- Email: [email protected]
Scattering by non-metallic spheres
Scattering
by non-metallic
P.D. Thorne”, Proudman L43 7RA, 1_Physics
L. Hayhurst?
Oceanographic
spheres
and V.F. Humphrey1
Laboratory,
Bidston
Observatory,
Birkenhead,
Merseyside
UK Department,
UK 1 School
of Physics,
Received
28 May
Liverpool University
Polytechnic, of Bath,
Byrom
Claverton
Street, Down,
Liverpool, Bath, Avon
L3 3AF,
Merseyside BA2 7AY,
UK
1991
In recent years acoustic backscattering has been employed to monitor the transport of suspended sediments. To develop our understanding of this interaction, measurements on suspensions of glass spheres have been and are being conducted. Generally these studies are underpinned by the form function description of sphere scattering. It is therefore necessary to consider the scattering by single glass spheres and this is examined here. The form function is considered in terms of its rigid and resonant components and this approach is employed to provide an understanding of the form function structure. The effect upon the form function of having a particle size distribution within a suspension is also appraised and its implications for scattering from suspensions are discussed. Keywords:
scattering;
spheres;
sediments
The use of acoustics as an investigatory method for measuring sedimentary transport processes in the marine environment is steadily gaining increasing acceptance within the sedimentology fraternity’ -3. The main acoustic technique employed for suspension measurements is a downward looking megahertz transceiver, usually mounted a metre or so above the bed, which measures suspended sediments by recording the backscattered signal. The advantage of the acoustic method over more traditional techniques is that it offers the potential of using a single instrument to obtain, non-intrusively, high spatial and temporal resolution estimates of the nearbed concentration profile; a goal sedimentologists have been pursuing for a number of years. Recent publications1~4~7 have described the use at sea of such acoustic devices and these have illustrated the detailed sediment transport processes that can be observed using acoustics. The approach commonly employed to extract information on the marine suspension from the acoustic data has been to painstakingly calibrate the acoustic system for a range of sediment sizes and at a number of frequencies’x4,* - 1‘. An empirical algorithm is then developed from the calibration exercises to translate the acoustic measurements into suspended sediment concentration profiles, which are employed for examining sedimentary transport mechanisms. There is an element of uncertainty in this empirical procedure which has led to a degree of circumspection in defining the accuracy of the final concentrations calculated. The development of a full theoretical explanation for the scattering processes * Present address (for the next year) : Applied Research Laboratories, The University of Texas at Austin, PO Box 8029, Austin, TX 78713-8029, USA
0041-624X/92/01 0015-06 @ 1992 Butterworth-Hememann
Ltd
would therefore be worthwhile; it should make the calibration exercises unnecessary and provide further confidence in the sedimentary parameters estimated from the acoustic data. A step in resolving the problem of the interaction of sound with a suspension composed of marine sediments would be to examine the scattering of sound by a suspension of spherical scatterers. Progress has already been made in this direction, with attenuation measurements of glass spheres in suspension having been conductedt2 and resulting in good agreement between theory and experiment. Also limited backscattering measurements on a suspension of glass spheres have been reported13. The reason for using glass spheres is based on the similarity of the mechanical and acoustical properties of glass to those of silica, which is the most common component of non-cohesive marine sediments. The cornerstone of these analyses of the interaction of sound with a suspension is the description of the sound scattered by a single sphere. The classical problem of scattering by a sphere can be described theoretically’4-‘6 and measurements on metallic spheres have validated the accuracy of the description’7-19. However, no single sphere scattering results on amorphous non-crystalline material are available. There is, therefore, the need to consider scattering by single glass spheres and this is examined here. The description of sound scattering by a sphere is formulated in terms of the form functionL5, which characterizes the scattering properties of a sphere. Consideration is given here to resolving the composition of the form function in terms of the backscattering from a rigid mobile sphere and the resonance response of the sphere. Comparisons of the predictions are made with a
Ultrasonics
1992
Vol 30 No 1
15
Scattering
by non-metallic
spheres:
P.D.
Thorne
et al
number of backscattering measurements conducted over a range of particle sizes and covering a broad frequency band. The effect of having a particle size distribution upon the structure of the form function is also considered.
Theory The backscattering from a sphere within the beam of a transceiver can be expressed experimentally as D2(e) -f,e 2r2
P, = aP,r,
-Zawrei(2kr
g
(2n+
f
as
=$ n-Tiov l)(-l)“brig)
(rig)
n +
m
To examine the resonant interaction of the sound with a sphere the background scattering is coherently subtracted from the full solution to yield the resonant constituent of the form function. This is given by fpS)=k
- 0s)
+ l)(-l)“(b,
zo(2n
-brig’)
1)(-l)%
Approaching the backscattered signal in this manner allows a better understanding of the essential features of the form function.
Experimental
arrangement
The apparatus used to measure the acoustic backscattering from a sphere is shown in Figure 1. A transceiver was
LINE
F
~
(2)
-WATER
n-0
where the time dependence coefficient b, is given by16
has been suppressed.
