolution that can be obtained in a long-range guided wave inspection system. A technique is presented for quickly predicting the rate of spreading of a dispersive wave-packet as it propagates. It is shown that the duration of a wave-packet increases linearly with propagation distance. It is also shown that the duration of a wave-packet after a given propagation distance can be minimised by optimising the input signal. A dimensionless parameter called minimum resolvable distance (MRD) is de®ned that enables a direct comparison to be made between the resolution attainable at different operating points. Some conclusions are made concerning the resolution of various operating points for the case of Lamb waves in an aluminium plate. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Dispersion; Guided waves; Lamb waves; Long-range testing
1. Background 1.1. Guided waves and non-destructive inspection Much work has been published on the use of Lamb waves and other guided waves for inspection purposes and a comprehensive review of applications may be found in Ref. [1]. Very broadly speaking, the use of guided waves for non-destructive inspection purposes falls into two categories depending on the distance of propagation. Firstly, there are short-range applications, where guided waves are used to obtain information about a specimen that cannot be readily obtained by more conventional means. These areas include the determination of the elastic properties of materials [2,3], the detection of defects near to interfaces such as in the inspection of adhesive joints [4] and air coupled ultrasonic inspection of thin specimens [5]. In these cases, sensitivity is of key importance and generally this is the main criterion for selecting a suitable guided wave mode. The effect of dispersion is relatively unimportant as the propagation distances are small. This paper is concerned with the second area of guided wave applications where the propagation distance is large. These include the detection of delaminations in rolled steel * Corresponding author. Tel.: 144-0171-594-7227; fax: 144-0171-5801560. E-mail address: [email protected] (P. Wilcox).
[6,7] and composites [8], pipeline [9±11] and plate [12] inspection. In long-range applications, the aim is to inspect large areas of a structure rapidly. 1.2. Long-range inspection using guided waves In long-range guided wave testing applications, the guided waves are excited by a short burst of energy (the input signal) applied by a suitable transducer at one location on a structure. The excitation causes a packet of guided waves (the wave-packet) to propagate away from the transducer into the surrounding structure. Then either the same transducer or a second transducer is used to detect signals caused by re¯ections of energy in the wave-packet from surrounding structural features or defects. The problems associated with the use guided waves for inspection purposes are well documented [13]. In summary, multiple modes of guided wave propagation are possible in most structures and these modes are generally dispersive (i.e. their velocities are frequency-dependent). In order to obtain useful data from a guided wave inspection system, it is necessary to selectively excite and detect a single guided wave mode while suppressing coherent noise due to other modes of guided wave propagation. For this reason, the design of the transducer and the input signal are tailored so that the excitation energy is targeted at a single point on a suitable guided wave mode at a suitable frequency. This point is called the operating point.
0963-8695/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0963-869 5(00)00024-4
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P. Wilcox et al. / NDT&E International 34 (2001) 1±9
Fig. 1. (a) Numerical simulation of the space±time map illustrating the dispersive propagation of the S0 mode in a 1-mm thick aluminium plate when the input signal is a 5-cycle Hanning windowed toneburst with a centre frequency of 2 MHz. Below are numerical predictions from the same model that show the timetraces that would be received, (b) close to the source, (c) 50 mm from the source and (d) 100 mm from the source.
Examples of transducers that may be used to excite and detect guided waves include inter-digital or comb transducers that operate by either piezoelectric [14,15] or electromagnetic mechanisms [16,17]. Alternatively, conventional plane bulk wave transducers may be used in conjunction with a coupling wedge in the angle incidence con®guration [18,19]. Suitable input signals are windowed tonebursts with a precise centre frequency and a limited bandwidth. The reason for using a limited bandwidth input signal is twofold. Firstly, it helps to prevent the excitation of undesired modes at other frequencies and secondly it reduces the effect of dispersion on the propagation of the desired mode [13]. The studies presented here are concerned solely with the effect of dispersion. For this reason, it is implicitly assumed throughout that a suitable transducer can be designed so that single mode excitation can be attained at any operating point on any guided wave mode. The practicalities of actually achieving this are beyond the scope of this paper.
