The effect of dispersion on long-range inspectionusing 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 dened 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 cat-

egories 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 ma-

terials [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

[6,7] and composites [8], pipeline [911] 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 trans-

ducer into the surrounding structure. Then either the same

transducer or a second transducer is used to detect signals

caused by reections 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.

NDT&E International 34 (2001) 19

0963-8695/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved.

PII: S0963-8695(00)00024-4

www.elsevier.com/locate/ndteint

* Corresponding author. Tel.: 144-0171-594-7227; fax: 144-0171-580-

1560.

E-mail address: p.wilcox@ic.ac.uk (P. Wilcox).

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 inci-

dence conguration [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 disper-

sion 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 wave-

packet 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 spacetime map and the means by which it is

calculated is summarised in the following section. The x-

axis on the spacetime map represents time measured from

the moment the excitation signal starts and the y-axis is the

distance of propagation measured from the excitation loca-

tion. 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 displace-

ment of the surface of the plate. The propagation and

P. Wilcox et al. / NDT&E International 34 (2001) 192

Fig. 1. (a) Numerical simulation of the spacetime 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 time-

traces that would be received, (b) close to the source, (c) 50 mm from the source and (d) 100 mm from the source.

spreading of the wave-packet in space and time can be

clearly seen on the spacetime map. The striped effect

within the area of the wave-packet is due to the displace-

ments from individual wave peaks and troughs. The points

at which its envelope amplitude falls below a certain thresh-

old level dene the boundaries of the wave-packet, and any

points outside this are coloured white in the spacetime

map. The denition 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 cross-

section parallel to the time axis through the spacetime 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 time-

traces, 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

reection 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 plate-

like structures such as pressure vessels, multi-layered struc-

tures 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

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 occa-

sions where it is necessary to consider a single operat-

ing 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 numer-

ical prediction of the propagation. This technique was used

to generate the spacetime 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 transdu-

cer) 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 dened 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 indenitely in the y-direction. The origin of the x-

axis is dened 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 variationof 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 knowl-

edge 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 spacetime map of thefunction u(x,t), which describes the propagation of the

guided wave mode in space x and time t. If the phase vel-

ocity dispersion curve data for the system is available, then

P. Wilcox et al. / NDT&E International 34 (2001) 19 3

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:

Uv 12p

Z12 1

ut e2iax dt 1

where v is angular frequency. The value of u(x,t) associatedwith an individual spectral component of U(v )is given bythe wave equation:

Uv eikvx2vt 2where k(v ) is the circular wavenumber that may be obtainedfrom the phase velocity n ph(v ) dispersion curve data by:

kv vnphv 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, U2v Upv: Hence the integra-tion over the negative frequency range is equal to that over

the positive frequency range and the expression for u(x, t)

may be written:

ux; t 2 ReZ1

0Uv eikvx2vt dv

4

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 spacetime

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 envel-

ope and this can be readily computed using the Hilbert

transform method, whereby Eq. (4) is replaced by:

Envelopeux; t 2Z1

0Uv eikvx2vt dv

5

The boundaries of the wave-packet can then be computed

using one of the denitions given below.

2.2. Denition of the duration of a wave-packet

To quantify dispersion, it is necessary to be able to

measure the duration of a wave-packet and this requires a

means of dening whereabouts in time a wave-packet

begins and ends. The easiest way in which to do this is to

dene the duration by the points in time at which the envel-

ope of the wave-packet falls below a particular reference

level. The problem that then arises is how to dene the

reference level and the two spacetime 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 spacetime map shown in Fig. 2(a), the amplitude

over the entire spacetime 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 dura-

tion of a wave-packet is dened in this manner then

although it initially increases, it will, after a sufcient propa-

gation 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 denition is clearly not suitable since the

rate will be dependent on the distance of propagation.

A more suitable denition of the reference level yields the

spacetime 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 disper-

sive propagation.

