Fundamentals of Ultrasonic Inspection

Fundamentals of Ultrasonic Inspection*Revised by Leonard J. Bond, Iowa State University

ULTRASONIC INSPECTION is a family ofnondestructive methods in which beams ofhigh-frequency mechanical waves are intro-duced into materials, using transducers, forthe detection and characterization of both sur-face and subsurface anomalies and flaws inthe material. The mechanical waves travelthrough the material with some attendant lossof energy (attenuation, including both scatter-ing and absorption) and interact with interfaces(reflection, transmission) and discontinuities,including flaws and other anomalies. Responsesignals are detected, displayed, and then ana-lyzed to give signatures that are used to definethe presence, location, and characteristics offlaws or other discontinuities.The degree of reflection/transmission depends

largely on the physical state of the materialsforming the interface and to a lesser extent onthe specific physical properties of the material.For example, ultrasonic waves are almostcompletely reflected atmetal-gas interfaces. Par-tial reflection occurs at a metal-liquid interfaceor at an interface between a metal and anothersolid, with the specific percentage of reflectedenergy depending mainly on the ratios of certainacoustic properties (acoustic impedance) of thematerial on opposing sides of the interface andthe angle of incidence.Cracks, delaminations, shrinkage cavities,

bursts, flakes, pores, disbonds, and other dis-continuities that produce reflective interfaces,and which are large compared to the wave-length of the ultrasound, can, in general, beeasily detected. Inclusions, voids, and otherinhomogeneities also can be detected by caus-ing partial reflection or scattering of the ultra-sonic waves or by producing some otherdetectable effect on the ultrasonic waves,which, in general, is a function of the wavetype and wavelength employed in relation tofeature size.Most ultrasonic inspection instruments

detect flaws by monitoring one or more ofthe following:

� Reflection of ultrasound from interfacesconsisting of material boundaries or discon-tinuities within the metal itself

� Velocity of the waves that control the timeof transit of an ultrasound wave throughthe testpiece

� Attenuation of ultrasound waves causedby absorption and scattering within thetestpiece

� Features seen in the spectral response foreither a transmitted or a reflected signal

Most conventional ultrasonic inspection isperformed using frequencies between 1.0 and25 MHz, which are well above the range ofhuman hearing, which is approximately 20 Hzto 20 kHz. Ultrasonic waves are mechanicalvibrations, where the amplitudes of vibrationsin metal parts being ultrasonically inspectedimpose stresses that are well below the elasticlimit, thus preventing permanent effects on theparts. Many of the characteristics described inthis article for ultrasonic waves, especially inthe section “General Characteristics of Ultra-sonicWaves,” also apply to audible soundwavesand to wave motion in general.Ultrasonic inspection is one of the most

widely used families of methods employedfor nondestructive testing (NDT). Its primaryapplication in the inspection of metals is thedetection and characterization of internalflaws. It also is used to detect surface flaws,to define bond characteristics, to measure thethickness and extent of corrosion, and (muchless frequently) to determine physical proper-ties, structure, grain size, and elastic constants.

Basic Equipment

Most ultrasonic inspection systems includethe following basic equipment, although suchunits are increasingly integrated into a digitalsystem and in many cases now employ a phasedarray, rather than a single-element transducer:

� An electronic signal generator that producesbursts of voltage (a negative spike or a squarewave) when electronically triggered (pulsed)

� A transmitting transducer (probe or searchunit) that can be a single element or anarray that emits a beam of ultrasonic waveswhen bursts of voltage are applied to it

� A couplant to transfer the input energy inthe beam of ultrasonic waves from thetransmitting transducer to the testpiece

� A couplant to transfer the output ultrasonicwaves (acoustic energy) from the testpieceto the receiving transducer

� A receiving transducer (can be the same asthe transducer initiating the ultrasound orit can be a separate one) to accept and con-vert the output of ultrasonic waves from thetestpiece to corresponding bursts of alternat-ing voltage. In most systems, a single trans-ducer or array alternately acts as sender(transmitter) and receiver

� An electronic device to amplify and, if nec-essary, demodulate or otherwise modify thesignals from the transducer (receiver)

� A display or indicating device to character-ize and/or record the output from the test-piece. The display device now is mostcommonly a computer screen, which is anintegral part of many units; a full matrixof data commonly is recorded and kept forsubsequent or repeat analysis.

