The determination of dynamic moduli at high frequencies

267

Journal of Non-Newtonian Fluid Mechanics, 42 (1992) 267-282

Elsevier Science Publishers B.V., Amsterdam

The determination

of dynamic moduli at high frequencies

P.R. Williams and D.J.A. Williams Department of Chemical Engineering, University College of Swansea, Singleton Park, Swansea SA2 8PP (UK)

(Received November 26, 1990; in revised form October 17, 1991)

Abstract A new technique for the measurement of dynamic moduli at high frequencies is presented, based upon the properties of a virtual gap formed by two unequal shear wave path lengths. Measuring procedures and underlying theory are presented which overcome many of the difficulties inherent in the high-frequency rheometry of concentrated dispersions. Use of the technique is illustrated with results obtained on dispersions of a commercially available model clay colloid. Keywords: colloidal dispersions;

high frequencies;

shear waves

1. Introduction A comprehensive characterisation of the dynamic rheological properties of many systems of theoretical and commercial importance requires testing in frequency ranges above those provided by commercially available I-heometers such as the Weissenberg Rheogoniometer or the many controlled-stress instruments. In rheological studies of concentrated dispersions, consideration of the relative time scales of particle-fluid displacement and interfacial polarisation mechanisms indicates a range of about 102-lo4 Hz is necessary [l]. In addition, controlled length scales of deformation must be employed to facilitate attempts to relate measured mechanical properties to fundamental aspects of particle interactions. The use of high frequencies in conjunction with the often mechanically weak materials under consideration may compromise the accuracy of Correspondence to: P.R. Williams, Department of Chemical Engineering, of Swansea, Singleton Park, Swansea SA2 8PP, UK. 0377-0257/92/$05.00

University College

0 1992 - Elsevier Science Publishers B.V. Ail rights reserved

268 dynamic measurements based upon assumptions of “gap-loading” conditions with negligible (sample) inertial forces [2]. In addition, the requisite ratio of shearing gap to shear wavelengths may conflict with the gap size requirement for adequate bulk measurements. However, shear wave propagation techniques which assume “surface’‘-loading conditions (involving freely propagating waves) will rarely be practicable, being limited to experimental systems whose attenuative properties prevent interfering wave effects complicating the analysis of measurements [3]. The prevailing conditions in high-frequency dynamical measurements on systems, such as concentrated dispersions, can involve a complicated shear-wave field throughout the sample, while the dissipative characteristics of the systems serve to further complicate interpretation. Under such conditions, the magnitude and relative phase of sample deformation can differ considerably from that calculated from the motion of the bounding surfaces of rheometer measuring geometries. In such cases, it is appropriate to develop rheometry based upon the characteristics of the shear-wave field within the sample. This paper describes the development and use of such rheometry. 2. Theoretical We consider the propagation of plane shear waves in a viscoelastic medium of density p. It may be readily shown [4], that

where G* is the complex shear modulus, w( = 2rf) is the angular frequency in rad s-l, f is the ordinary cyclic frequency in Hertz and I is the complex wave propagation constant, which is conveniently written

,2+?_ X0

237’

where A is the shear wave length, and x0 is the distance in the direction of wave propagation over which the wave amplitude decreases by a factor e and i = \r-i-. Writing G * = G’ + iG”, we have 47r2 pm2 7-2 G’=

1

I

4,rr2 (- A2 ‘2

I

i

2 ’ 1

(24

269

G” =

WI

2'

where G’ is the storage modulus modulus and

(or dynamic rigidity), G” is the loss

G”

tan a=~, where 6 is the phase angle between shear stress and shear strain. Equations (2a) and (2b) take on simplified forms for special cases, e.g. for non-ideal solids with low attenuation such that A/x, K 1. Then it is assumed that G’ may be closely approximated by G,, a high-frequency limiting wave rigidity modulus G,; G, =pu2,

(4)

where u is the shear wave phase velocity and A = u/f = VT, where T is the shear wave period. Thus for linear viscoelastic systems, G’ and G” may be obtained from eqns (2a) and (2b) by measurement of A and x0 for known test frequency and sample density. 3. The virtual gap geometry: determination

of Y and 6

Let A, B and C be parallel plates in a linear viscoelastic medium (Fig. 1). A undergoes continuous forced harmonic displacement in its own plane, generating plane shear waves which travel towards B and C. B and C are at distances x1 and x2 from A such that x1 > x2. Thus a “virtual gap” CB of width Ax is created where Ax =x1 -x2 and Ax <
Fig. 1. The virtual gap geometry. The virtual gap (width Ax) is formed by difference between wave paths of length x1 and x2.