2.3. Prediction of dispersive propagation using group

velocity dispersion curves

The Fourier decomposition procedure described

previously is an inefcient 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 denition 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) 194

Fig. 2. The relative envelope amplitude of the wave-packet from the space

time map in Fig. 1(a) plotted using two different denitions for the refer-

ence 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.

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 wave-

packet at one point in space are known then the duration of

the wave-packet at any other point in space can be calcu-

lated. 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.

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 nmin tonmax, then the temporal limits of the wave-packet as it passesthe second point may be expressed in terms of these vel-

ocities. 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 nmax to travel thedistance 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 nmin to travel the distance lstarting at time t2. Hence the start and end of the wave-

packet after it has propagated a distance l are:

t3 t1 1 lnmax t4 t2 1l

nmin6

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 maxi-

mum 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 ofthe 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 atthe 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 andthese are the values used for nmin and nmax. If t1 is set equal tozero, 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 spacetime 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 benets 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 denition of the band-

width of the input signal. This is a shortcoming that the

P. Wilcox et al. / NDT&E International 34 (2001) 19 5

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 indi-

cated.

Fig. 4. Comparison between group velocity and Fourier decomposition

methods for predicting the boundary of the wave-packet when the S0Lamb wave mode is excited in a 1-mm thick aluminium plate by a 5-

cycle Hanning windowed toneburst with a centre frequency of 2 MHz.

Fourier decomposition technique does not suffer from, as

that technique uses contributions from all the spectral

components of the input signal with their appropriate ampli-

tudes.

However, it should be stressed that neither of the techni-

ques 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 char-

acteristics 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 justica-

tion 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. Quantication of dispersive propagation

3.1. Resolvable distance

Having developed a tool for predicting the rate of wave-

packet 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

specied, which leads to the problem of how to dene a

standard input signal at different frequencies. One possi-

bility is to dene the standard input signal as being a wind-

owed toneburst containing a xed number of cycles at all

frequencies. From this denition, 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 dene

the standard input signal as a windowed toneburst contain-

ing a number of cycles proportional to the centre frequency

so that its duration is constant at all frequencies. From this

denition, a second set of values for the rate of wave-packet

spreading could be calculated for every point on the disper-

sion curves of a system. Unfortunately, the results obtained

using these two denitions of input signal are signicantly

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 7where 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 l1=nmin 2 1=nmax 8In order to obtain a measure of the spatial resolution asso-

ciated with the wave-packet, its temporal duration Tout is

multiplied by a nominal group velocity n 0. For the purposesof this study, n 0 is dened as the group velocity at the centrefrequency of the wave-packet, since in practice this is vel-

ocity 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

dened as the resolvable distance:

Resolvable distance Toutn0d

n0dl1=nmin 2 1=nmax1 Tin 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) 196

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.

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 signicant. As the

number of cycles in the input signal is increased, its band-

width 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 win-

dowed 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 denes the minimum

resolvable distance (MRD):

MRD n0dl1=nmin 2 1=nmax1 Tinumin 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

P. Wilcox et al. / NDT&E International 34 (2001) 19 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.

higher order modes) at high frequencies. From the point of

view of long-range guided wave testing, the region of inter-

est 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 oper-

ate 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 signicant 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 benet

(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 funda-

mental 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 out-

of-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 benet from the point of view of obtaining good

resolution at operating at any point other than that on the S0mode 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 par-

ticular 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 propa-

gates through a structure. If an appropriate denition for the

duration of a wave-packet is made, then it has been shown

that the spreading of a wave-packet is linear with propaga-

tion distance. A parameter called minimum resolvable

distance (MRD) has been introduced that enables a compar-

ison 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 benet 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 struc-

ture [20].

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P. Wilcox et al. / NDT&E International 34 (2001) 198

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) 1.47 0.15 2.67 4.05 6.27 6.78

Optimum cycles in input signal 8 1 27 41 69 52

MRD 26 51 56 65 53 48

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