A large ultrasonic bounded acoustic pulse transducer for

A large ultrasonic bounded acoustic pulse transducerfor acoustic transmission goniometry: Modelingand calibration

Youcef Bouzidia� and Douglas R. Schmittb�

Institute for Geophysical Research, Department of Physics, University of Alberta, Edmonton,Alberta T6G 2J1, Canada

�Received 24 June 2005; revised 11 October 2005; accepted 12 October 2005�

A large, flat ultrasonic transmitter and a small receiver are developed for studies of materialproperties in acoustic transmission goniometry. While the character of the wave field produced bythe transmitter can be considered as a plane wave as observed by the receiver, diffraction effects arenoticeable near critical angles and result in the appearance of weak but detectable arrivals.Transmitted ultrasonic waveforms are acquired in one elastic silicate glass and two visco-elasticacrylic glass sample plates as a function of the angle of incidence. Phase velocities are determinedfrom modeling of the shape of curves of the observed arrival times versus angle of incidence. Thewaveform observations are modeled using a phase propagation technique that incorporates full wavebehavior including attenuation. Subtle diffraction effects are captured in addition to the mainbounded pulse propagation. The full propagation modeling allows for various arrivals to beunambiguously interpreted. The results of the plane wave solution are close to the full wavepropagation modeling without any corrections to the observed wave field. This is an advantage asit places confidence that later analyses can use simpler plane wave solutions without the need foradditional diffraction corrections. A further advantage is that the uniform bounded acoustic pulseallows for the detection of weak arrivals such as a low energy edge diffraction observed in ourexperiments. © 2006 Acoustical Society of America. �DOI: 10.1121/1.2133683�

PACS number�s�: 43.20.Gp, 43.20.Jr, 43.20.Ye, 43.35.Cg �YHB� Pages: 54–64

I. INTRODUCTION

A great deal of information on the mechanical propertiesof solids is obtained by studying transmitted and reflectedacoustic waves. Observations of the angle-of-incidence-dependent variations of the complex transmission and reflec-tion coefficients can be used to quantitatively determine elas-tic constants, wave velocities, and attenuation. As such,acoustic goniometry, in which a fluid-loaded plate �i.e., aplate completely immersed in a liquid� is insonified withacoustic energy at a variety of angles of incidence and ana-lyzed in either the time or frequency domain, has long beenpopular in material property studies. Transmission goniom-etry techniques, in particular, have been used for studies ofhigh-resolution imaging,1 complex wave propagation invisco-elastic and anisotropic,2–4 thin,5,6 and saturatedporo-elastic7–11 media.

Ultrasonic goniometry experiments are carried out witha variety of geometries. One problem ubiquitous to all thesegeometries, however, is that the transmitting and receivingtransducers are of finite dimension and limited frequencybandwidth. As a result, the analysis of real measurementsrequires that transducer diffraction,12,13 such as beam diver-gence, edge effects, and off-axis wave-number components,be considered in the experimental design and analysis. The

a�Now at Geo-X Systems Ltd., Suite 500, 440 2nd Avenue S.W. Electronicmail: [email protected]

b�Author to whom correspondence should be addressed. Electronic mail:
[email protected]
54 J. Acoust. Soc. Am. 119 �1�, January 2006 0001-4966/2006/1

energy propagating from such transducers is containedwithin a continuous bounded beam or a transient boundedpulse. The most dramatic example of bounded beam or pulseeffects is nonspecular reflection produced near criticalangles14,15 that results in broadening of reflectivity coeffi-cient maxima and apparent lateral shifting of the reflectedbeam upon reflection near the Rayleigh critical angle. Assuch, the influence of such diffraction affects on both trans-missivity and reflectivity; the phase must be considered par-ticularly as the S-wave critical angle is approached.

Most theoretical developments in acoustics are con-structed with the assumption that monochromatic plane-wave fronts propagate through a medium. Ideally, experi-mental measurements consequently strive to reduce theimperfect, diffraction-limited observations to a form fromwhich a plane wave interpretation can be employed. This isespecially pertinent for more fundamental measurements ofreflection and transmission coefficients,16 intrinsicattenuation,17 and inversion of anisotropic properties.18 A va-riety of strategies are used to overcome this limitation. Themost common is to correct the observed amplitudes wave-forms for changes due to beam characteristics.16,19–23 An-other approach is to mimic plane wave transducers by super-position of the time domain waveforms acquired during thesystematic scanning of a transmitted wave field24 or by syn-thetic aperture arrays25,26 in which the effective dimensionsof the transducers are artificially increased.

In principal, the same effects can be obtained by use of
large transducers directly. The larger the transducer relative
© 2006 Acoustical Society of America19�1�/54/11/$22.50

to the distance to the receiver and the wavelength of thepropagating acoustic energy, the more the wave field ap-proaches plane wave behavior. Construction of large aperturetransducers is problematic and there is little discussion in theliterature of their use in material characterization. However,Hosten and co-workers27,28 have more recently employedspecially constructed large �40�80 mm2� high-frequency�3.2 MHz� transducers. Using these transducers, the fre-quency dependence of transmissivity of isotropic glass wasmeasured at a variety of angles of incidence and these agreedwith the predictions calculated using simpler plane wavetheory. In contrast, the transmissivities obtained using a pairof smaller transducers ��19 mm diameter� differed substan-tially from the plane wave theory even after laterally shiftingthe receiver to correct for refraction of the beam within theplate. Consequently, the information obtained from the near-plane wave could be inverted for material properties withouthaving to make transducer diffraction corrections, correc-tions that would increase the uncertainty of the measure-ments. This was found to be particularly important in thedetermination of the imaginary components of the complexmoduli that indicate the attenuation.