I
The WEIGHT
~
SPHERE
-___
(3) OPA(XIE PLASTIC
LY11
=
x:%p(x)
cx21
=
-
CONTAINER
I
-T 1.2m
-
-xh;“‘(x)
P, CI 12
=
CWfl
a32
=
2[jn(xd
!.I 13
=
Wn
L-i 33 --
+
+
x3iixJ - %jXxJ a22= xddx,)
1)
-
-
xljb(xl)l
1 )Cx,L(x,)
2x,jL(x,)
+ [x:
-
k3 = n(n+ l)j,(x,)
j,(xJl
- 2n(n + 1) + 2]j,(x,)
where Filter
x = ka,
x,=xz
r
x,-w’ ct
Cl
where: j, is the spherical Bessel function; hh” is the Hankel function; the prime denotes differentiation with respect to the argument; c is the velocity of sound in the fluid; c, and c1 are the shear wave and compressional wave velocities, respectively, in the sphere; p is the density of the fluid; and ps is the density of the sphere. It is informative16 when considering the form function characteristics of a sphere subject to acoustic insonification to decompose the backscattered signal into two components: the background scattered radiation associated with a rigid movable sphere and the resonance response which is identified with the modes of vibration of the sphere. The response of a rigid movable sphere is obtained by allowing c, + cc and c, -+ co. The form function can
16
Ultrasonics
(5)
n-
where: P, is the transmitted pressure at range rO (usually 1 m); r is the range to the particle; D( 0) is the transceiver directivity function (axisymmetry assumed); 8 is the angle the sphere subtends to the acoustic axis; a is the particle radius; k is the wavenumber; o is the angular frequency; CC,,, is the attenuation due to water absorption; andf, is the form function. Theoretically the farfield backscattered form function can be described by /_=&
then be expressed
1992 Vol 30 No 1
Low noise Amp
Broadband Transducer Figure
1
Experimental
arrangement
-
Scattering by non-metallic spheres: P.D. Thorne et al.
placed at the bottom of a water-filled tank and sound pulses were directed upwards. Measurements were taken using three transceivers covering the frequency ranges 0.35-0.95, 0.6-1.4 and 2.6-3.4 MHz. The transmit and receive sensitivities of the transceivers were calibrated using a PVDF membrane hydrophone and absolute pressure measurements were employed to measure the form function. The transceivers were driven by a sinusoidal signal, typically of 120 ps duration. The backscattered signal was amplified, filtered and, where necessary, averaged to improve the signal-to-noise ratio. The spheres employed were suspended in the farfield of the transceiver by a 80 pm diameter nylon line attached to the sphere by a thin film of soft adhesive. This arrangement of positioning the sphere above the transceiver minimized the influence of the mounting on the backscattered signal. The materials employed were tungsten carbide, lead glass and soda glass. The particles ranged in diameter from 0.7 to 9.0 mm. The tungsten carbide measurements were taken to provide a benchmark test for the experimental arrangement. Two glasses were examined to look at the differences in form function for nominally similar materials. The parameters used to compute the form function are shown in Table 1. The density of the spheres was measured directly and the Table 1
compression and shear wave velocities were obtained from the literature’2*20, with final adjustments inferred from the scattering measurements.
Results A comparison of the data collected for tungsten carbide with the theoretical predictions obtained using Equation (2) is shown in Figure 2~. Two sets of data taken by different observers are presented and each shows good agreement with the computations in terms of both the absolute level and the features of the form function. Previous studies17 have shown similar agreement and the concordance of prediction and observation obtained here is used as confirmation of the experimental accuracy. An interesting aspect of the result is the dips in the form function at nominally regular values of ka. This feature of the results is examined by decomposing the backscattered signal into its components. The results of computing Equation (4) are shown in Figure 2b, where it can be seen that for ka approximately greater than unity the backscattered form function for a mobile rigid sphere has a gradually damped oscillatory behaviour, tending towards a constant value of unity.
Values of the sphere and water parameters used to compute the form function Compressional velocity (ms-‘)
Shear velocity (ms-‘)
1000
1476
_
14935 2893 2586
6860 4860 5550
4140 3065 3545
Density (kg m-? Water Tungsten carbide Lead glass Soda glass
a
10.0
8.0
ka
b
ka Figure 2 (a) Comparison of the form function for tungsten carbide with two independent data sets. (b) (c) Resonance form function
Rigid mobile form function.
Ultrasonics 1992 Vol 30 No 1
17
Scattering
by non-metallic
spheres: P.D.
Thorne et
al.