1.3. Manifestation of dispersion effects The effect of dispersion is that the energy in a wavepacket propagates at different speeds depending on its frequency. This manifests itself as a spreading of the wave-packet in space and time as it propagates through a structure. This is illustrated in Fig. 1(a), which shows the propagation of the S0 Lamb wave mode in a 1 mm thick aluminium plate after excitation with a 5-cycle Hanning windowed toneburst with a centre frequency of 2 MHz. This type of graphical representation of dispersion will be referred to as a space±time map and the means by which it is calculated is summarised in the following section. The xaxis on the space±time map represents time measured from the moment the excitation signal starts and the y-axis is the distance of propagation measured from the excitation location. The greyscale level indicates the value of a suitable quantity that is affected by the passage of the guided wave. In this case the quantity used is the out-of-plane displacement of the surface of the plate. The propagation and
P. Wilcox et al. / NDT&E International 34 (2001) 1±9
spreading of the wave-packet in space and time can be clearly seen on the space±time map. The striped effect within the area of the wave-packet is due to the displacements from individual wave peaks and troughs. The points at which its envelope amplitude falls below a certain threshold level de®ne the boundaries of the wave-packet, and any points outside this are coloured white in the space±time map. The de®nition of this threshold level will be discussed later. A propagating wave-packet is detected by a transducer that is able to monitor a suitable quantity associated with the wave-packet, such as the out-of-plane surface displacement, as a function of time (a time-trace). A time-trace is a crosssection parallel to the time axis through the space±time map of that parameter. Example time-traces that would be measured after propagation distances of 0.1, 50 and 100 mm are shown in Fig. 1(b), (c) and (d). In these timetraces, the effect of dispersion appears as an increase in the duration of the wave-packet in time and a decrease in its amplitude. 1.4. Why dispersion is undesirable The effects of increasing wave-packet duration and decreasing amplitude due to dispersion are both undesirable in long range guided wave testing. The spreading of a wave-packet in space and time reduces the resolution that can be obtained. This problem is frequently encountered when attempting to detect defects in close proximity to structural features, such as welds. In such a case, the defect can only be reliably detected if its re¯ection can be resolved from that due to the feature. The reduction in amplitude of a dispersive wave-packet reduces the sensitivity of the testing system. Although, the studies presented here are primarily concerned with the increase in temporal duration of a wave-packet, the decrease in wave-packet amplitude can be estimated by using energy conservation. On this basis and neglecting other losses, it can be assumed to a ®rst approximation that the amplitude of a wave-packet will decrease in proportion to the square root of the increase in its duration. 1.5. Overview The ®rst part of this paper describes a simple method for quantitatively predicting the rate of wave-packet spreading due to dispersion at any operating point on the dispersion curves for a particular structure. The second part of the paper explains how the practical effect of dispersion at different operating points can be compared. The procedures described in this paper may be used for any structure to which long-range guided wave inspection techniques can be applied. The list includes metallic platelike structures such as pressure vessels, multi-layered structures such as adhesive joints and cylindrical systems such as pipes. The only requirement is that the dispersion curve data (phase and group velocity) is available for the mode or
3
modes over the frequency range of interest. For the purposes of illustrating the techniques proposed in this paper, the simple case of Lamb waves in a 1-mm thick aluminium plate in vacuum will be used as an example. On occasions where it is necessary to consider a single operating point, the S0 Lamb wave mode at a frequency of 2 MHz will be used.