A similar experimental configuration consisting of a�102�76 mm2� transmitter but with a much smaller �1.9�1.9 mm2� receiver is described in this contribution, an ul-timate goal of which is to have a system sufficiently sensitivefor the study of attenuation and wave propagation in com-plex materials. The large transmitter is intended to create anacoustic pulse the character of which approaches that of aplane wave. Conversely, the small receiver is intended tosense the pressure at a point in the transmitted wave field.The purpose of this is to eliminate any corrections due todiffraction effects of the transmitted wave field that lead to anonuniform distribution of pressure over the surface area ofthe receiver. So, in this context the somewhat subjective termof “large” is taken to mean that which will yield a transmit-ted wave front that is flat over dimensions greater than theaperture of the “small” receiver. This combination allows forsensitive detection of weak and unanticipated arrivals. Theexperimental configuration and transducer characterizationsare described in Sec. I. In Sec. II a procedure for modeling ofthe bounded pulse produced in the tests is described. Theangle of incidence dependence of the wave fields transmittedthrough isotropic silica and acrylic glass plates is comparedto the wave field modeling in Sec. III, some diffraction ef-fects are noted even for this experimental configuration. Thecontribution concludes with some directions for future workin Sec. IV.

II. EXPERIMENTAL APPARATUS

A. Experimental configuration

A familiar experimental configuration for measuring thetransmissivity of plates is employed.7 The experiments areconducted in a tank filled with de-ionized water �Fig. 1� toavoid deposition of calcium deposits from normal tap waterin our region. The large �102�76 mm2� transmitting sourceand the small �1.9�1.9 mm2� receiver are coaxially
mounted 17 cm apart with the alignment confirmed using a
J. Acoust. Soc. Am., Vol. 119, No. 1, January 2006 Y

laser mounted in the same position as the receiver both be-fore and after measurements were made. The sample plate ismounted vertically �Fig. 2� on a horizontal rotating goniom-eter table between the transmitter and the receiver. Duringmeasurements the plate is manually rotated to successiveangles of incidence �; the plate can be rotated to both nega-tive and positive � with a precision better than 0.1°. Thelarge dimensions of the transmitter typically limit the rangeof incidence angles to ±50° and the wave field is sampled atincrements of 1° with 101 records obtained in a given trans-mission test.

During the transmission goniometry measurements, thetransmitter and receiver are aligned and then left stationary.Both, however, are mounted with a manual scanner that hasthe capacity to sample the wave field in three dimensions toa precision of better than 0.5 mm over 15 cm in the x-y planeand 2 mm over 35 cm along the z axis. Wave field scanningwas also used in characterization of the transducers and toprovide the initial input for modeling as is described shortly.

FIG. 1. Photograph of measurement system.

FIG. 2. Simplified diagram of the experimental configuration as seen fromabove. Coordinate system x-z has origin at the center of the face of the largetransmitter T. Receiver R mounted on alignment rod and sample plate Smounted on goniometer table G are both centered on z axis. The angle ofincidence � is equivalent to the angle that the sample face normal makeswith respect to the z axis; measurements are made at both negative and
positive �.
. Bouzidi and D. R. Schmitt: Large transducer for goniometry 55

The transmitter and receiver were specially constructedfor this study. The transmitter �Fig. 3�a�� consists of a 2.54-mm-thick rectangular sheet �102�76 mm2� of piezo-electricceramic �PZT-840, American Piezo Ceramics Inc., Mack-ayville, PA� that has a 780-kHz free resonant frequency inthe longitudinal thickness mode. The back face of the ce-ramic is bonded to a copper sheet and backed by a thickshaped block of a urethane-tungsten mixture that mechani-cally damps the oscillations producing a broadband signal.The front face of the ceramic is covered with a thin layer ofconductive epoxy which is in turn isolated from the sur-rounding water by a thin layer of electronic lacquer �G CElectronics polystyrene Q dope�. For protection and furtherisolation from the water, this assembly is packed into a plas-tic housing.

In contrast, the receiver �Fig. 3�b�� is constructed from asmall �1.9�1.9 mm2� and thin �2.01 mm� piezo-electricpiece �PZT-850, American Piezo Ceramics, Mackayville,PA� prepared in a manner similar to the transmitter. Thesedimensions are of the order of the ultrasonic wavelengths inwater and as such allow this transducer to be considered as asmall receiver. The transmitted wave field, particularly nearits center, is uniform over distances much larger than thewidth of the receiver. Consequently, no additional correc-tions are necessary to account for spatial variations in thewave field intensity. The area of the receiver ��3.6 mm2� isonly 0.046% that of the transmitter �7752 mm2�. The re-ceived signal is increased two to four times by a preamplifiermounted near the ceramic in order to reduce noise. Tests onthe receiver have shown that it is highly directional andsenses mainly wave arrivals with wave fronts parallel to the

FIG. 3. Cut-away views of the �a� large transmitter and �b� small receiver.

surface of the transducer; this minimizes the effect of nondi-

56 J. Acoust. Soc. Am., Vol. 119, No. 1, January 2006

rected scattered energy. The use of this small transducer iseffective because of the large energy provided by the trans-mitter as demonstrated later.

The experiments here are conducted in a pulsed modeand as such the wave field produced by the transmitter isconsidered to be a bounded pulse being limited both spatiallyand temporally. The pulse is created by a 200-V, 5-ns risetime square step with energies between 12.5 to 100 �J�Panametrics Pulse Model 5800� supplied to the transmitter.The receiver senses the pressure and its preamplified signalis digitally obtained �Tektronix TDS 420A� at a 40-ns sam-pling period after stacking 300 individual pulses. The digitalrecords are immediately stored to a computer for later analy-sis. The received waveform pulse in the water has a durationof approximately 8 �s �Fig. 4�a�� and significant bandwidthfrom 200 kHz to 1.2 MHz �Fig. 4�b��.