The sharp dips characteristic of Figure 2a for an elastic sphere are not observed for the rigid case. The results calculated using Equation (5) are presented in Figure 2c. These results clearly show the resonant characteristics of the backscattered radiation, with a series of relatively high broad peaks and smaller narrower resonant structures. The nomenclature for the resonances’ ‘, (n, 1),corresponds to the modes of vibration of a free elastic sphere, where n is the nth partial wave in Equation (2) and 1 is the radial structure number. The wider peaks (n, 1) are associated with Rayleigh wave resonances which quickly radiate into the fluid, while the narrower band structures arise from whispering gallery wave modes which are poorly coupled to the fluid and take correspondingly longer to radiate into the fluid. Examination of the resonance structure clearly shows the origins of the form function structure shown in Figure 2~. The backscattered signal, in general, is close in form to the rigid case. However, in the regions near to the sphere resonances, the form function differs significantly from the background case. Figures 2b and c show this is due to the interactions of the two constituent components of the backscattered signal. The dips in the form function for ,tungsten carbide are associated with regions near sphere resonance radiating out of phase with the specular reflection, resulting in destructive interference in the backscatter direction. The scattering from tungsten carbide spheres was pursued to assess the experimental arrangement and because interpretation of the form function in terms of resonance modes and background scattering is well understood. The form function is, however, sensitive to the material composition of the sphere and it was,
Zl
Figure 3
(a) Comparison
of the form function
function
18
Ultrasonics
1992
Vol
30 No 1
3,1
therefore, necessary to move closer to the more relevant case of studying spheres with properties more similar to those of sedimentary material; hence, as previously mentioned, two glasses were examined. The results for the lead glass measurements are shown in Figure 3~. It can readily be seen that there is a significant difference between the form function for lead glass and tungsten carbide. The main feature of the form function is again the nominally regular reductions in amplitude with ka: however, the dips in the form function for lead glass are noticeably broader than for the case of tungsten carbide. This is principally due to the factor of six reduction in sphere density. There is also a decrease in the values of ka at which the troughs occur owing to the lower shear wave velocity for lead glass. Comparison of the theoretical predictions with the experimental data generally show good agreement, with the differences between theory and experiment generally being comparable to those in Figure 2~. For ka < 4 there is a greater divergence near the peaks which appeared genuine, although for the smaller size spheres, of the order of 1 mm diameter, the mounting may have been having a detectable influence on the backscattered signal. The computed background and resonance radiation components are shown in Figures 3b and c. The background component is essentially the same as that for tungsten carbide for ka > 2 with some amplitude difference below this value. However, the resonance constituents are substantially different. The resonant components are still present; however, the bandwidth of the peaks is greater, which is commensurate with the observations in Figure 3~. The interaction between the background and resonant constituents is less clear-cut than for tungsten carbide, although the resonance
431
591
‘391
for lead glass with the collected data. (b) Rigid mobile form function.
(c) Resonance form
Scattering
by non-metallic
spheres.. P.D.
et al.
Thorne
a
b
8.0
10.0
12.0
:4.3
16.0
l8.0
23.0
10.0
12.0
Iri.0
16.0
18.0
20.0
ka
8.0
ka Figure 4 (a) Comparison of the form function for soda glass with two independent data sets. (b) Comparison of the form function for soda glass of slightly different composition (see text for details) with the collected data. (c) Resonance form function
radiation still tends to reduce the amplitude of the backscattered signal. The results for soda glass are shown in Figures 4a-c. Two data sets were again taken. The manufacturing source for the soda glass spheres changed for sphere diameters above 7 mm, and Figure 4b shows the results for spheres from the second source, taken on the middle frequency range transducer. The shear wave velocity was decreased by 5.5% to calculate the form function in Figure 4b. The form functions in Figures 4a and b are seen to be comparable with those of lead glass and, as shown in Figure 4c, this is due to the similarities in the resonance component of the form function. The rigid components were essentially identical. The main discernable difference between the form functions for the lead glass and soda glass is that the resonances occur at the higher ka values in soda glass, and this is associated with its 15.7% increase in shear wave velocity. For backscattering measurements taken using suspensions of particles there will almost certainly be a size distribution within the suspension. To give an indication of the consequences this would have on the effective form function for a suspension, the average form function was computed using
(Ifm(%)l>
=&
Ifm(X)I som
x exp[ - (x - x,)*/202]dx
bolder line was calculated from Equation (6). For the 5% case, the structure in the form function is still partially in place, although the excursions have been reduced. However, by what would be a modest standard deviation for a natural suspension of marine sediments, 20%, the structure of the form function has been completely lost, with the form function having the shape of a simple high-pass filter response. It seems highly likely that Figure .5c would be representative of the form function for any marine suspensions of non-cohesive sediments that consist of nominally spheroidal particles. (The effect of non-sphericity on the form function has been examined22 for tungsten carbide spheroids and this causes an admixture in the resonance structure.) This high-pass filter approach has been very successfully adopted23 to explain measurements24 of attenuation by suspended sediments, although it should be noted that for the case of attenuation the backscattering cross-section does not exhibit the oscillations observed in the generally” backscattered form function and a smoothed mobile rigid sphere delineation described as a high-pass filter response was applied.
Conclusions
(6)
A Gaussian probability function has been used to represent the distribution, with x0 being the mean value and ~7the standard deviation. Figures 5a and b show results for c = 0.05~~ and 0.2x, for soda glass. The,fine line shows the form function for a single sphere and the
Measurements of the backscattered form function have been taken using tungsten carbide and glass spheres; the former because it has been studied by a number of previous investigators and provides a benchmark test, and the latter owing to the interest in modelling the interaction of sound with marine suspended sediments by using glass spheres. To aid the interpretation of the form function it was decomposed into its background
Ultrasonics
1992
Vol
30
No
1
19
Scattering
by non-metallic
spheres: P.D.
et al.