� An electronic clock, or timer, to control theoperation of the various components of thesystem, to serve as a primary reference timepoint, and to provide coordination for theentire system

In many cases the pulser, receiver, display,and clock are a single integrated unit thatmay be digital or interfaced to an externalcomputer. Traditionally, many ultrasonicinspections used single transducers in a pulse-echo mode, while others operated with twotransducers in transmission, with the objectunder test between them. An example of a

ASM Handbook, Volume 17, Nondestructive Evaluation of MaterialsAquil Ahmad and Leonard J. Bond, editors

Copyright # 2018 ASM InternationalW

All rights reservedwww.asminternational.org

*Adapted from P.G. Kenny, Ultrasonic Inspection, online update to ASM Handbook, Vol 17, Nondestructive Evaluation and Quality Control; and Ultrasonic Inspection by Yoseph Bar-Cohenet al., in Nondestructive Evaluation and Quality Control, Vol 17, ASM Handbook, 1989, p 231–277

https://www.asminternational.org/handbooks/-/journal_content/56/10192/33375464/PUBLICATION

two-transducer NDT system is shown in sche-matic form in Fig. 1 (Ref 1). Recent years haveseen increasing adoption of phased arrays;these are considered in the article “PhasedArray Ultrasound” in this Volume.

Advantages and Disadvantages

The principal advantages of ultrasonic inspec-tion, as compared to other methods used for non-destructive inspection of parts, are:

� Superior penetrating power, which allowsthe detection of flaws deep in the part

� High sensitivity, permitting the detection ofextremely small flaws

� Greater accuracy than other nondestructivemethods in determining the position of inter-nal flaws, estimating their size, and character-izing their orientation, shape, and nature

� Only one surface needs to be accessible.� Operation is electronic, which provides

almost instantaneous indication of flaws.� Withmost systems, a permanent digital record

of inspection data can bemade for subsequentoff-line review and for future reference.

� Volumetric scanning ability, enabling theinspection of a volume of metal extendingfrom front surface to back surface of a part

� It is nonhazardous to operators or to nearbypersonnel and has no effect on equipmentand materials in the vicinity.

� Portability (in many implementations)� It provides an output that can be processed

digitally in the test unit or by an externalcomputer to characterize defects and todetermine material properties.

The disadvantages of ultrasonic inspectioninclude:

� Manual implementations require carefulattention by experienced technicians.

� Extensive technical knowledge is required forthe development of inspection procedures.

� Parts that are rough, irregular in shape, verysmall or thin, or not homogeneous are moredifficult to reliably inspect.

� Discontinuities that are present in a shallowlayer immediately beneath the surface (deadzone) may not be detectable.

� For many types of piezoelectric-basedtransducers, couplants are needed to pro-vide effective transfer of ultrasonic waveenergy between transducers and parts beinginspected.

� Reference standards are needed in manyapplications, both for calibrating the equip-ment and for characterizing flaws.

Ultrasonic inspection is routinely applied tolarge parts, such as those with thicknesses ofmore than a meter, and in the axial inspectionof parts such as long steel shafts or rotor for-gings to ranges/thicknesses of 6 m (20 ft) andmore, particularly when using guided waves.A diverse range of systems has now beendeveloped, and these make the method in itsvarious implementations suitable for data digi-tization, immediate interpretation, automation,rapid scanning, in-line production monitoring,and process control. Some of the couplingrequirements can be overcome, in some cases,by using gas/air-coupled, laser, and electro-magnetic acoustic transducer (EMAT) ultra-sonics, which are all addressed in articles ontransduction. The issues related to referencestandards and defect characterization are con-sidered in the article “Ultrasonic Imaging andSizing” in this Volume.