2. Modelling dispersive propagation 2.1. Numerical prediction of dispersive propagation using Fourier decomposition In order to make quantitative measurements of dispersive wave propagation, it is necessary to be able to model how a guided wave-packet excited by an arbitrary input signal propagates when dispersion is present. In particular, it is desirable to be able to predict the rate at which the packet spreads out with propagation distance. An obvious way to do this is to make use of the phase velocity (or wavenumber) dispersion data available for the structure and then to use a Fourier decomposition technique to make an exact numerical prediction of the propagation. This technique was used to generate the space±time map and the time-traces shown in Fig. 1 and it is summarised below. Consider the case when a suitable guided wave transducer (e.g. an angle incidence device or an inter-digital transducer) is used to excite guided waves in a structure. For the purposes of this study, it is assumed that the transducer is ideal, in that it excites only one guided wave mode and in only one direction. Cartesian axes are de®ned using the same convention as Viktorov [18] for straight crested waves with wave propagation in the positive x-direction and the z-axis normal to the plane of the structure. The wave-crests are orientated parallel to the y-axis and these, as well as the structure and the transducer, are assumed to extend inde®nitely in the y-direction. The origin of the xaxis is de®ned so that the front edge of the transducer is located at x 0: The transducer is supplied with an electrical signal of ®nite duration V(t), which is converted into acoustic energy. This energy propagates away from the transducer in the positive x-direction as a single guided wave mode. It is assumed that at the transducer position, x 0; the variation of a parameter in the plate, such as out-of-plane surface displacement u with time t is directly proportional to V(t). Although this assumption is not strictly true, it is the only reasonable approximation that can be made without knowledge of the transducer characteristics and the excitability function [20] of the guided wave mode. The known function u(t) can be considered as being a `slice' taken at x 0 through the space±time map of the function u(x,t), which describes the propagation of the guided wave mode in space x and time t. If the phase velocity dispersion curve data for the system is available, then
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The boundaries of the wave-packet can then be computed using one of the de®nitions given below. 2.2. De®nition of the duration of a wave-packet
Fig. 2. The relative envelope amplitude of the wave-packet from the space± time map in Fig. 1(a) plotted using two different de®nitions for the reference level: (a) reference (0 dB level) is the peak amplitude of the signal at distance equal to zero; (b) reference is re-calculated at each distance as the peak amplitude of the wave-packet at that distance.
u(x, t) can be calculated at any other point in time and space in the manner described below. First u(t) is Fourier transformed to obtain its frequency spectrum: U v
1 Z1 u t e2iax dt 2p 2 1
1
where v is angular frequency. The value of u(x,t) associated with an individual spectral component of U(v )is given by the wave equation: U v ei k vx2vt
2
where k(v ) is the circular wavenumber that may be obtained from the phase velocity n ph(v ) dispersion curve data by: k v
v nph v
3
The overall value of u(x, t) due to the propagation of the input wave-packet u(t) is given by the integration of the contributions from all the spectral components of U(v ). To perform this integration, use is made of the fact that because u(t) was real, U 2v U p v: Hence the integration over the negative frequency range is equal to that over the positive frequency range and the expression for u(x, t) may be written: Z1 U v ei k vx2vt dv 4 u x; t 2 Re 0
Hence the time signal that would be obtained if u(x, t) was measured at any location can be computed. The Fourier decomposition method was used to predict the space±time map and the time-traces shown in Fig. 1. For the purposes of predicting the boundaries of the wave-packet, it is more convenient to work with its envelope and this can be readily computed using the Hilbert transform method, whereby Eq. (4) is replaced by: Z 1 5 Envelope u x; t 2 U v ei k vx2vt dv 0
To quantify dispersion, it is necessary to be able to measure the duration of a wave-packet and this requires a means of de®ning whereabouts in time a wave-packet begins and ends. The easiest way in which to do this is to de®ne the duration by the points in time at which the envelope of the wave-packet falls below a particular reference level. The problem that then arises is how to de®ne the reference level and the two space±time maps that are plotted in Fig. 2 illustrate this. These are both for the same example as that used in Fig. 1, except that in this ®gure the greyscale indicates the amplitude of the signal envelope rather than the individual wave peaks and troughs. In the space±time map shown in Fig. 2(a), the amplitude over the entire space±time map is referenced to a ®xed value, and this value (i.e. 0 dB) is taken as being the peak level of the wave-packet envelope at the source. If the duration of a wave-packet is de®ned in this manner then although it initially increases, it will, after a suf®cient propagation distance, begin to