270

Fig. 2. (a) The determination of u and 6 using the virtual gap geometry: curve 1, stress at plate A (calculated); curve 2, displacement of plate A (measured); curve 3, stress at plate C (measured); curve 4, stress at plate B (measured). (b) The determination of maximum local shear strain amplitude (-ym,): curve 1, displacement of plate A, x = 0,(measured); curve 2, stress at x = x’, (calculated).

response to the harmonic motion of A, the common source of shear waves. The displacement of A may be measured (e.g. by a non-contacting displacement transducer). Under steady-state conditions, the time-varying form of the stress response at B and C (also harmonic for a linear viscoelastic material) will be as shown in Fig. 2(a). The shear wave phase velocity (i.e. the propagation velocity of the fluid mechanical stress) in terms of the virtual gap AX is given by v = Axw/$,

(5)

where @r is the phase angle between curves (3) and (4) (see legend Fig. 2(a)) and A = 27rAx/@,. Writing the stress at plate B (X =x1) as c = a0 e’“‘,

(6)

then the phase of the stress at the driving plate A(x = 0) is eiw(t+xl/v) (see curve (l), Fig. 2(a)). In the vicinity of the driving plate (x --) 0) the complex shear strain y is given by

(see Ref. 4), and thus y = 0 when z = 0 (i.e. plate displacement and shear strain at the plate surface are instantaneously in phase). Accordingly, the phase angle S between shear stress (from the instantaneous zero stress of

271 waveform (1) of Fig. 2(a)) and shear strain (for instantaneous zero displacement of plate A) may be found where 0 I 6 I 90”. Linearity of viscoelastic response may be tested by varying the displacement amplitude of A and noting the ranges of applied displacement which produce no change in @i and 6. Within the linearity limit thereby established for the medium in both virtual gap Ax and physical gaps x1 and x2, G’ and G” are obtained from eqns. (2a), (2b) and (3), knowing tan 6 and A. From eqns. (2a) and (2b)

where

Equation (7) may be solved for x0 and thus G’ and G” are found. Note that for A/x,, -x 1, x0 = h/rr tan 6, which may provide a ready estimate of X0*

The maximum local shear strain amplitude y,, is found as follows (Fig. 2(b)). Curve (2) of Fig. 2(b) is calculated from U, x1 and x’. The phase of instantaneous zero displacement at x =x’ is then calculated from curve (2) using 6. In terms of Fig. 2(b), the maximum synoptic difference in transverse displacement, D (calculated with the displacement of the lamina at x’ instantaneously zero), over a distance x’, allows ymax to be obtained from D Y

=Inax

xt)

where x’ is chosen such that the theoretical gap-loading conditions (p +Z G’/(x’)~~~ G”/(x’)~~~), are satisfied (see Refs. 2 and 3). In the present analysis the phase angle 6 is obtained for stress and strain at the same surface, the wave-generating plate. Thus the phase error due to finite gap size of conventional applied displacement oscillation techniques is avoided. However, in contrast to conventional techniques where force and displacement are measured at the same surface [5] no inertial correction for motion of the driven surface is required as the force is actually measured at surfaces with negligible motion (B and C of Fig. 1). No corrections for compliance of the drive mechanism are required, nor are the phase characteristics of the driving mechanism (the induced phase lag between voltage and force in an electro-mechanical system). It should be noted that absolute determination of force is not required, only the phase

272 angle between the force measured at the two bounding surfaces of the virtual gap. Plane waves are assumed in the derivation of eqns. (2a) and (2b). Deviations from a one-dimensional disturbance due to the presence of bounding surfaces and driving surface dimensions when h/x, is small can cause overestimates of attenuation in direct measurements. Accurate determination of G” is most susceptible to such errors. In the technique described here, A and tan S (and G’ and G” therefrom) are determined by the more reliable method of direct measurement of phase velocity [6]. Further aspects of shear wave propagation exploited in the virtual gap geometry are now discussed. 4. Shear wave reflection and attenuation Consider the gap of width x2 between plates A and C of Fig. 1. Let A be undergoing forced harmonic displacement with amplitude a = a, cos(wt). The wave amplitude constants a = a, exp(iq,) and b = b, exp - (i*J for outgoing and reflected shear waves (travelling in the positive and negative x-direction) have phase angles ?P1 and qZ respectively. Wave reflection at plate C produces the resultant shear wave field between A and C with complex amplitude function

l(k,

X) = 2b cos(kx) + (a -b)

exp(iluc)