Applicability

Ultrasonic inspection is conducted princi-pally for the detection of discontinuities. The

various implementations of the method canbe used to detect internal flaws in manyengineering materials. Bonds produced bywelding, brazing, soldering, and adhesionalso can be ultrasonically inspected. In-linetechniques have been developed for monitor-ing and classifying material as acceptable,salvageable, or scrap, and for process control.Systems range from those that are line-powered with automated transducer move-ment that may include scanning and robotics,to battery-operated commercial equipmentand systems that in many cases enable manualinspection in laboratory, shop, warehouse,and field with applications at all points inthe life cycle.Figure 2 shows examples of field portable

ultrasonic units with a single transducer and aphased array, as well as a large-scale inspec-tion unit.Ultrasonic inspection is used for quality

control and materials inspection in all majorindustries. This includes electrical and elec-tronic component manufacturing; productionof metallic and composite materials; and fabri-cation of structures such as airframes andengines, piping and pressure vessels, ships,bridges, motor vehicles, machinery, and jetengines. In-service ultrasonic inspection is usedfor detecting anomalies (e.g., cracks and corro-sion) with potential for failure of railroad-rolling-stock axles, press columns, earth-movingequipment, mill rolls, mining equipment, nuclearsystems, airframes and engines, pipelines, andmany other machines and components.Some of the major types of equipment that

are ultrasonically inspected for the presenceof flaws are:

� Mill components: Rolls, shafts, drives, andpress columns

� Power equipment: Turbine forgings, gener-ator rotors, pressure piping, weldments,pressure vessels, nuclear fuel elements,and other reactor components

� Jet engine parts: Turbine and compressorforgings, and gear blanks

� Aircraft components: Forging stock, framesections, and honeycomb sandwich assemblies

� Machinery materials: Die blocks, toolsteels, and drill pipe

� Pipelines: Cracks in seams and welds� Railroad parts: Axles, wheels, track, and

welded rail� Automotive parts: Forgings, ductile cast-

ings, and brazed and/or welded components

Flaws that are detected include voids, cracks,inclusions, pipe delaminations, debonding,bursts, and flakes that occur in a wide range ofsizes. They may be inherent in the raw material,result from fabrication and heat treatment, oroccur in service due to stressors that causefatigue, impact, abrasion, corrosion, or othercauses of degradation.Government agencies and standards-

making organizations have issued inspectionFig. 1 Schematic for a two-transducer nondestructive testing (NDT) system

156 / Sonic and Ultrasonic Techniques

procedures, acceptance codes, standards, andrelated documentation. These documents aremainly concerned with the detection of flawsin specific manufactured products, but theyalso can serve as the basis for characterizingflaws in many other applications and are com-monly used when specifying the required qual-ity in an item received from a supplier (Ref 2).Ultrasonic inspection can also be used to

measure the thickness of materials. Thicknessmeasurements are made for applications atfabrication and then in service for items thatare as diverse as refinery and chemical proces-sing equipment, shop plate, steel castings, sub-marine hulls, aircraft sections, and pressurevessels. A variety of ultrasonic techniques areavailable for thickness measurements, includ-ing the one shown in Fig. 3, and most nowuse units with a digital readout and, increas-ingly, data recording. Performance dependson the frequency used, but for structural mate-rial samples ranging in thickness from 0.2 mm(a few thousandths of an inch) to severalmeters can be measured with accuracies of bet-ter than 1%. Ultrasonic inspection methods areparticularly well suited to the assessment ofloss of thickness from corrosion inside closedsystems, such as chemical processing equip-ment. Such measurements often can be madewithout shutting down the process.Special ultrasonic techniques and equip-

ment have been used to operate at elevatedtemperatures and on such diverse problemsas the rate of growth of fatigue cracks, detec-tion of borehole eccentricity, measurement ofelastic moduli, study of press fits, determina-tion of nodularity in cast iron, and metallurgicalresearch on phenomena such as structure, hard-ening, and inclusion count in various metals.For the successful application of ultrasonic