decrease and will ultimately reach zero. This can be seen in Fig. 2(a), where the contours of the greyscale form closed lobes. The reason for this is that the increase in wave-packet duration must be accompanied by a decrease in its amplitude as noted earlier. For the purposes of predicting the rate of wave-packet spreading due to dispersion, this de®nition is clearly not suitable since the rate will be dependent on the distance of propagation. A more suitable de®nition of the reference level yields the space±time map shown in Fig. 2(b). Here the amplitude at a particular propagation distance is referenced to the peak value of the wave-packet envelope at that distance. In this way, the measured duration of the dispersive wave-packet monotonically increases as it propagates. It can be seen from Fig. 2(b) that this increase is actually a linear function of the propagation distance. This is a useful fact and forms the basis of the technique described below for predicting dispersive propagation. 2.3. Prediction of dispersive propagation using group velocity dispersion curves The Fourier decomposition procedure described previously is an inef®cient and time-consuming method of obtaining the rate of spread of the wave-packet and it is not suitable for the type of iterative calculation that will be described later in this paper. This problem motivated the development of an alternative technique for predicting the duration of a received wave-packet after an arbitrary propagation distance. It was shown above that a dispersive wave-packet spreads out linearly in space and time as it propagates if a suitable de®nition for the boundary of the wave-packet is used. This means that the ends of the envelope of the dispersive wave-packet can be regarded as
P. Wilcox et al. / NDT&E International 34 (2001) 1±9
Fig. 3. Group velocity dispersion curves for a 1-mm thick aluminium plate in vacua. The vertical lines represent the bandwidth (based on the 220 dB points of the spectrum) of a 5-cycle Hanning windowed toneburst with a centre frequency of 2 MHz. The portion of the S0 mode that falls within this bandwidth is emboldened and the extrema of its group velocity are indicated.
propagating with two different velocities, and it is this difference in velocities that causes the envelope to become longer. If these two velocities and the duration of the wavepacket at one point in space are known then the duration of the wave-packet at any other point in space can be calculated. The method described here takes the values for these two velocities from the group velocity dispersion curve for the guided wave mode. Consider a wave-packet as it passes a point in a structure.
Fig. 4. Comparison between group velocity and Fourier decomposition methods for predicting the boundary of the wave-packet when the S0 Lamb wave mode is excited in a 1-mm thick aluminium plate by a 5cycle Hanning windowed toneburst with a centre frequency of 2 MHz.
5
At this point, a signal due to the wave-packet is recorded as a function of time. It is found that the wave-packet begins to pass the point at time t1 and ®nishes passing the point at the later time t2. The same wave-packet is then recorded as it passes a second point a distance l beyond the ®rst point. The wave-packet starts to pass the second point at time t3 and ®nishes passing at time t4. If the packet of guided waves contains waves with a range of group velocities from n min to n max, then the temporal limits of the wave-packet as it passes the second point may be expressed in terms of these velocities. The time t3, when the wave-packet ®rst reaches the second point cannot be earlier than the time taken for waves propagating at the maximum velocity n max to travel the distance l starting at time t1. Similarly, the time t4 at which the wave-packet ®nishes passing the second point cannot be later than the time taken for waves propagating with the minimum velocity n min to travel the distance l starting at time t2. Hence the start and end of the wavepacket after it has propagated a distance l are: t3 t1 1
l nmax
t4 t2 1
l nmin
6
Again the example of the propagation of the S0 mode in a 1 mm thick aluminium plate after excitation with a 5-cycle Hanning windowed toneburst at a centre frequency of 2 MHz is considered. The bandwidth of such a signal is obtained by ®nding the frequencies at which the spectral amplitude falls a certain number of decibels below its maximum value at the centre frequency. The choice of this value is somewhat arbitrary but in this case a value of 20 dB is used which yields a frequency range from 1.32 to 2.68 MHz. The group velocity dispersion curves for a 1-mm thick aluminium plate are shown in Fig. 3. These were obtained using the software suite Disperse [21]. The bandwidth of the input signal is indicated and the dispersion curve for the S0 mode is emboldened in this region. The group velocity over this region ranges from a minimum of 1.791 mm/ms at the dip in the curve at 2.48 MHz to a maximum of 4.827 mm/ms at the lower limit of the bandwidth at 1.32 MHz and these are the values used for n min and n max. If t1 is set equal to zero, then t2 is the initial wave-packet duration of 2.5 ms (i.e. ®ve times the period of one cycle). With this information and the velocity limits just calculated, a space±time map showing just the boundary of the propagating wave-packet can be plotted as shown by the dotted lines in Fig. 4. For comparison, the signal envelope amplitude predicted using the Fourier decomposition method that was shown in Fig. 2(b) is also plotted in Fig. 4. It can be seen that boundaries of the wave-packet that are predicted by the two methods are in reasonably good agreement. It is important to stress that the bene®ts of the group velocity technique are its speed and simplicity and its use is for making comparative measurements of wave-packet spreading. It is not suitable for making absolute predictions because it is highly sensitive to the de®nition of the bandwidth of the input signal. This is a shortcoming that the