B. Samples

Three isotropic blocks composed of soda-lime silicateand acrylic �polymethyl methacrylate� glasses were used astest pieces. The low attenuation of silicate glass allows it tobe considered as a nearly elastic medium and the qualityfactors of pure silica glasses are high29 �Q�106� in the fre-quencies used here and as such it is difficult even to obtainreliable values for them; it is assumed in this study that theglass is representative of an elastic medium. In contrast, thevisco-elastic properties of acrylic glass are well studied withrepresentative quality factors30 Q of �50 and �10 for thelongitudinal and shear waves, respectively. The faces of allthe samples were of the same dimensions of 22�22 cm2 butof variable thicknesses �Table I�. The two acrylic glasssamples were taken from sheets of differing thicknesses and

FIG. 4. Typical waveform recorded 2 cm along the axis of the large trans-ducer in the �a� time domain and �b� normalized amplitude spectrum.

consequently were not from the same batch as made apparent

Y. Bouzidi and D. R. Schmitt: Large transducer for goniometry

by the differences in the longitudinal VP and shear VS wavespeeds. The greater thickness of the larger plate did not allowfor as great a range of angles of incidence to be sampled. Thewave speed values given in Table I were measured using P-and S-wave transducers in direct contact with the plates in athrough transmission experiment.

III. WAVE FIELD CHARACTERIZATION ANDMODELING

Proper interpretation of any experimental results re-quires that the wave field and any diffraction limitations bewell understood. In particular, it is important to know howflat the bounded acoustic pulse will be and the distances towhich the bounded acoustic pulse retains a planar character.This was accomplished by both scanning and modeling thepropagating bounded acoustic pulse.

Below, some of the quantitative amplitude informationis given in the form of the peak value of the magnitude of theanalytic signal as determined using a Hilbert transform. Theevaluation of the amplitude envelope of the signal is ob-tained via the analytical signal given by

fc�t� = s�t� + iHt�s�t�� , �1�

where s�t� is a real signal and Ht denotes the Hilbert trans-form. The amplitude envelope A�t� is simply the magnitudeof the analytic signal

A�t� = �fc�t�fc*�t� , �2�

where * denotes the complex conjugate.

A. Observations of the free bounded acoustic pulsepropagation

Here we take a free bounded acoustic pulse to be onethat is freely propagating through the water only. The re-sponse of the receiver to the free bounded pulse is firstsampled along x and y lines in increments of 1 and 2 mm,respectively, and at z=2 cm from the transmitter face at adigitization period of 40 ns. This recorded wave field is re-constructed by plotting the recorded waveforms, normalizedwith respect to the direct arrival recorded with the sampleremoved, with only the shorter vertical dimension shown�Fig. 5�a��. The observed bounded acoustic pulse generatedhas a uniform arrival time and a nearly constant amplitude in

TABLE I. Sample characteristics.

SampleSoda-lime

glass Thick acrylic Thin acrylic

Thickness �cm� 1.9 3.64 2.46Density �g/cm3� 2492 1153±35 1153±35VP �m/s� 5900 2765±80 2675±30VS �m/s� 3400 1390±10 1390±10�P �Np/m� �0 9.4±0.9 9.4±0.9�S �Np/m� �0 26.5±2.7 26.5±2.7Vl �m/s� 1500±10 1510±10 1490±10�c

P �°� 14.7 33.1 33.9�c

S �°� 26.2 N/A N/A

a x-y large area centered around the axis of propagation.


Although not shown, the bounded acoustic pulse is wider inthe horizontal x direction than in the vertical y direction ac-cording to the lateral dimensions of the source. Also in Fig.5�a�, some lower frequency coherent noise is noticeable attimes greater than 15 �s. These events have been generatedby reverberations between the transmitter and the receiver.Hyperbolic diffractions from each edge of the transducer arealso clearly apparent. However, at the center of the boundedacoustic pulse the diffractions and reverberations do not in-terfere with the planar front which has uniform amplitudeand phase over a distance greater than 9 cm �Fig. 6�a�� in thex direction.

In contrast to the near field �Fig. 5�a��, the edge diffrac-tions become increasingly important with distance. Despiteappearing flat over �7 cm width at 35 cm �Fig. 5�b��, theedge diffractions now contaminate the main front; there areno longer any sharp edges to the bounded pulse in contrast toits early character �Fig. 5�a��. These edge effects are alsoquantitatively apparent when variations in amplitudes acrossthe bounded pulse are measured �Fig. 6�b��. The edge dif-fractions do not significantly influence the observed ampli-tude along the z axis of propagation until approximately adistance of z=28 cm �Fig. 6�c��.

B. Modeling of free bounded acoustic pulsepropagation

As noted earlier, it is important that we understand thebounded acoustic pulse. One critical aspect of this is being

FIG. 5. �Color online� Images of diffraction behavior of the large transducerin water. Scans of the bounded pulse recorded with the receiver along theshort dimension of the large transducer at distances from its surface of �a�2 cm and �b� 35 cm. �c� Modeled scan at 35 cm calculated using the input of�a�.

able to predict how the pulse will change with propagation


distance. Using as a starting point the early observed wavefield of Fig. 5�a�, the evolution of the wave field as it propa-gates through the water is modeled. This modeling will becompared to the observations at greater distances in order toassist in characterization of the bounded pulse and to developa predictive tool for the analysis of the more complicatedreflection and transmission experiments. Specifically, thewave field shape and amplitude at z=35 cm are calculatedand compared to that observed.