(9)

and propagation constant k = 27r/A. Thus the wave field in either gap (between plates A and B or A and C> can be represented as the sum of standing and travelling waves. With a = b in (9) the standing wave amplitude varies from zero at the nodes to 2a, at the antinodes and the distance between successive nodes (or antinodes) is h/2. In the region between two nodes, the sign of cos(kx) is constant while crossing a node produces a change of phase angle @* of 180”. In practice, due to wave attenuation and imperfect reflection b
exp( -X/X,),

b = b, exp - (iq*) exp( -x/x,), Then e(k, x) =

2X/Xo)[2b cos(kx) + (u -b) (’+e2 cosh( x/x,,)

exp(ikx)]

(10)

273

For large values of x/x0 the variation in amplitude the driving plate is of the form

E with distance x from

E = Ed exp( -X/X,),

(11) so that a plot of log E against x should be linear and of slope -0.434/x,. Equations (9) and (10) provide the basis for alternative experimental procedures based upon standard interferometric methods [7] for the direct determination of A, and x,, (and G’, G” therefrom, see Section 7). These procedures serve herein to provide direct comparison of results with those obtained with the virtual gap geometry as direct comparison with conventional rheometers is clearly not possible in the frequency range under consideration. 5. Selection of gap widths for the virtual gap geometry Errors in phase velocity measurement will occur in the method described in Section 3 if plates B and C occupy adjacent regions of space where the phase angle of motion (due to the standing wave component of the wave field) @)2 is the same, but the plates are not located between the same nodes (or minima). This is overcome by setting x1 =x2 at maximum amplitude (Q,, = 0) and decreasing the gaps fractionally to obtain optimum response. One gap is then decreased further (to width x2 in terms of Fig. 2(a)) care being taken not to invoke 180” phase change. Both plates are then situated between the same nodes (or minima), both within h/4 of the same node. They may be situated within h/2 of the same node for increased phase resolution. The upper limiting test frequency f,, for velocity determination for a virtual gap Ax is simply calculated, as At between waveform reference points must be less than T, the wave period. Limiting frequencies are shown in Fig. 3 for a range of Ax and u (assuming G, = pu2, for the sake of illustration). The limiting frequency condition is Ax ___

&Z/P

< T.

The gap-setting criteria and procedure described here have the advantage of providing a ready estimate of U, as the distance between minima and maxima is u/4f. Thus Ax may be set appropriately with regard to the required frequency range and an accurate determination of u made. It may be seen from the foregoing that, by holding Ax small, one virtual gap configuration will serve to determine a wide range of dynamical conditions. The gap-loading criteria (see Section 3) reveal that the virtual gap geometry is especially suited to the study of weak materials at high

274 f",,IHzl IO'-

10-l

IO0

10'

Fig. 3. Limiting frequencies for a range of virtual gap widths.

frequencies (low A) where relatively large particulate constituents necessitate large X. Where sample characteristics permit, the physical gaps (x1, x2 in Fig. 1) may be set such that the gap-loading conditions are satisfied. 6. Experimental The apparatus described in this Section is based upon, and intended to illustrate, the use of the foregoing theory. A perspex stalk is attached by screw fitting to a bearing plate suspended between a set of electromagnetic coils (see Figs. 4 and 5). The electromagnetic drive coils and attached stalk (the drive assembly) are fixed in a vertical pillar by a gimballed mount, providing a range of vertical and horizontal displacements. During operation the lower (wave-generating) surfaces of the stalk (corresponding to A in Fig. 1) lie between the upper halves of two steel discs (corresponding to B and C of Fig. 1) in direct contact with a matched pair of piezocrystals (BSR x 5H). The requisite virtual gap configuration is achieved by micrometer-driven positioning of disc B and a facility to move

275

-

r

r

Fig. 4. Experimental apparatus, showing: (1) sample reservoir, (2) receiving/reflecting plates: (3) piezo-ceramic transducers; (4) sample cell and water jacket; (5) signal output to microcomputer; (6) micrometer; (7) water inlet to (4) (above); (8) cover plates; (9) drive assembly; (10) support pillar; (11) wave-generating surfaces; (12) electromagnetic coils.