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P. Wilcox et al. / NDT&E International 34 (2001) 1±9
Fig. 5. Example, in this case for the S0 mode at 2 MHz in a 1-mm thick aluminium plate, showing the variation of the duration of the received signal with the number of cycles in the input signal (a Hanning windowed toneburst) for various propagation distances.
Fourier decomposition technique does not suffer from, as that technique uses contributions from all the spectral components of the input signal with their appropriate amplitudes. However, it should be stressed that neither of the techniques that have been described makes a perfectly accurate prediction of the wave-packet duration since they are both subject to the same initial approximation. This is the assumption that the spectrum of the excited wave-packet in the structure is the same as that of the input signal supplied to the transducer. This is not true due to the characteristics of the transducer itself and the excitability of a guided wave mode. Both of these are functions of frequency that cause the spectrum of the wave-packet to be distorted compared to the spectrum of the input signal. The justi®cation is that over a limited bandwidth these factors are suf®ciently slowly varying functions of frequency for their effects to be ignored. For the reasons mentioned earlier, the input signals used in long range guided wave testing are usually of limited bandwidth, so this approximation is reasonable.
3. Quanti®cation of dispersive propagation 3.1. Resolvable distance Having developed a tool for predicting the rate of wavepacket spreading and therefore the duration of a received wave-packet, the next stage is to obtain a useful quantity by which to make comparative measurements of the dispersion at different operating points on the dispersion curves for a particular system. An obvious quantity to compare is the rate of wave-packet spreading with propagation distance.
However, this can only be computed if the input signal is speci®ed, which leads to the problem of how to de®ne a standard input signal at different frequencies. One possibility is to de®ne the standard input signal as being a windowed toneburst containing a ®xed number of cycles at all frequencies. From this de®nition, it is straightforward to calculate the rate of wave-packet spreading at every point on the dispersion curves for a particular system using the technique described above. A second possibility is to de®ne the standard input signal as a windowed toneburst containing a number of cycles proportional to the centre frequency so that its duration is constant at all frequencies. From this de®nition, a second set of values for the rate of wave-packet spreading could be calculated for every point on the dispersion curves of a system. Unfortunately, the results obtained using these two de®nitions of input signal are signi®cantly different. Because of the ambiguity in how to specify the input signal, the rate of wave-packet spreading alone is not a suitable quantity for comparing different operating points. This motivated the development of the alternative procedure to compare different operating points described below. Instead of considering the rate of signal spreading at a particular operating point, the best resolution that can be obtained is examined. If the initial temporal duration of a wave-packet is Tin then after propagating a distance l the new temporal duration, Tout, will be: Tout Tdisp 1 Tin
7
where Tdisp is the increase in wave-packet duration due to dispersion. Using the technique for predicting dispersion based on group velocity described previously, this can be written as: Tdisp l 1=nmin 2 1=nmax
8
In order to obtain a measure of the spatial resolution associated with the wave-packet, its temporal duration Tout is multiplied by a nominal group velocity n 0. For the purposes of this study, n 0 is de®ned as the group velocity at the centre frequency of the wave-packet, since in practice this is velocity that is used when converting the arrival times of signals to propagation distances. In order to make the spatial resolution dimensionless, it is divided by a characteristic thickness dimension d of the system to give what will be de®ned as the resolvable distance: Resolvable distance
Tout n0 d
n0 l 1=nmin 2 1=nmax 1 Tin d
9
3.2. Minimum resolvable distance and optimised input signal It can be seen from Eqs. (7) and (9) that the duration of a wave-packet and hence the resolvable distance is governed by two terms, the ®rst term being due to the increase in
P. Wilcox et al. / NDT&E International 34 (2001) 1±9
7
Fig. 6. (a) MRD curves for Lamb waves in a 1-mm thick aluminium plate and a propagation distance of 1000 mm and (b) the associated curves illustrating the optimum number of cycles required in the input signals in order to attain the MRD at each point. The circles indicate points of maximum group velocity.