Thorne
a
ka
L-
‘1.0
-
~~
6.0
-‘J_,_
1
8.0
:3.0
ka Figure
5
Averaged
form function:
(a) standard deviation
5% of the mean; (b) standard deviation
and resonant components to help provide an understanding of the underlying structure of the form function. The variability in shape of the form function for differing materials can be seen to be principally associated with the change in form of the resonant structure. Comparison of theoretical predictions with the data sets for tungsten carbide, lead glass and soda glass showed good agreement, both in terms of the form function structure and its absolute level. This provides evidence that the form function description is appropriate for non-crystalline amorphous materials, and suspension studies conducted on this basis are appropriate. The consequences of having a particle size distribution on the form function is, in effect, to smooth out the resonant structure and, for even modest standard deviations, it results in a profile similar to that of a simple high-pass filter response. This shows that when making theoretical comparisons with suspension measurements, due allowance needs to be made for the variation in particle size, even for a case in which the particles could be considered nominally uniform in size. As far as scattering from natural sediments is concerned, the size distribution effect will normally be a dominant feature of the scattering response and a form function similar to that shown in Figure 5b will frequently be the pertinent curve to employ.
7
8 9
10 11
12 13
14 15 16 17 18
References Vincent, C.E., Huntley, D.A. and Clarke, T.L. Acoustic measurements of suspended sand concentration in the C’S’ experiment at Stanhope Land, Prince Edwards Island Marine Geology (1988) 81 185-196 Tborne, P.D., Williams, J.J. and Heathershaw, A.D. In situ acoustic measurements of marine gravel threshold and transport Sedimentology (1989) 36 61-74 Lowe, R.L. Acoustic bedload sensor Proc 17th Int Conf on Coastal Engineering Sydney, Australia (March 1980) 215-216 Hanes, D.M. and Vincent, C.E. Detailed dynamics of nearshore suspended sediments Proc Coastal Sediments 87 Conf Am Sot Chem Eng, New York, USA (1987) 285-299 Libicki, C., Bedford, K.W. and Lynch, J.F. The interpretation and evaluation of a 3 MHz acoustic backscatter device for measuring benthic boundary layer sediment dynamics J Acoust Sot Am ( 1989) 85 1501-1511 Vincent, C.E. and Green, M.O. Field measurements of the suspended sand concentration profiles and fluxes and of the resuspension coefficient y0 over a rippled bed J Geophys Res (1989) Hnnes, D.M.,
20
Ultrasonics
1992 Vol 30 No 1
19
20 21 22 23
24
20% of the mean (see text for details)
95 11591-11601 Thorne, P.D., Vincent, C.E., Hardcastle, P.J., Rehman, S. and Pearson, N. Measuring suspended sediment concentration using
acoustic backscatter devices Marine Geology (1991) 98 7-16 Young, R.A., Merrill, J.T., Clarke, T.L. and Proni, J.R. Acoustic profiling of suspended sediments in the marine bottom boundary layer Geophys Res Lett (1982) 9(2) 175-178 Crickmore, M.J., Shepherd, I.E., Dore, P.M. A field instrument for measuring the concentration and size of fine sand suspensions Proc Conf on Measuring Techniques London, UK (April 1986) 425-442 Jansen, R.H.J. The in-situ measurement of sediment transport by means of ultrasound scattering, Report 203, Delft Hydraulics Laboratory, Delft, The Netherlands (1978) Schaafsma, AS. and Kindereo, W.J.G.J. Ultrasonic instruments for the continuous measurement of suspended sand transport Proc IAHR Symp on Measuring Techniques in Hydraulic Research (Ed Wessels, A.C.E.) A.A. Balkema, Rotterdam, The Netherlands (1986) 125-136 Hay, A.E and Schaafsma, AS. Resonance scattering in suspensions J Acoust Sot Am (1989) 85(3) 1124-1138 Theme, P.D., Hayburst, L. and Campbell, S.C. Measurements of scattering from a suspension of spherical scatterers Proc Inst Acoustics: Acoustics ‘90 Vol 12, Inst Acoustics, St Albans, UK (1990) Pt 1, 115-122 Faran, J.J. Sound scattering by solid cylinders and spheres J Acoust Sot Am (1951) 23(4) 405-418 Hickling, R. Analysis of echoes from a solid elastic sphere in water J Acoust Sot Am (1962) 34(10) 1582-1592 Gaunaurd, G.C. and Uberall, H. RST analysis of monostatic and bistatic acoustic echoes from an elastic sphdre J Acoust Sot Am (1983) 73(l) 1-12 Neubauer, W.G., Vogt, R.H. and Dragonette, L.R. Acoustic reflections from elastic spheres: I. Steady-state signals J Acoust Sot Am (1974) SS(6) 1123-1129 Dragonette,
L.R., Vogt,
R.H.,
Flax,
L. and Neubauer,
W.G.
Acoustic reflections from elastic spheres and rigid spheres and spheroids: II. Transient analysis J Acoust Sot Am (1974) 5y6) 1130-1936 Humphrey, V.F., Murphy, C. and Moustafa, A.H.A. Wideband backscattering measurement using a parametric array Proc UI 87 Butterworths, Guildford, UK (1987) 265-270 Kaye, G.W.C. and Laby, T.H. Tables of Physical and Chemical Constants Longman, London, UK (1973) . Vogt, R.H. and Neubauer, W.G. Relationship between acoustic reflection and vibrational modes of elastic spheres J Acoust Sot Am (1976) 60(l) 15-22 Werby, M.F., Uberall, H., Nagl, A., Brown, S.H. and Dickey, J.W.