A phase advance technique is employed to carry out thecalculations in the two-dimensional plane �x ,z�. In this con-tribution the wave front is considered two-dimensional be-cause the measurements are taken within the transmitter’scentral x-z plane at y=0. This allowed selection of a conve-nient coordinate system �Fig. 2� such that only the compo-nents of the wave number in the x-z plane are necessary inmodeling. In this two-dimensional propagation, the out-of-plane components are ignored and the freely propagatingwave field ��x ,z , t� is described using an integral,

��x,z,t� =1

4�2−�

+� −�

+�

��kx,0,�eikxxeikzzeit dkx d , �3�

where ��kx ,0 ,� is the wave field recorded on the plane z=0 once transformed into the Fourier domain �kx ,�. kx andkz are the horizontal �x� and the vertical �z� wave numbers,respectively. Essentially, this equation describes the superpo-sitioning of monofrequency plane waves with different fre-

FIG. 6. Quantitative comparison of observed and model peak envelope am-plitudes. �a� Peak pulse amplitudes observed 2 cm from large transducersurface along its long and short dimensions. Comparison of observed andmodeled peak pulse amplitudes �b� along the long transducer dimensionperpendicular to the bounded pulse propagation axis 35 cm from the surfaceof the large transducer, and �c� with distance from the transmitter surfacealong the bounded beam axis.

quencies and propagation directions within the x-z plane that


form the final bounded pulse. These wave numbers are re-lated to each other by kx

2+kz2=2 /v2, where v is the speed of

the wave in the medium under consideration. ��x ,z , t� is thewave field everywhere in the space �x ,z� and at any time t.Note that ��x ,z , t� can be any component �i.e., pressure,particle displacement, etc.� describing the wave as it propa-gates. This is simply the phase shift method of forward mod-eling often used in seismic migration corrections.31 Theimplementation of this procedure is briefly described below.

Using Eq. �3�, the wave field recorded 2 cm above thesource is propagated to a distance of 35 cm. This is accom-plished by first taking the double Fourier transform of thewave field recorded at z=2 cm in time and x, phase shiftingthis result in the frequency domain by multiplication witheikzz where z=0.33 m, and finally taking the double inverseFourier transform to obtain the wave field at z=35 cm. Tospeed up the process the fast Fourier transform algorithm canbe used. Depending on memory availability the process canbe done in one step or in steps in the z direction that sum upto the desired distance �here 33 cm�.

The wave fields observed and modeled at a distance of35 cm are compared in Fig. 5. We notice that these are inexcellent agreement despite the reverberations present in themodel. This distance of 35 cm is greater than the source-receiver distance of 17 cm used in the transmission experi-ments. We notice that the “reverberation events” are presentin the model above 250 �s but not in the observed data. Thisfurther supports the contention that these result from theproximity of the source and receiver �2 cm� when the inputwave field data are acquired.

The maximum value of the amplitude envelope of boththe observed and the modeled wave fields at distances of 2and 35 cm are compared Fig. 6. At 2 cm both the verticaland horizontal wave fields show a large area around the axisof propagation where the amplitude is nearly constant. How-ever, the horizontal bounded acoustic pulse is wider than thevertical one. At 35 cm the model and the observed ampli-tudes are in agreement despite a misalignment of about2 mm; this is due to the precision of the scanner positioningsystem at the 35-cm distance. The edges of the boundedacoustic pulse increase in amplitude and migrate inwards to-wards the center as predicted by the theoretical model. Thisis evident in the amplitude envelope of the cross section ofthe acoustic bounded acoustic pulse as it propagates in thefluid as shown in the composite image of Fig. 7. The wavefield as predicted by the model given by Eq. �1� spreadsenergy both outwards and inwards by diffraction as it propa-gates. The amplitude at the center of the acoustic boundedacoustic pulse as a function of propagation distance is givenin Fig. 6�c� with the model superimposed. We see that theyare in very good agreement. This confirms that the forwardmodeling of the bounded acoustic pulse using Eq. �1� ad-equately and effectively describes the wave field at any pointin front of the acoustic source and places further confidencein the theoretical model.

Equation �3� still holds for an acoustic bounded wavefield transmitted through a solid material plate immersed in a
fluid and recorded by a point receiver on the opposite side of

the source provided that the appropriate transmission coeffi-cients were used. Indeed Eq. �3� can be modified for suchexperiment with thick plates as follows:

��x,z,t� =1

4�2−�

+� −�

+�

T1T2��kx,0,�eikxxeikzzeit dkx d ,

�4�

where T1 and T2 are the transmission coefficients at inter-faces 1 and 2, respectively. For the sake of completeness, forthe reflected wave field the equation can be written as

��x,z,t� =1

4�2−�

+� −�

+�

R1��kx,0,�eikxxeikzzeit dkx d , �5�

where R1 is the reflection coefficient at the first interface. Itneeds to be noted here that both Eqs. �4� and �5� apply to athick plate in the sense that we are describing only the pri-mary reflections and ignoring later events that are well sepa-rated in time. We leave studies of reflected pulses for latercontributions due to the added difficulties encountered atcritical angles.32,33 It is important to note that the transmis-sion and reflection coefficients in Eqs. �4� and �5� should beproperly calculated to include the material properties particu-larly the P-wave and shear-wave attenuations.

Equations �3�–�5� describe precisely the propagation, thetransmission, and the reflection of a bounded acoustic pulse.These are tools in modeling such experiments with variousmaterials provided that the reflection and transmission coef-ficients are properly calculated for the materials under con-sideration. The reflection and transmission coefficients forelastic isotropic materials can be found in many text books.It is most useful to write these coefficients in terms of thewave numbers that implicitly contain the angle of incidence.