inspection, the inspection physics and itsimplementation into a system must be wellunderstood, performance requirements defined,and then be suitable for the type of inspectionbeing performed, and the operator must besufficiently trained and experienced. If theseprerequisites are not met, there is a high potentialfor gross error in the inspection results. Forexample, with inappropriate equipment or witha poorly trained operator, discontinuities havinglittle or no bearing on product performance maybe deemed serious, or potentially damaging

discontinuities may go undetected or be deemedunimportant.The term flaw as used in this article means a

detectable lack of continuity or an imperfec-tion in a physical or dimensional attribute ofa part. The fact that a part contains one ormore flaws does not necessarily imply thatthe part is nonconforming to specification orunfit for use. Modern design philosophies willincorporate damage tolerance (Ref 3), anapproach that enables living with some flawsthat are not considered structurally deleterious,because materials are not perfectly isotropic orhomogeneous. It is critical that codes and stan-dards be established so that decisions to acceptor reject parts are based on the probable effectthat a given flaw will have on service life orproduct safety under particular operating con-ditions. Once such standards are established,ultrasonic inspection can be used to character-ize flaws in terms of impact on structural integ-rity, or life estimate, using a fracture mechanicsand life assessment, rather than some arbitraryworkmanship standard that may impose uselessor redundant quality requirements. This is anarea where NDT has evolved to leverage quanti-fication of nondestructive evaluation (NDE) cap-abilities required by design tools that relate NDEneeds to the ability of an item to safely operateuntil repaired or replaced. This performance ofNDT is typically considered in terms of a proba-bility of detection (POD) that becomes justpart of a quality chain (Ref 4) which connectsresearch and development to standards and pro-cedures, the equipment used, as well as person-nel training and certification that combine andseek to optimize performance and at the sametime minimize the effects of human factors oninspection performance.

General Characteristics ofUltrasonic Waves

Ultrasonic waves are mechanical waves (incontrast to, for example, light or x-rays, whichare electromagnetic waves) that consist ofoscillations or vibrations of the atomic ormolecular particles of a substance about theequilibrium positions of these particles. Ultra-sonic waves can propagate in an elasticmedium, which can be solid, liquid, or

gaseous, but not in a vacuum, and essentiallybehave the same as audible sound waves. Theyalso are fundamentally similar to the seismicwaves encountered in geophysics (Ref 1, 5).In some respects, a beam of ultrasound is

similar to a beam of light: both are wavesand obey a general wave equation. Each tra-vels at a characteristic velocity in a givenhomogeneous medium—a velocity thatdepends on the properties of the medium andthe wave type (Ref 5). Like beams of light,ultrasonic waves are reflected from surfaces,refracted when they cross a boundary betweentwo substances that have different characteris-tic acoustic velocities, and are diffracted atedges and around obstacles. Scattering byrough surfaces or particles reduces the energyof an ultrasonic beam, which is analogous tothe manner in which scattering reduces theintensity of a light beam.Analogy with Waves in Water. The gen-

eral characteristics of sonic or ultrasonic wavesare conveniently illustrated by analogy withthe behavior of waves produced in a body ofwater when a stone is dropped into it. Casualobservation may lead to the erroneous conclu-sion that the resulting outward radial travel ofalternate crests and troughs represents themovement of water away from the point ofimpact. The fact that water is not thus trans-ported is readily deduced from the observationthat a small object floating on the surface ofthe water does not move away from the pointof impact but instead merely bobs up anddown. The waves travel outward only in thesense that the crests and troughs (which canbe compared to the compressions and rarefac-tions of mechanical waves in an elasticmedium) and the energy associated with thewaves propagate radially outward. The waterparticles remain in place and oscillate up anddown from their normal positions of rest.Continuing the analogy, the distance

between two successive crests or troughs isthe wavelength, l. The fall from a crest to atrough and subsequent rise to the next crest(which is accomplished within this distance)is a cycle. The number of cycles in a specificunit of time is the frequency, f, of the waves.The height of a crest or the depth of a troughin relation to the surface at equilibrium is theamplitude of the wave.