Bistatic scattering and identification of the resonances of elastic spheroids J Acoust Sot Am (1988) f&l(4) 1425-1436 Sheng, J. and Hay, A.E. An examination of the spherical scatterer approximation in aqueous suspensions of sand J Acoust Sot Am (1988) 83(2) 598-610 Flammer, G.H. Ultrasonic measurement of suspended sediments Geological Survey Bulletin No. 1141-A US GPO, Washington DC, USA (1962)
by non-metallic
P.D. Thorne”, Proudman L43 7RA, 1_Physics
L. Hayhurst?
Oceanographic
spheres
and V.F. Humphrey1
Laboratory,
Bidston
Observatory,
Birkenhead,
Merseyside
UK Department,
UK 1 School
of Physics,
Received
28 May
Liverpool University
Polytechnic, of Bath,
Byrom
Claverton
Street, Down,
Liverpool, Bath, Avon
L3 3AF,
Merseyside BA2 7AY,
UK
1991
In recent years acoustic backscattering has been employed to monitor the transport of suspended sediments. To develop our understanding of this interaction, measurements on suspensions of glass spheres have been and are being conducted. Generally these studies are underpinned by the form function description of sphere scattering. It is therefore necessary to consider the scattering by single glass spheres and this is examined here. The form function is considered in terms of its rigid and resonant components and this approach is employed to provide an understanding of the form function structure. The effect upon the form function of having a particle size distribution within a suspension is also appraised and its implications for scattering from suspensions are discussed. Keywords:
scattering;
spheres;
sediments
The use of acoustics as an investigatory method for measuring sedimentary transport processes in the marine environment is steadily gaining increasing acceptance within the sedimentology fraternity’ -3. The main acoustic technique employed for suspension measurements is a downward looking megahertz transceiver, usually mounted a metre or so above the bed, which measures suspended sediments by recording the backscattered signal. The advantage of the acoustic method over more traditional techniques is that it offers the potential of using a single instrument to obtain, non-intrusively, high spatial and temporal resolution estimates of the nearbed concentration profile; a goal sedimentologists have been pursuing for a number of years. Recent publications1~4~7 have described the use at sea of such acoustic devices and these have illustrated the detailed sediment transport processes that can be observed using acoustics. The approach commonly employed to extract information on the marine suspension from the acoustic data has been to painstakingly calibrate the acoustic system for a range of sediment sizes and at a number of frequencies’x4,* - 1‘. An empirical algorithm is then developed from the calibration exercises to translate the acoustic measurements into suspended sediment concentration profiles, which are employed for examining sedimentary transport mechanisms. There is an element of uncertainty in this empirical procedure which has led to a degree of circumspection in defining the accuracy of the final concentrations calculated. The development of a full theoretical explanation for the scattering processes * Present address (for the next year) : Applied Research Laboratories, The University of Texas at Austin, PO Box 8029, Austin, TX 78713-8029, USA
0041-624X/92/01 0015-06 @ 1992 Butterworth-Hememann
Ltd
would therefore be worthwhile; it should make the calibration exercises unnecessary and provide further confidence in the sedimentary parameters estimated from the acoustic data. A step in resolving the problem of the interaction of sound with a suspension composed of marine sediments would be to examine the scattering of sound by a suspension of spherical scatterers. Progress has already been made in this direction, with attenuation measurements of glass spheres in suspension having been conductedt2 and resulting in good agreement between theory and experiment. Also limited backscattering measurements on a suspension of glass spheres have been reported13. The reason for using glass spheres is based on the similarity of the mechanical and acoustical properties of glass to those of silica, which is the most common component of non-cohesive marine sediments. The cornerstone of these analyses of the interaction of sound with a suspension is the description of the sound scattered by a single sphere. The classical problem of scattering by a sphere can be described theoretically’4-‘6 and measurements on metallic spheres have validated the accuracy of the description’7-19. However, no single sphere scattering results on amorphous non-crystalline material are available. There is, therefore, the need to consider scattering by single glass spheres and this is examined here. The description of sound scattering by a sphere is formulated in terms of the form functionL5, which characterizes the scattering properties of a sphere. Consideration is given here to resolving the composition of the form function in terms of the backscattering from a rigid mobile sphere and the resonance response of the sphere. Comparisons of the predictions are made with a
Ultrasonics
1992
Vol 30 No 1
15
Scattering
by non-metallic
spheres:
P.D.
Thorne
et al
number of backscattering measurements conducted over a range of particle sizes and covering a broad frequency band. The effect of having a particle size distribution upon the structure of the form function is also considered.
Theory The backscattering from a sphere within the beam of a transceiver can be expressed experimentally as D2(e) -f,e 2r2
P, = aP,r,
-Zawrei(2kr
g
(2n+
f
as
=$ n-Tiov l)(-l)“brig)
(rig)
n +
m
To examine the resonant interaction of the sound with a sphere the background scattering is coherently subtracted from the full solution to yield the resonant constituent of the form function. This is given by fpS)=k
- 0s)
+ l)(-l)“(b,
zo(2n
-brig’)
1)(-l)%
Approaching the backscattered signal in this manner allows a better understanding of the essential features of the form function.
Experimental
arrangement
The apparatus used to measure the acoustic backscattering from a sphere is shown in Figure 1. A transceiver was
LINE
F
~
(2)
-WATER
n-0
where the time dependence coefficient b, is given by16
has been suppressed.