C. Pattern of transmission mode arrival times

The results below are presented as composite images ofthe amplitudes of adjacent time traces versus the angle ofincidence here referred to as a �-t plot; as this is not a con-

FIG. 7. �Color online� Modeled bounded pulse energy along the propagationpath as represented by the maximum of the Hilbert amplitude envelope.

ventional format some discussion of what is to be expected is


useful. The propagation of the bounded pulse from the trans-mitter to the receiver through the fluid-loaded plate is influ-enced by both refraction and mode conversion. Despite thefact that the transducers are held fixed, the pulse’s transittime necessarily changes with angle of incidence due tovariations in the length of the path through the fluid and solidplate. Along the plate axis and in the near field such that theside diffractions do not interfere, the transit time can readilybe determined using Snell’s law ray tracing. When viewed asa function of the angle of incidence, the travel time loci �Fig.8� curve upwards or downwards depending on whether thewave speed in the plate is respectively greater or less thanthat of the surrounding fluid �speed V1�. For example, in amaterial with VPVlVS the longitudinal wave arrivals willbe concave upwards while the converted shear wave arrivalis concave downwards. When the wave speed in the solidequals that of the surrounding liquid, the arrival time curve is

FIG. 8. �Color online� Observed �a� and modeled �b� �-t composite imagesfor the thin �2.46 cm� acrylic glass. Each trace is normalized to its rms valueand amplitudes greater than 1

3 of the maximum amplitude are clipped inorder for small amplitudes to display properly.

flat. In later figures, the calculated travel time loci are super-


imposed on the �-t images. The designations P and S indi-cate the directly arriving longitudinal and converted shearwave modes, although it must be kept in mind that all ofthese arrivals are sensed by the receiver in the water as alongitudinal mode. Primary multiple reverberations are indi-cated by three letters, for example PPS represents those ar-rivals with two passes of the longitudinal wave and one witha converted shear wave through the plate.

IV. TRANSMISSION RESULTS

A. Thin acrylic glass plate

A number of classic plane-wave features are seen in the�-t image �Fig. 8� for the thin acrylic glass plate. The mostnotable and strongest are the direct longitudinal �P� and con-verted shear �S� wave arrivals. The faster longitudinal arrivalcurves upwards with angle of incidence due to the longerpath through the faster material. VS�Vl and this slowerspeed of the converted shear wave is manifest as a weaklyconcave downward curve. As VS�Vl�VP only the longitu-dinal critical angle �c

P exists as apparent by the loss of thestrong longitudinal arrival at ±33.9° after which the strengthof the S arrival is dominant. As anticipated, the convertedshear arrival is weak at normal incidence ��=0� and is notobservable near this angle in the later multiple reverberationsPSS and PPS. Phase shifts near �c

P appear to give the S locusan apparent discontinuity.

Diffraction effects of the finite transmitter still add com-plications. First, the locus of the direct P arrivals extends toangles greater than �c

P although this post critical energy isweak. Second, another weak arrival denoted by “?” appearsbetween those for P and S and may exist at all angles ofincidence although it is lost in the strong P arrival near nor-mal incidence. This latter arrival complicated our prelimi-nary interpretations of �-t images acquired in more complexmaterials than discussed here; it will shortly be demonstratedthat it is a transducer diffraction effect. The modeled trans-mitted wave field �Fig. 8�a�� is close to the observed wavefield �Fig. 8�b��.

B. Thick acrylic glass plate

The �-t images for both acrylic plates essentially displaythe same information including the “?” arrival but with somedifferences. The most obvious are the P and S mode traveltimes that are respectively advanced and delayed for thethick plate �Fig. 9�. The P mode arrives earlier because of itshigher speed in the acrylic relative to the water. In contrastthe S mode is retarded because the shear wave speed is lowerthan that in the water �Table I�. The pulse propagates throughmore acrylic material in the thicker plate and consequently ismore attenuated. This is evident in the lengthening of theobserved pulse waveform and weakening of the amplitudesrelative to the thin plate. Here again the modeled transmittedwave field �Fig. 9�a�� is close to the observed wave field�Fig. 9�b�� and it is important to note that attenuation needed
to be considered in the modeling.

C. Thin soda-lime glass plate

The elastic properties of the soda-lime glass plate areessentially elastic and hence simpler to model than those forthe acrylic. This same material simplicity together with thelarger wave speeds �Table I�, however, makes for a morecomplex �-t image �Fig 10�. In this case VPVSVl andtwo critical angles exist �Table I�, both of which are appar-ent. Due to the high velocities, the P, the S, and the latermultiple reverberations overlap and are difficult to separate;the loci shown in Fig. 10 are calculated using the knownwave speeds. The amplitudes of the S post �c

P are substan-tially smaller in a relative sense than those for the acrylicsdue to the greater loss of energy on reflection from the firstsurface of the glass due to its higher elastic impedance rela-tive to the surrounding water. One of the arrivals is labeledby “??” that cannot be explained by simple direct or multiplyreflected longitudinal or shear waves within the plate is alsohighlighted. However, in this case it is a diffracted shear

FIG. 9. �Color online� Observed �a� and modeled �b� �-t composite imagesfor the thick �3.64 cm� acrylic glass. Each trace is normalized to its rmsvalue and amplitudes greater than 1

3 of the true amplitudes are clipped inorder for small amplitudes to display properly.

wave. At post-shear-critical angle of incidence this wave ap-


pears in same manner of that seen in the acrylic samples as aP wave. The P wave at P-post-critical angle labeled “?” ofincidence is diluted within the S wave first arrival and con-sequently cannot be clearly identified and shows better in themodel. Other similar types of arrivals are seen at later timesin this low attenuation material and are multiple reflectionsof this diffracted mode.

V. DISCUSSION

In this section, the modeling of the bounded pulse in theplate transmission experiments is developed and used tocompare with the observed wave fields. The modeling tech-nique used is able to explain without ambiguity the “?” ar-

FIG. 10. �Color online� Observed �a� and modeled �b� �-t composite imagesfor the soda-lime �1.9 cm� glass plate. Each trace at angles less than the Scritical angle is normalized with respect to its rms value and traces at anglesgreater than the S critical angle to three times their rms values.

rivals.