wave-packet length due to dispersion and the second term being the length of the input signal. At a particular frequency and with a small number of cycles in the input signal, Tin will be small, but its bandwidth will be large and dispersion effects will therefore be signi®cant. As the number of cycles in the input signal is increased, its bandwidth decreases and the size of the dispersion term in Eq. (9) also decreases. However, the Tin term will increase. At some point, an optimum input signal will be found that minimises the duration of the wave-packet and therefore the resolvable distance. To illustrate this, the example of the S0 Lamb wave mode in a 1 mm thick aluminium plate around a centre frequency of 2 MHz will again be considered. Fig. 5 shows a graph of the duration of wave-packet vs. number of cycles in the input signal for various distances of propagation. In all cases, the input signal is assumed to be a Hanning windowed toneburst. It can be seen that the curve for each propagation distance has a characteristic `tick' shape. The minimum of each curve represents the number of cycles in the optimum input signal for that propagation distance. It can be seen that as the propagation distance is increased, so does the number of cycles in the optimum input signal and the associated minimum duration of the wave-packet. The locus of this minimum is indicated by the dotted line. Although this line appears to be approximately straight, there is no reason why it should be since the group velocity curves are not linear functions of frequency. The minimum duration of wave-packet that can be achieved at an operating point for a given propagation distance de®nes the minimum
resolvable distance (MRD): MRD
n0 l 1=nmin 2 1=nmax 1 Tin umin d
10
The MRD curves for Lamb waves in the example structure can now be calculated and the results for a propagation distance of 1000 mm are shown in Fig. 6(a). The procedure used to calculate the value of MRD at each point on these curves is an iterative one whereby the number of cycles in the input signal is optimised to minimise the duration of the wave-packet after a propagation distance of 1000 mm. A side product of the calculation of MRD is the optimum number of cycles in the input signal. Hence a second set of curves showing the optimum number of cycles in the input signal may be obtained. These are plotted for the example structure in Fig. 6(b). 3.3. Discussion of results for Lamb waves in a 1-mm thick aluminium plate in vacuum For a long-range guided wave inspection system, it is desirable to operate at a point where the MRD is as low as possible. It can be seen from the curves shown in Fig. 6(a) that as the frequency is increased the MRD for each Lamb wave mode passes through an initial minimum followed by a number of further maxima and minima before eventually decreasing monotonically. This last portion occurs as the velocity of each Lamb wave mode tends to a constant value (equal to the Rayleigh wave velocity for the fundamental modes and the bulk shear wave velocity for
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P. Wilcox et al. / NDT&E International 34 (2001) 1±9
Table 1 Centre frequency, MRD and number of cycles required in optimum input signal at various points of low MRD on the ®rst six Lamb wave modes in a 1-mm thick aluminium plate. A propagation distance of 1000 mm is assumed Mode
A0
S0
A1
S1
A2
S2
Frequency (MHz) Optimum cycles in input signal MRD
1.47 8 26
0.15 1 51
2.67 27 56
4.05 41 65
6.27 69 53
6.78 52 48
higher order modes) at high frequencies. From the point of view of long-range guided wave testing, the region of interest on each mode is at lower frequencies. For this reason, it is desirable that such testing takes place at an operating point on or close to one of the minima on the MRD curve for a particular mode. It is also generally preferable to operate at a point where the group velocity is a maximum rather than a minimum. Reference to the group velocity dispersion curves for a 1-mm thick aluminium plate will indicate that the maximum in group velocity for a mode corresponds to the lowest frequency minimum in MRD for that mode. For the ®rst six Lamb wave modes, these points are indicated by circles in Fig. 6(a) and (b), and