I
The WEIGHT
~
SPHERE
-___
(3) OPA(XIE PLASTIC
LY11
=
x:%p(x)
cx21
=
-
CONTAINER
I
-T 1.2m
-
-xh;“‘(x)
P, CI 12
=
CWfl
a32
=
2[jn(xd
!.I 13
=
Wn
L-i 33 --
+
+
x3iixJ - %jXxJ a22= xddx,)
1)
-
-
xljb(xl)l
1 )Cx,L(x,)
2x,jL(x,)
+ [x:
-
k3 = n(n+ l)j,(x,)
j,(xJl
- 2n(n + 1) + 2]j,(x,)
where Filter
x = ka,
x,=xz
r
x,-w’ ct
Cl
where: j, is the spherical Bessel function; hh” is the Hankel function; the prime denotes differentiation with respect to the argument; c is the velocity of sound in the fluid; c, and c1 are the shear wave and compressional wave velocities, respectively, in the sphere; p is the density of the fluid; and ps is the density of the sphere. It is informative16 when considering the form function characteristics of a sphere subject to acoustic insonification to decompose the backscattered signal into two components: the background scattered radiation associated with a rigid movable sphere and the resonance response which is identified with the modes of vibration of the sphere. The response of a rigid movable sphere is obtained by allowing c, + cc and c, -+ co. The form function can
16
Ultrasonics
(5)
n-
where: P, is the transmitted pressure at range rO (usually 1 m); r is the range to the particle; D( 0) is the transceiver directivity function (axisymmetry assumed); 8 is the angle the sphere subtends to the acoustic axis; a is the particle radius; k is the wavenumber; o is the angular frequency; CC,,, is the attenuation due to water absorption; andf, is the form function. Theoretically the farfield backscattered form function can be described by /_=&
then be expressed
1992 Vol 30 No 1
Low noise Amp
Broadband Transducer Figure
1
Experimental
arrangement
-
Scattering by non-metallic spheres: P.D. Thorne et al.
placed at the bottom of a water-filled tank and sound pulses were directed upwards. Measurements were taken using three transceivers covering the frequency ranges 0.35-0.95, 0.6-1.4 and 2.6-3.4 MHz. The transmit and receive sensitivities of the transceivers were calibrated using a PVDF membrane hydrophone and absolute pressure measurements were employed to measure the form function. The transceivers were driven by a sinusoidal signal, typically of 120 ps duration. The backscattered signal was amplified, filtered and, where necessary, averaged to improve the signal-to-noise ratio. The spheres employed were suspended in the farfield of the transceiver by a 80 pm diameter nylon line attached to the sphere by a thin film of soft adhesive. This arrangement of positioning the sphere above the transceiver minimized the influence of the mounting on the backscattered signal. The materials employed were tungsten carbide, lead glass and soda glass. The particles ranged in diameter from 0.7 to 9.0 mm. The tungsten carbide measurements were taken to provide a benchmark test for the experimental arrangement. Two glasses were examined to look at the differences in form function for nominally similar materials. The parameters used to compute the form function are shown in Table 1. The density of the spheres was measured directly and the Table 1
compression and shear wave velocities were obtained from the literature’2*20, with final adjustments inferred from the scattering measurements.
Results A comparison of the data collected for tungsten carbide with the theoretical predictions obtained using Equation (2) is shown in Figure 2~. Two sets of data taken by different observers are presented and each shows good agreement with the computations in terms of both the absolute level and the features of the form function. Previous studies17 have shown similar agreement and the concordance of prediction and observation obtained here is used as confirmation of the experimental accuracy. An interesting aspect of the result is the dips in the form function at nominally regular values of ka. This feature of the results is examined by decomposing the backscattered signal into its components. The results of computing Equation (4) are shown in Figure 2b, where it can be seen that for ka approximately greater than unity the backscattered form function for a mobile rigid sphere has a gradually damped oscillatory behaviour, tending towards a constant value of unity.
Values of the sphere and water parameters used to compute the form function Compressional velocity (ms-‘)
Shear velocity (ms-‘)
1000
1476
_
14935 2893 2586
6860 4860 5550
4140 3065 3545
Density (kg m-? Water Tungsten carbide Lead glass Soda glass
a
10.0
8.0
ka
b
ka Figure 2 (a) Comparison of the form function for tungsten carbide with two independent data sets. (b) (c) Resonance form function
Rigid mobile form function.
Ultrasonics 1992 Vol 30 No 1
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Scattering
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Thorne et
al.