the support pillar relative to the test cell. Figure 4 shows the drive assembly in the measuring position with driving surfaces and discs forming a small sub-cell within the much larger sample reservoir (length 8 cm; diameter 4.5 cm). The shape of the drive member was chosen to minimise the generation of extraneous resonances in the test cell by limiting shear wave paths to those directed towards the receiving discs. Shear wave paths outside the driver-receiver sub-cell are relatively large (e.g. from driver to cell base is often > 20A) and in operation are found to have no detectable influence on results. The sample cell is enclosed within a thermostatically controlled waterjacket, fitted with a removable perspex cover for fitting, cleaning, inspec-

d i

Fig. 5. The drive member, showing: (a) screw attachment to electromagnetic wave-generating surfaces; (c) diameter (11 mm); (d) height (9 mm).

coils; (b)

276 tion, etc. It incorporates two sliding cover plates which are recessed to accommodate the perspex stalk. Sinusoidal voltages from a signal generator to the drive coils cause the stalk to execute small-amplitude harmonic oscillation over a wide range of frequency (100-800 Hz) and displacement amplitude. The shear waves thereby generated travel across the gaps of width x1 and x2 through viscoelastic samples and are detected by the receiving discs. Signal outputs from the disc transducers (typically a few hundred millivolts) are fed to a twin-channel adjustable notch filter with 40 dB gain per chanel, thence to a 50 kHz 12-bit precision analogue-to-digital convertor and PC-AT microcomputer for data analysis. Fourier analysis is used to check for harmonic distortion, and to measure the requisite phase angles. Electronically generated differential phase shifts are accounted for in the gap-setting procedure (see Section 5) with x1 =x2 = 0 (maximum amplitude settings). Curvature of the wave-generating surfaces, which are cut from a hollow cylinder of perspex, produces spherical waves. At large distances from these surfaces the curvature of the spherical wavefront is extremely small and the wave can be approximated locally by a plane wave, i.e. in the far field where kx > 1. When frequency is low, the test materials of interest have shear wavelengths of a millimetre or so. In practice, gaps of about 5-10 mm are used so that kx is in the range 30 I kx I 70 and the far-field condition can be readily satisfied under all likely test conditions. In the case of more elastic materials (larger A) there is generally a concomitant increase in useable path length. At high frequencies (> 400 Hz) kx often considerably exceeds 102. In the virtual gap geometry, where the attenuative properties of the sample permit, gap widths x1 and x2 can be made very large compared with Ax, thereby ensuring luc is large without sacrificing accuracy of measurement. Additional measurements are reported here for comparison using a modified Rank Pulse Shearometer system with a nominal operating frequency of 220 Hz. Details of this instrument and its use have been given elsewhere [8]. Equation (4) (this paper) was used to convert shear wave velocity to G,. The test material is a synthetic smectite colloid called Laponite (grade RD) (Laporte Industries, U.K.) which forms viscoelastic gels in aqueous systems. Laponite has been accepted as a model clay colloid for fundamental studies; some aspects of its dynamic rheological properties have been reported [9]. Samples were prepared by successive addition of small amounts of dry Laponite powder to the aqueous phase using ultrasonification to effect complete dispersal of the powder. The pH of the resulting dispersions was

277 9.8. Where samples were made up over a range of molar&y all chemicals used were AnalaR grade. All measurements were performed at 20°C. 7. Results and discussion The following results were obtained using the interferometric method of Section 4 and are intended to serve for comparison with results obtained using the virtual gap geometry and associated procedures. The magnitude of relevant experimental errors is indicated on each curve, being typically the order of a few percent. Figure 6 shows the linearity of viscoelastic response obtained by the interferometric method of Section 4 for a sample dispersed in twice-distilled water (2 X H,O) with solids concentration 0.06 g ml-’ at a test frequency of 500 Hz. The results in Fig. 6 were obtained by moving the micrometer-driven receiving plate (see Fig. 4) by successive incremental amounts (in the x-direction) from the wave-generating surface. The maximum equilibrium amplitude response was noted at each successive position. The response was recorded over a range of driving-plate amplitude at each (constant) frequency, and over a range of frequency at each drivingplate amplitude. Records such as Fig. 6 in conjunction with attenuation plots (see Fig. 8) are a sensitive test of the linearity of viscoelastic response. Note that no change occurs in recorded peak spacing for a range of driving plate displacement amplitude. Linearity is therefore established for the spatially distributed shear strain amplitude range invoked in the experiment. A (and hence v> was determined directly from the recorded peak spacing and x,, 700-