they are tabulated in Table 1. It is interesting to observe that the MRD at these points does not exhibit any signi®cant upward or downward trend with frequency. This is not the same as conventional bulk wave ultrasonic testing where high frequencies are associated with short wavelengths and high resolutions. In long-range guided wave testing, the resolution is determined by the length of the wave-packet and not by the wavelength of individual waves. The exception to this is at frequencies below the ®rst minimum in MRD on the S0 mode. Here the group velocity is almost constant with frequency and hence dispersion is negligible. For this reason, the optimum input signal is made as short as possible, which is toneburst containing a single cycle. Hence in this low frequency region on this mode, the MRD tends to a value equal to one wavelength. The conclusion from this study is that there is no bene®t (in terms of resolution) in using higher order Lamb wave modes for long-range testing. An operating point on one of the fundamental modes below the cut-off frequency of the A1 mode is attractive in practice, since only the two fundamental Lamb wave modes can propagate. Furthermore, the fundamental modes are well separated in phase velocity in this frequency region. Both of these factors make the design of modally selective transducers considerably easier. For a long-range testing application, the operating point at the minimum in MRD on the A0 mode at 1.47 MHz is especially attractive since the mode at that point is very easily excitable and detectable by any transduction method that couples to out-of-plane surface displacement [20]. The only problem with the A0 mode is that it is highly attenuated if the plate is immersed in a liquid. This is because the outof-plane component of the displacement at the surface of the plate couples to the surrounding liquid and causes energy to be radiated away from the plate in the form of bulk compres-
sion waves. For this reason, the A0 mode is not suitable for the inspection of structures such as liquid-®lled tanks and pressure vessels. In these situations, it is desirable to operate at a point where the attenuation of a guided wave mode due to leakage is small. Although several of the operating points in Table 1 can be shown to satisfy this requirement [20], there is no bene®t from the point of view of obtaining good resolution at operating at any point other than that on the S0 mode at 0.15 MHz. Here, the mode is again well separated from other modes in phase velocity, so the suppression of unwanted modes is straightforward. It should be stressed that all the results presented here relate purely to the propagation of guided waves and do not take any account of how a mode interacts with a particular feature or defect. For this reason, it should not be assumed that a point of good resolution on a mode is necessarily a point of high sensitivity to the defects it is desired to detect. 4. Conclusion A simple technique has been presented for predicting the spreading of a dispersive packet of guided waves as it propagates through a structure. If an appropriate de®nition for the duration of a wave-packet is made, then it has been shown that the spreading of a wave-packet is linear with propagation distance. A parameter called minimum resolvable distance (MRD) has been introduced that enables a comparison to be made between the effect of dispersion at different operating points. In the case of a 1-mm thick aluminium plate in vacuum, the MRD has been used to show that the best operating point in terms of resolution is at 1.47 MHz on the fundamental anti-symmetric mode A0. It has also been shown that there is no bene®t in terms of resolution from operating on a higher order mode at higher frequency. Resolution, attenuation and defect sensitivity are three of the criteria that make up a general rationalised strategy developed by the authors for selecting the operating point for a guided wave inspection system for a particular structure [20]. References [1] Chimenti DE. Guided waves in plates and their use in materials characterization. Appl Mech Rev 1997;50:247±84. [2] Chimenti DE, Nayfeh A. Ultrasonic re¯ection and guided wave
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