Fig. 2 Examples of portable ultrasonic units (a) with transducer for pulse-echo measurement and (b) with phased array. Reprinted with permission from Olympus Corporation.(c) system for testing round bars: ROWA B-130 (left), ROWA B-260 (right). Courtesy of Timken

Fundamentals of Ultrasonic Inspection / 157

The velocity of a wave, of a particular type,and the rates at which the amplitude andenergy of a wave decrease as it propagatesare constants that are characteristic of themedium in which the wave is propagating.Stones of equal size and mass striking oil andwater with equal force will generate waves thattravel at different velocities in the two media.Stones impacting a given medium with greaterenergy will generate waves having greateramplitude and energy but the same wavevelocity.The aforementioned attributes apply simi-

larly to both audible and ultrasonic waves pro-pagating in an elastic medium. The particles of

the elastic medium move, but they do notmigrate from their initial spatial orbits; onlythe energy travels through the medium. Theamplitude and energy of mechanical waves inthe elastic medium depend on the amount ofenergy supplied. The velocity and attenuation(loss of amplitude and energy) of the acousticwaves depend on the properties of the mediumin which they are propagating.Wave Propagation. Ultrasonic waves (and

other mechanical waves) propagate to someextent in any elastic material. When the atomicor molecular particles of an elastic material aredisplaced from their equilibrium positions byany applied force, internal stresses act to

restore the particles to their original positions.Because of the interatomic forces betweenadjacent particles of material, a displacementat one point induces displacements at neigh-boring points and so on, thus propagating astress-strain wave. The actual displacement ofmatter that occurs in ultrasonic waves isextremely small. The amplitude, vibrationmode, and velocity of the waves differ due tovariation in elastic properties in solids, liquids,and gases because of the large differences inthe mean distance between particles in theseforms of matter. These differences influencethe forces of attraction between particles andthe elastic properties and behavior of thematerials.The concepts of wavelength, cycle, fre-

quency, amplitude, velocity, and attenuationdescribed in the preceding section “Analogywith Waves in Water” in this article apply, ingeneral, to ultrasonic waves and other mechan-ical waves. The relationship between to fre-quency and wavelength is given by:

V ¼ fl (Eq 1)

Where V is velocity (in meters per second), f isfrequency (in hertz), and l is wavelength (inmeters for one cycle). Other consistent unitsof measure can be used for the variables inEq 1, where convenient.On the basis of the mode of particle dis-

placement, ultrasonic waves are classified aslongitudinal waves, transverse (shear) waves,surface (Rayleigh) waves, and Lamb waves.These four types of waves are described inthe following sections (Ref 1, 5).Longitudinal waves, sometimes called com-

pression waves, are the type of ultrasonicwaves most widely used in the inspection ofmaterials. These waves travel through materi-als as a series of alternate compressions andrarefactions in which the particles transmittingthe wave vibrate back and forth in the direc-tion of travel of the waves.Longitudinal ultrasonic waves and the

corresponding particle oscillation and resultantrarefaction and compression are shown sche-matically in Fig. 4(a); a plot of amplitude ofparticle displacement versus distance of wavetravel, together with the resultant rarefactiontrough and compression crest, is shown inFig. 4(b). The distance from one crest to thenext (which equals the distance for one com-plete cycle of rarefaction and compression) isthe wavelength, l. The vertical axis in Fig.4(b)could represent pressure instead of particle dis-placement. The horizontal axis could representtime instead of travel distance, because thespeed of sound/ultrasound is constant in agiven homogeneous material, and because ofthis the relationship is used in the measure-ments made in ultrasonic inspection.Longitudinal ultrasonic waves are readily

propagated in liquids and gases as well as inelastic solids. The mean free paths of the mole-cules of liquids and gases at a pressure of

Fig. 3 (a) Example of a handheld ultrasonic thickness gage. Courtesy of DeFelsko Corporation. (b) Table showingtransducer type, accuracy, and wave form. Used with permission from Olympus Corporation