A. Modeling of bounded pulse transmission

Modeling of the plate transmission is accomplished inthe same manner as described earlier for the free propagationof the bounded pulse but with the added complications ofrefraction and attenuation via Eq. �4�. All the transmittedwave modes detected by the small receiver including alltypes of multiples can be modeled via the phase shift eikzz

and by appropriately calculating the transmission and reflec-tion coefficients at each boundary and including the P and Swave attenuations of the material when applicable. The PPSmultiple, for example, can be modeled by first taking theFourier transform of the wave field generated by the source�at the angle of incidence under consideration� both in timeand space �x direction�. Let kp and ks denote the wave num-bers of the P and S waves in the solid, respectively, and let kdenote the wave number in the fluid. It is important here tonote that the x component of any wave mode is equal to theprojection of the wave number in the fluid on the x axis�Snell’s law�. Then the following steps are to be performedto obtain the wave field at the small receiver.

�1� Propagation in the fluid to the first interface via eikzz

where kz= �k2−kx2�1/2.

�2� Multiplication by the transmission coefficient �incidentP� between the fluid and the solid to obtain the P wavetransmitted into the solid.

�3� P-wave propagation in the solid to the second interfacevia eikzz where kz= �kp

2 −kx2�1/2.

�4� Multiplication by the reflection coefficient �incident P�between the solid and the fluid to obtain the reflected Pwave from the second interface.

�5� P-wave propagation in the solid to the first interface viaeikzz where kz= �kp

2 −kx2�1/2.

�6� Multiplication by the reflection coefficient �incident P�between the solid and the fluid to obtain the reflected Swave from the solid-fluid interface.

�7� S-wave propagation in the solid to the second interfacevia eikzz where kz= �ks

2−kx2�1/2.

�8� Multiplication by the transmission coefficient �incidentS� between the solid and the fluid to obtain the transmit-ted P wave into the fluid.

�9� P-wave propagation in the fluid to the receiver via eikzz

where kz= �k2−kx2�1/2.

These steps are performed for each wave mode. Thewave field at the receiver is then recorded as the modeledtrace and the procedure is repeated for each angle of inci-dence. Temporal snapshots of the wave field at a series oftimes during the propagation of the pulse 20° incident to thethin elastic plate highlight the precritical behavior �Fig. 11�and illustrate the modeling. Only the P and S transmittedarrivals are considered in the modeling of Fig. 11; multipleshave not been included. The first panel �Fig. 11�a�� shows thebounded pulse immediately prior to contact with the plate. Ata time 42 �s later half the pulse front has intersected the firstsurface of the plate and the P and S �Fig. 11�b�� and both theP and S arrivals are propagating through the plate while
another 64 �s later the main direct portions of the P and S

arrivals have left the sample. It is important to note that thenew P component is shifted to greater values of x because ofrefraction in the plate.

This procedure is very accurate in adequately modelingall arrivals with the correct amplitude and phase includingthe arrival labeled “?” in Figs. 8–10. The modeling is re-peated for a postcritical incidence of 40° �Fig. 12�. In orderto explain the origin of the “?” arrival, recall that the spa-tially and temporally limited bounded pulse will have a widerange of wave numbers in its Fourier domain decomposition.

FIG. 11. �Color online� Modeled propagation of the bounded pulse precriti-cally incident at 20° upon the thin �2.46 cm� acrylic glass plate immersed inwater. Display is shown to scale and amplitudes greater than half of themaximum true amplitude are clipped in order for small amplitudes to dis-play properly. The critical angle in this case is �P

C=33.9°. The origin of thecenter bounded acoustic pulse is not shown and is located at z=−6.6 cm andx=−2.85 cm. The direct distance between the source and the receiver is0.17 cm. Visualizations of some of the components of the wave field attimes of �a� 42 �s with bounded pulse in water incident to first surface, �b�64 �s partway through contact of pulse with first surface �note generation ofa reflected and transmitted P and a converted transmitted S arrival, and �c�at 93 �s showing nearly completed reflected and transmitted pulses.

Past the critical angle for the dominant wave numbers some


components with wave numbers smaller than the criticalangle are still refracted into the solid and then to the water atthe second interface. Snell’s law does not explain the transittimes of these arrivals. To establish its ray path it is impor-tant to consider the width of the original spatially boundedpulse and how it evolves in space as it propagates in thewater. The flat part of the acoustic wave field shrinks as itpropagates through the medium and its edges contain most ofthe wave numbers deviating from the main propagationangle. Consequently, the path of the edge diffractions mustbe taken into account with the point upon which the incidentpulse first intersects the first surface taken as the departurepoint for rays refracted into the solid and then emerging intothe water through the second interface. Even when this pro-

FIG. 12. �Color online� Modeled propagation of the bounded pulse post-critically incident at 40° upon the thin �2.46 cm� acrylic glass plate im-mersed in water that includes only the edge diffraction arrivals; the primaryreflection from the first surface and any induced multiples are not includedin the model. Display is shown to scale and amplitudes greater than one-third of the maximum true amplitude are clipped in order for small ampli-tudes to display properly. The origin of the center of the bounded pulse islocated at z=−3.7 cm and x=−4.14 cm. �a� 42 �s at a time after edge 1 hascontacted the first surface with the diffractions transmitted into the plate andthe underlying water. �b� 80 �s at a time after the diffracted arrival fromedge 2 has contacted the first surface with the diffractions transmitted intothe plate and the underlying water. �c� 106 �s after both edge diffractionshave propagated through the plate. See Fig. 11 for “edge 1 and edge 2labels.”

cedure is followed it is difficult to predict the behavior of the


diffractions and therefore ray tracing would not lead to theexact path. The results from this modeling show the devel-opment of the edge diffractions from edges 1 and 2 in Figs.12�a� and 12�b�, respectively. It is important to note that thestrongest parts of the edge 1 diffraction do not propagate tothe receiver and the energy that does arrive there is consid-erably weakened such that it is not reliably observed in thereal �-t images �Figs. 8–10�. In contrast portions of the edge2 diffraction are much stronger at the receiver and are readilydetected as the “?” arrival �Figs. 8–10�. This modeling con-clusively demonstrates that the “?” arrival is yet another dif-fraction effect that must be properly accounted for in analy-sis of the goniometric records.