soos E

LOO-

d 2 c -z

300 -

-200

g % 3 c z

2

5 200 --1

-100

100 O-+ 01

I 02

I 03

I 04

I 05

I 06

Fig. 6. Variation of dynamical response Laponite RD (0.06 g ml- ‘, 2 X H,O).

I 07

I 08

dtcml

with wave amplitude

at 500 Hz; test sample

278

600 -

500 ~ 9 5 d 3

400 -

2

300 -

4E 200 -

100 -

O,I-i

01

02

03

04

05

06

07

08

dfcml

Fig. 7. Variation of dynamical response at 300 and 500 Hz; test sample as in Fig. 6.

from eqn. (11). Figure 7 shows the variation of wavelength directly determined between 300 and 500 Hz for this sample. Attenuation measurements derived from the envelope of peak amplitude vs. plate separation on this sample and another at 0.05 g ml-’ are shown in Fig. 8. The variation of linear dynamic response (in terms of wavelength) at a fixed frequency of 500 Hz and fixed solids concentration (0.06 g ml-‘) in the presence of NaCl is shown in Fig. 9. Little change is detected between suspensions in 2 X H,O and low3 M NaCl; the wavelength clearly increases significantly in the presence of lo-* M NaCl. Figure 10 compares the results obtained using the interferometric method of Section 4, the pulse-propagation (Shearometer) technique and those found with the virtual gap geometry. In the last, x0 was determined by the method of Section 4. Good agreement is found in all cases between the

logA

1 --_==:

I

3LI 72

73

II 71,

75

II 76

I 77

70

I 79

I 80

I

121 I

81

82

ximml

Fig. 8. Attenuation plots for two samples; test for linearity. Curve (1) Laponite RD (0.06 g ml-‘, 2xH,O); curve (2) Laponite RD (0.06 g ml-‘, 2xHzO).

279

1000

200 0 01

02

03

04

05

06

07

08

09

10

11

12

13 dlcml

Fig. 9. Results of tests at fixed frequency (500 Hz) over a range of electrolyte concentration. Sample is Laponite RD, solids concentration 0.06 g ml-‘; (0) lo-* M NaCI, (0) 10m3 M NaCl, CO), 2xH,O.

virtual gap and interferometric techniques, and the limited results at 220 Hz using the Shearometer. It was noticed, however, that in all cases the Shearometer gave slightly lower values of u (by between 5 and 10%) than the other techniques. This may be due in part to the nature of the pulse technique, in which strains are largely uncontrolled and precise velocity determination is impossible. The dynamic moduli determined by all three techniques are found to increase with increasing electrolyte concentration in agreement with other published work on this material (see Ref. 91, and allowing for the higher frequencies used in the present work, are commensurate in magnitude. A plateau modulus is clearly found between 200 and 500 Hz for dispersions in 2 X H,O and 10e3 M salt. The frequency dependence of the sample in lop2 M salt is interesting and possibly related to a change in the mode of

FrequencyiHzI

Fig. 10. Comparison of results (G’ vs. frequency) using three techniques; (o), (0) and (A) virtual gap; (A) interferometry; W pulse propagation-Shearometer. Curve (0, lo-* M NaC1; curve (21, lop3 M NaCl; curve (3), 2x H,O; all 0.06 g ml-‘.

280 v(m Is1 16,

Frequency

(Hz)

Fig. 11. Comparison of velocity measurements using two techniques: (o), (0) virtual gap; (0) (0) interferometry. Samples are Laponite RD. Curve (l), 2XH,O, 0.05 g ml-‘; curve (2), 2X H,O, 0.03 g ml-‘.

particle interaction and enhanced interfacial polarisation mechanisms due to electrical double layer compression [9,10]. Figure 11 shows a direct comparison of u obtained by two methods on samples of solids content 0.03 and 0.05 g ml-’ in 2 X H,O. Again good agreement is found between the standard interferometric method and the results obtained with the virtual gap geometry. 8. Concluding

remarks

The virtual gap geometry (and associated measurement technique) has been found to provide a convenient and rapid characterisation of the dynamic viscoelastic properties of concentrated dispersions at high frequencies. Notable features of the technique are that (in general) one geometric configuration of the measuring system serves to characterise an unusually wide dynamic range while permitting the testing of suspensions containing relatively large particulate or aggregate constituents. The technique is being further developed for the study of concentrated dispersions and other materials where detailed consideration of the length and time scales of high-frequency deformation are required. Future work will incorporate the measurement of wave-generating surface displacement to allow the direct determination of 6. Acknowledgement

PRW is grateful for the support of an SERC Advanced Fellowship.