The sharp dips characteristic of Figure 2a for an elastic sphere are not observed for the rigid case. The results calculated using Equation (5) are presented in Figure 2c. These results clearly show the resonant characteristics of the backscattered radiation, with a series of relatively high broad peaks and smaller narrower resonant structures. The nomenclature for the resonances’ ‘, (n, 1),corresponds to the modes of vibration of a free elastic sphere, where n is the nth partial wave in Equation (2) and 1 is the radial structure number. The wider peaks (n, 1) are associated with Rayleigh wave resonances which quickly radiate into the fluid, while the narrower band structures arise from whispering gallery wave modes which are poorly coupled to the fluid and take correspondingly longer to radiate into the fluid. Examination of the resonance structure clearly shows the origins of the form function structure shown in Figure 2~. The backscattered signal, in general, is close in form to the rigid case. However, in the regions near to the sphere resonances, the form function differs significantly from the background case. Figures 2b and c show this is due to the interactions of the two constituent components of the backscattered signal. The dips in the form function for ,tungsten carbide are associated with regions near sphere resonance radiating out of phase with the specular reflection, resulting in destructive interference in the backscatter direction. The scattering from tungsten carbide spheres was pursued to assess the experimental arrangement and because interpretation of the form function in terms of resonance modes and background scattering is well understood. The form function is, however, sensitive to the material composition of the sphere and it was,
Zl
Figure 3
(a) Comparison
of the form function
function
18
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therefore, necessary to move closer to the more relevant case of studying spheres with properties more similar to those of sedimentary material; hence, as previously mentioned, two glasses were examined. The results for the lead glass measurements are shown in Figure 3~. It can readily be seen that there is a significant difference between the form function for lead glass and tungsten carbide. The main feature of the form function is again the nominally regular reductions in amplitude with ka: however, the dips in the form function for lead glass are noticeably broader than for the case of tungsten carbide. This is principally due to the factor of six reduction in sphere density. There is also a decrease in the values of ka at which the troughs occur owing to the lower shear wave velocity for lead glass. Comparison of the theoretical predictions with the experimental data generally show good agreement, with the differences between theory and experiment generally being comparable to those in Figure 2~. For ka < 4 there is a greater divergence near the peaks which appeared genuine, although for the smaller size spheres, of the order of 1 mm diameter, the mounting may have been having a detectable influence on the backscattered signal. The computed background and resonance radiation components are shown in Figures 3b and c. The background component is essentially the same as that for tungsten carbide for ka > 2 with some amplitude difference below this value. However, the resonance constituents are substantially different. The resonant components are still present; however, the bandwidth of the peaks is greater, which is commensurate with the observations in Figure 3~. The interaction between the background and resonant constituents is less clear-cut than for tungsten carbide, although the resonance
431
591
‘391
for lead glass with the collected data. (b) Rigid mobile form function.
(c) Resonance form
Scattering
by non-metallic
spheres.. P.D.
et al.
Thorne
a
b
8.0
10.0
12.0
:4.3
16.0
l8.0
23.0
10.0
12.0
Iri.0
16.0
18.0
20.0
ka
8.0
ka Figure 4 (a) Comparison of the form function for soda glass with two independent data sets. (b) Comparison of the form function for soda glass of slightly different composition (see text for details) with the collected data. (c) Resonance form function
radiation still tends to reduce the amplitude of the backscattered signal. The results for soda glass are shown in Figures 4a-c. Two data sets were again taken. The manufacturing source for the soda glass spheres changed for sphere diameters above 7 mm, and Figure 4b shows the results for spheres from the second source, taken on the middle frequency range transducer. The shear wave velocity was decreased by 5.5% to calculate the form function in Figure 4b. The form functions in Figures 4a and b are seen to be comparable with those of lead glass and, as shown in Figure 4c, this is due to the similarities in the resonance component of the form function. The rigid components were essentially identical. The main discernable difference between the form functions for the lead glass and soda glass is that the resonances occur at the higher ka values in soda glass, and this is associated with its 15.7% increase in shear wave velocity. For backscattering measurements taken using suspensions of particles there will almost certainly be a size distribution within the suspension. To give an indication of the consequences this would have on the effective form function for a suspension, the average form function was computed using
(Ifm(%)l>
=&
Ifm(X)I som
x exp[ - (x - x,)*/202]dx
bolder line was calculated from Equation (6). For the 5% case, the structure in the form function is still partially in place, although the excursions have been reduced. However, by what would be a modest standard deviation for a natural suspension of marine sediments, 20%, the structure of the form function has been completely lost, with the form function having the shape of a simple high-pass filter response. It seems highly likely that Figure .5c would be representative of the form function for any marine suspensions of non-cohesive sediments that consist of nominally spheroidal particles. (The effect of non-sphericity on the form function has been examined22 for tungsten carbide spheroids and this causes an admixture in the resonance structure.) This high-pass filter approach has been very successfully adopted23 to explain measurements24 of attenuation by suspended sediments, although it should be noted that for the case of attenuation the backscattering cross-section does not exhibit the oscillations observed in the generally” backscattered form function and a smoothed mobile rigid sphere delineation described as a high-pass filter response was applied.
Conclusions
(6)
A Gaussian probability function has been used to represent the distribution, with x0 being the mean value and ~7the standard deviation. Figures 5a and b show results for c = 0.05~~ and 0.2x, for soda glass. The,fine line shows the form function for a single sphere and the
Measurements of the backscattered form function have been taken using tungsten carbide and glass spheres; the former because it has been studied by a number of previous investigators and provides a benchmark test, and the latter owing to the interest in modelling the interaction of sound with marine suspended sediments by using glass spheres. To aid the interpretation of the form function it was decomposed into its background
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1992
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Scattering
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Thorne
a
ka
L-
‘1.0
-
~~
6.0
-‘J_,_
1
8.0
:3.0
ka Figure
5
Averaged
form function:
(a) standard deviation
5% of the mean; (b) standard deviation
and resonant components to help provide an understanding of the underlying structure of the form function. The variability in shape of the form function for differing materials can be seen to be principally associated with the change in form of the resonant structure. Comparison of theoretical predictions with the data sets for tungsten carbide, lead glass and soda glass showed good agreement, both in terms of the form function structure and its absolute level. This provides evidence that the form function description is appropriate for non-crystalline amorphous materials, and suspension studies conducted on this basis are appropriate. The consequences of having a particle size distribution on the form function is, in effect, to smooth out the resonant structure and, for even modest standard deviations, it results in a profile similar to that of a simple high-pass filter response. This shows that when making theoretical comparisons with suspension measurements, due allowance needs to be made for the variation in particle size, even for a case in which the particles could be considered nominally uniform in size. As far as scattering from natural sediments is concerned, the size distribution effect will normally be a dominant feature of the scattering response and a form function similar to that shown in Figure 5b will frequently be the pertinent curve to employ.