A complete modeling of all the angles of incidence mea-sured for the thin acrylic glass sample �Fig. 12�b�� displaysall the same behavior observed in the recorded traces �Fig.12�a��. The travel times for the various arrivals are shown asdetermined from the modeling also.

B. Advantage of large transmitter

The experimental method described can detect wavemodes that would otherwise be lost within the backgroundnoise if a small source transmitter is used. Here a model forthe waves propagating through the thin acrylic plate�2.46 cm thick� transmitted from a transducer of 2 mm indiameter is presented. It is assumed that the source wouldproduce as much energy per surface as the large transmitterused in these experiments. The resulting direct arrival to thesame distance of 17 cm �Fig. 13�b�� is shown as a true am-plitude normalized with respect to the amplitude of the freepulse at 2 mm away from the source �Fig. 13�a��. As is ex-pected, the peak modeled amplitude of Fig. 13�b� is muchweaker �approximately four orders of magnitude� than that

FIG. 13. �a� Observed wave field produced by the large transmitter. �b�Modeled wave field propagated to a distance of 17 cm for a small2-mm-diam spherical source for illustration. �c� Recorded passive noise.

obtained using the large transmitter �Fig. 13�a��. Indeed, the


background noise measured during passive listening pro-duced amplitudes significantly above those produced using asmall source �Fig. 13�c��.

One additional advantage of the larger transmitter is thatthe wave field smearing due to diffractions are also substan-tially reduced. As such, critical angle phenomena are clearlyapparent.

Finally, while the full modeling of the wave fields isuseful here in interpreting the observations, and in particularthe edge diffraction “?” arrivals, this full modeling is notneeded in all cases to determine the effective transmissivityof a plate of material. An original motivation for use of thelarge transmitter was to reduce the level of complexity re-quired to appropriately model and interpret observed wave-forms by allowing use of simpler plane wave theory withoutthe need to account for diffractions. A comparison of theobserved P and S amplitudes transmitted through the thinacrylic plate with those calculated using visco-elastic planewave theory agree well �Fig. 14�. In the precritical range, theobserved and calculated amplitudes differ by a small amounton average. At �c

P, however, some residual diffracted P re-mains that cannot exist for the pure plane wave solution.This residual observed P rapidly vanishes and good agree-ment between the observed and modeled S amplitudes exist.The elastic response �i.e., neglecting attenuation� curves ob-viously diverge from the visco-elastic ones.

VI. CONCLUSION

A new laboratory method to probe various materialswith acoustic waves was developed. The method utilizes alarge area source piezoelectric transmitter and a small-receiver piezoelectric receiver. The source produces anacoustic wave field bounded both spatially and temporally.The amplitude along the axis of propagation stays nearlyconstant to distances up to 30 cm away from the source. Thelarge amount of energy carried by the wave field further

FIG. 14. �Color online� Comparison of the P and S transmitted amplitudesboth observed and calculated using plane wave assumptions both with andwithout attenuation for the thin acrylic glass sample.

makes the detection of subtle arrivals, even in the absence of


amplification, events that would be difficult to observe with asmall source transducer. The source-receiver developed herewas well characterized and can be used with great confidencein various experiments such as in transmission-reflectionlaboratory tests on various materials.

In the event that not all characteristics of the wave fieldare understood, the entire wave field may be modeled using aphase propagation procedure. The advantage of this proce-dure is that the evolution of independent components of thewave field may be studied. However, once details of thewave field are sufficiently known, the transmissivity of platesmay be understood using less onerous calculations involvingonly plane wave concepts with some care exercised in thevicinity of critical angles.

The goniometer has been used in the characterization ofsaturated porous materials. Use of the same system in reflec-tivity studies will be forthcoming in a companion contribu-tion.

ACKNOWLEDGMENTS

This work would not have been possible without thededication of L. Tober, G. Lachat, A. Paget, and P. Zimmer-man. Funding for this work was provided by NSERC and theCanada Research Chairs program. DRS thanks the ResearchSchool of Earth Science, Australian National University andin particular Professor I. Jackson for access to facilities dur-ing the completion of this manuscript.

1Y. E. Shigong, J. Wu, and J. Peach, “Ultrasound shear wave imaging forbone,” Ultrasound Med. Biol. 26, 833–837 �2000�.

2S. I. Rokhlin and W. Wang, “Double through-transmission bulk wavemethod for ultrasonic phase velocity measurement and determination ofelastic constants of composite materials,” J. Acoust. Soc. Am. 91, 3303–3312 �1992�.

3B. Hosten, “Ultrasonic through-transmission method for measuring thecomplex stiffness moduli of composite materials,” in Handbook of ElasticProperties of Solids, Liquids, and Gases, edited by M. Levy, H. E. Bass,and R. R. Stern �Academic, New York, 2001�, Vol. 1, Chap. 4, pp. 87–106.

4A. B. Temsamani, S. Vandenplas, and L. vanBiesen, “Experimental inves-tigation of bounded beam reflection from plane interfaces in the vicinity ofleaky wave angles,” Ultrasonics 38, 749–753 �2000�.