281 References 1 J.A. Helsen, R. Govaerts, G. Schoukens, J. DeGraeuwe and J. Mewis, J. Phys. E (Sci. Instrum.), 11 (1978) 139-144. J.L. Schrag, Trans. Sot. Rheol., 21(3) (1977) 399-413. J.D. Ferry, Viscoelastic Properties of Polymers, 3rd edn., Wiley, New York, 1980. R.W. Whorlow, Rheological Techniques, Halsted Press, London, 1980. M.H. Birnboim and J.D. Ferry, J. Appl. Phys., 32 (1961) 2305-2313. F.T. Adler, W.M. Sawyer and J.D. Ferry, J. Appl. Phys., 20 (1949) 1036-1041. R.W.B. Stephens and A.E. Bate, Acoustics and Vibrational Physics, 2nd edn., Arnold, London, 1966. 8 A.E. James, D.J.A. Williams and P.R. Williams, Rheol. Acta, 26 (1987) 437-446. 9 J.D.F. Ramsay, J. Colloid Interface Sci., 109 (1986) 441-447. 10 S.S. Dukhin and V.N. Shilov, Dielectric Phenomena and the Double Layer in Disperse Systems and Polyelectrolytes, Keter, Jerusalem, 1974. 11 W.J. Fry, J. Acoust. Sot. Am., 20 (1949) 17-28.

Appendix: Direct determination of shear wavelength and attenuation by frequency variation in the virtual gap geometry Due to wave reflection and attenuation, minima and maxima are recorded by plates B and C as their positions in the x direction are varied. The differential path lengths x1 and x2 produce a phase angle @r due to the travelling wave component of eqn. (10) across the virtual gap where one (or both) gaps is non-resonant. As the frequency is varied, maxima or minima will be recorded independently at plates B or C, as one or other corresponding fixed gaps passes through a resonant or anti-resonant condition; aI = 0 for these cases. A range of discrete frequencies will be encountered for which both gaps are simultaneously resonant or anti-resonant; then QI = 0, and x0 may be obtained directly by noting the recorded maxima at each plate. The position of either (or both) receiving plates may be adjusted to provide further data with aI = 0. x0 may then be calculated from eqns. (10) and/or (11) depending on the values of x1 and x2. The accuracy of direct attenuation measurements at large x/h and low standing wave ratio has been discussed by Fry [ll]. For successive minima recorded simultaneously at B and C with @r = 0 for frequencies fr and fi, then, with x1 and x2 simultaneously integral multiples of A/2 (see eqn. (9)), Ax is also an integral multiple of A/2 and so, in terms of the virtual gap Ax n4 Ax= - 2 u =&A,,

where n = 1, 2, 3,. . . ,

282

5 10-l r

510.'

510.)

5 10“ lo‘*

10-l

10' A,. mm

lop

Fig. Al. (l/F - 1) for typical values of wavelength and virtual gap width: (l), Ax = 10-l mm; (21, Ax = 10’ mm; (31, Ax = lo1 mm.

and AX = (’ +‘)‘IA2 7 where u =f 2 A2 at fi and f2 respectively. For non-dispersive systems fl A, = -A,, f2

so that

fl

where -=F. f2

Thus the wavelength A, is rapidly determined by measurement of virtual gap width and two frequencies. Successive maxima may be utilised with account taken for 180” phase change. The function (l/F - 1) is plotted for a range of Ax and A in Figure Al. It is seen that for the systems of interest here (given approximately by the shaded area) and typical virtual gaps of the order of a millimetre, only small percentage changes in frequency are required to effect measurement of wavelength by this technique. Thus the method may be extended with some success to weakly dispersive systems in which the dynamic moduli are only slowly varying functions of frequency.