7
8 9
10 11
12 13
14 15 16 17 18
References Vincent, C.E., Huntley, D.A. and Clarke, T.L. Acoustic measurements of suspended sand concentration in the C’S’ experiment at Stanhope Land, Prince Edwards Island Marine Geology (1988) 81 185-196 Tborne, P.D., Williams, J.J. and Heathershaw, A.D. In situ acoustic measurements of marine gravel threshold and transport Sedimentology (1989) 36 61-74 Lowe, R.L. Acoustic bedload sensor Proc 17th Int Conf on Coastal Engineering Sydney, Australia (March 1980) 215-216 Hanes, D.M. and Vincent, C.E. Detailed dynamics of nearshore suspended sediments Proc Coastal Sediments 87 Conf Am Sot Chem Eng, New York, USA (1987) 285-299 Libicki, C., Bedford, K.W. and Lynch, J.F. The interpretation and evaluation of a 3 MHz acoustic backscatter device for measuring benthic boundary layer sediment dynamics J Acoust Sot Am ( 1989) 85 1501-1511 Vincent, C.E. and Green, M.O. Field measurements of the suspended sand concentration profiles and fluxes and of the resuspension coefficient y0 over a rippled bed J Geophys Res (1989) Hnnes, D.M.,
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20 21 22 23
24
20% of the mean (see text for details)
95 11591-11601 Thorne, P.D., Vincent, C.E., Hardcastle, P.J., Rehman, S. and Pearson, N. Measuring suspended sediment concentration using
acoustic backscatter devices Marine Geology (1991) 98 7-16 Young, R.A., Merrill, J.T., Clarke, T.L. and Proni, J.R. Acoustic profiling of suspended sediments in the marine bottom boundary layer Geophys Res Lett (1982) 9(2) 175-178 Crickmore, M.J., Shepherd, I.E., Dore, P.M. A field instrument for measuring the concentration and size of fine sand suspensions Proc Conf on Measuring Techniques London, UK (April 1986) 425-442 Jansen, R.H.J. The in-situ measurement of sediment transport by means of ultrasound scattering, Report 203, Delft Hydraulics Laboratory, Delft, The Netherlands (1978) Schaafsma, AS. and Kindereo, W.J.G.J. Ultrasonic instruments for the continuous measurement of suspended sand transport Proc IAHR Symp on Measuring Techniques in Hydraulic Research (Ed Wessels, A.C.E.) A.A. Balkema, Rotterdam, The Netherlands (1986) 125-136 Hay, A.E and Schaafsma, AS. Resonance scattering in suspensions J Acoust Sot Am (1989) 85(3) 1124-1138 Theme, P.D., Hayburst, L. and Campbell, S.C. Measurements of scattering from a suspension of spherical scatterers Proc Inst Acoustics: Acoustics ‘90 Vol 12, Inst Acoustics, St Albans, UK (1990) Pt 1, 115-122 Faran, J.J. Sound scattering by solid cylinders and spheres J Acoust Sot Am (1951) 23(4) 405-418 Hickling, R. Analysis of echoes from a solid elastic sphere in water J Acoust Sot Am (1962) 34(10) 1582-1592 Gaunaurd, G.C. and Uberall, H. RST analysis of monostatic and bistatic acoustic echoes from an elastic sphdre J Acoust Sot Am (1983) 73(l) 1-12 Neubauer, W.G., Vogt, R.H. and Dragonette, L.R. Acoustic reflections from elastic spheres: I. Steady-state signals J Acoust Sot Am (1974) SS(6) 1123-1129 Dragonette,
L.R., Vogt,
R.H.,
Flax,
L. and Neubauer,
W.G.
Acoustic reflections from elastic spheres and rigid spheres and spheroids: II. Transient analysis J Acoust Sot Am (1974) 5y6) 1130-1936 Humphrey, V.F., Murphy, C. and Moustafa, A.H.A. Wideband backscattering measurement using a parametric array Proc UI 87 Butterworths, Guildford, UK (1987) 265-270 Kaye, G.W.C. and Laby, T.H. Tables of Physical and Chemical Constants Longman, London, UK (1973) . Vogt, R.H. and Neubauer, W.G. Relationship between acoustic reflection and vibrational modes of elastic spheres J Acoust Sot Am (1976) 60(l) 15-22 Werby, M.F., Uberall, H., Nagl, A., Brown, S.H. and Dickey, J.W.
Bistatic scattering and identification of the resonances of elastic spheroids J Acoust Sot Am (1988) f&l(4) 1425-1436 Sheng, J. and Hay, A.E. An examination of the spherical scatterer approximation in aqueous suspensions of sand J Acoust Sot Am (1988) 83(2) 598-610 Flammer, G.H. Ultrasonic measurement of suspended sediments Geological Survey Bulletin No. 1141-A US GPO, Washington DC, USA (1962)