5A. I. Lavrentyev and S. I. Rokhlin, “Ultrasonic study of environmentaldamage initiation and evolution in adhesive joints,” Res. Nondestruct.Eval. 10, 17–41 �1998�.

6J. Wu, “Determination of velocity and attenuation of shear waves usingultrasonic spectroscopy,” J. Acoust. Soc. Am. 99, 2871–2875 �1996�.

7T. J. Plona, “Observation of a second bulk compressional wave in a porousmedium at ultrasonic frequencies,” Appl. Phys. Lett. 36, 259–261 �1980�.

8P. N. J. Rasolofosaon, “Importance of interface hydraulic condition on thegeneration of the second bulk compressional wave in porous media,”Appl. Phys. Lett. 52, 780–782 �1988�.

9B. Gurevich, O. Kelder, and D. M. J. Smeulders, “Validation of the slowcompressional wave in porous media: comparison of experiments and nu-merical simulations,” Transp. Porous Media 36, 149–160 �1999�.

10D. L. Johnson, T. J. Plona, and H. Kojima, “Probing porous media withfirst and second sound. II. Acoustic properties of water saturated porousmedia,” J. Appl. Phys. 76, 115–125 �1994�.

11O. Kelder and D. M. J. Smeulders, “Observation of the Biot slow wave in


water-saturated Nivelsteiner sandstone,” Geophysics 62, 1794–1796�1997�.

12R. Bass, “Diffraction effects in the ultrasonic field of a piston source,” J.Acoust. Soc. Am. 30, 602 �1958�.

13P. H. Rogers and A. L. Van Buren, “An exact expression for the Lommeldiffraction correction integral,” J. Acoust. Soc. Am. 55, 724 �1974�.

14H. L. Bertoni and T. Tamir, “Unified theory of Rayleigh-angle phenomenafor acoustic beams at liquid-solid interfaces,” Appl. Phys. 2, 157–172�1973�.

15T. J. Plona, L. E. Pitts, and W. G. Mayer, “Ultrasonic bounded beamreflection and transmission effects at a liquid/solid-plate/liquid interface,”J. Acoust. Soc. Am. 59, 1324–1328 �1976�.

16L. Wang, A. I. Larentyev, and S. I. Roklin, “Beam and phase effects inangle-beam-through-transmission method of ultrasonic velocity measure-ment,” J. Acoust. Soc. Am. 113, 1551–1559 �2003�.

17K. W. Winkler and T. J. Plona, “Technique for measuring ultrasonic ve-locity and attenuation spectra in rocks under pressure,” J. Geophys. Res.87, 776–780 �1982�.

18M. Mah and D. R. Schmitt, Determination of the complete elastic stiff-nesses from ultrasonic phase velocity measurements, J. Geophys. Res.108�B1�, 2016 �2003�.

19W. Xu and J. J. Kaufman, “Diffraction correction methods for insertionultrasound attenuation estimation,” IEEE Trans. Biomed. Eng. 40, 563–570 �1993�.

20O. I. Lobkis, A. Safaeinili, and D. E. Chimenti, “Precision ultrasonic re-flection studies in fluid-coupled plates,” J. Acoust. Soc. Am. 99, 2727–2736 �1996�.

21O. I. Lobkis and D. E. Chimenti, “Three-dimensional transducer voltage inanisotropic materials characterization,” J. Acoust. Soc. Am. 106, 36–45�1999�.

22P. He and J. Zheng, “Acoustic dispersion and attenuation measurementusing both transmitted and reflected pulses,” Ultrasonics 39, 27–32�2001�.

23M. Arakawa, J. Kushibiki, and N. Aoki, “An evaluation of effective radi-uses of bulk-wave ultrasonic transducers as circular piston sources foraccurate velocity measurements,” IEEE Trans. Ultrason. Ferroelectr. Freq.Control 51, 496–501 �2004�.

24B. Hosten and M. Castings, “Transfer-matrix of multilayered absorbingand anisotropic media—Measurements and simulations of ultrasonicwave-propagation through composite-materials,” J. Acoust. Soc. Am.94�Part 1�, 1488–1495 �1993�.

25D. Fei, D. E. Chimenti, and S. V. Teles, “Material property estimation inthin plates using focused, synthetic-aperture acoustic beams,” J. Acoust.Soc. Am. 113, 2599–2610 �2003�.

26O. I. Lobkis and D. E. Chimenti, “Equivalence of Gaussian and pistonultrasonic transducer voltages,” J. Acoust. Soc. Am. 114, 3155–3166�2003�.

27P. Cawley and B. Hosten, “The use of large ultrasonic transducers toimprove transmission coefficient measurements on visco-elastic aniso-tropic plates,” J. Acoust. Soc. Am. 101, 1373–1379 �1997�.

28M. Castaings, B. Hosten, and T. Kundu, “Inversion of ultrasonic, plane-wave transmission data in composite plates to infer visco-elastic materialproperties,” NDT & E Int. 33, 377–392 �2000�.

29W. J. Startin, M. A. Beilby, and P. R. Saulson, “Mechanical quality factorsof fused silica resonators,” Rev. Sci. Instrum. 69, 3681–3689 �1998�.

30E. Juliac, J. Arman, and D. Harran, “Ultrasonic interferences in polymerplates,” J. Acoust. Soc. Am. 104, 1232–1241 �Part 1� �1998�.

31J. Gazdag, “Wave equation migration with the phase shift method,” Geo-physics 43, 1342–1351 �1997�.

32A. Schoch, “Schallrelexion, schallbrechung und schallbeugung,” Ergeb.Exakten Naturwiss. 23, 127–234 �1950�.

33L. C. Brekhovskikh, “Waves in layered media,” in Applied Mathematicsand Mechanics, edited by R. T. Beyer �Academic, New York, 1960�, pp.100–122.


Documents

A large ultrasonic bounded acoustic pulse transducer for