101 kPa (1 atmosphere or 14.7 psi) are so shortthat longitudinal waves can be propagated sim-ply by the elastic collision of one moleculewith the next. The velocity of longitudinalultrasonic waves is approximately 5960 m/s(20,000 ft/s) in steel, 1500 m/s (5000 ft/s) inwater, and 330 m/s (1080 ft/s) in air.Transverse waves, which also are called

shear waves, also are extensively used in theultrasonic inspection of materials. Transversewaves are visualized readily in terms of vibra-tions of a rope that is shaken rhythmicallyfrom side to side, in which each particle, ratherthan vibrating parallel to the direction of wavemotion as in the longitudinal wave, vibratesside to side or up and down in a plane perpen-dicular to the direction of propagation. A trans-verse wave is illustrated schematically inFig. 5, which shows particle oscillation,wave-front, direction of wave travel, and thewavelength, l, corresponding to one cycle.Unlike longitudinal waves, transverse waves

cannot be supported by the elastic collision ofadjacent molecular or atomic particles. Forthe propagation of transverse waves, it is nec-essary that each particle exhibit a strong forceof attraction to its neighbors so that as a parti-cle moves back and forth it pulls its neighborwith it, thus causing the energy to movethrough the material with the velocity asso-ciated with transverse waves. The velocity oftransverse waves is approximately 50% ofthe longitudinal wave velocity for the samematerial.Air and water will not support transverse

waves. In gases, the forces of attractionbetween molecules are so small that transversewaves cannot be transmitted. The same is trueof a liquid, unless it is particularly viscous or ispresent as a very thin layer.

Surface waves (Rayleigh waves) (Ref 1, 6)are another type of ultrasonic wave used in theinspection of materials. These waves travelalong the flat or curved surface of relativelythick solid parts. For the propagation of wavesof this type, the waves must be traveling alongan interface bounded on one side by the strongelastic forces of a solid and on the other sideby the practically negligible elastic forcesbetween gas molecules. Surface waves leakenergy into liquid and couplants, and use ofsuch leaky waves forms a special class ofinspections (Ref 1). In general, leaky surfacewaves do not exist for any significant distancealong the surface of a solid immersed in a liq-uid, unless the liquid covers the solid surfaceonly as a very thin film.Surface waves are subject to attenuation in a

given material, as are longitudinal or trans-verse waves. They have a velocity approxi-mately 90% of the transverse wave velocityin the same material. The region within whichthese waves propagate with effective energy isnot much thicker than approximately onewavelength beneath the surface of the metal,which in aluminum at 1 MHz is 2.9 mm(0.1 in.). At this depth, wave energy is approx-imately 4% of the wave energy at the surface,and the amplitude of oscillation decreasessharply to a negligible value at greater depths.Surface waves can follow contoured sur-

faces. For example, surface waves travelingon the top surface of a metal block arereflected from a sharp edge, but if the edge isrounded off, the waves continue down the sideface and are reflected at a defect or the loweredge, returning to the sending point. Surfacewaves will travel completely around a cube ifall edges of the cube are rounded off. Surfacewaves can be used to inspect many parts thathave complex contours.In surface waves, particle oscillation gener-

ally follows an elliptical orbit, as shown sche-matically in Fig. 6. The major axis of theellipse is perpendicular to the surface alongwhich the waves are traveling. The minor axisis parallel to the direction of propagation. Sur-face waves can exist in complex forms that arevariations of the simplified waveform illu-strated in Fig. 6. These waves, and their appli-cations, are discussed in the article “RayleighWave Nondestructive Evaluation for Defect

Detection and Materials Characterization” inthis Volume.Lamb waves, also known as plate or guided

waves (Ref 7, 8), are another type of ultrasonicwave used in the nondestructive inspection ofmaterials. Lamb waves are propagated inplates (made of composites or metals) thatare only a few wavelengths thick. A Lambwave consists of a complex pattern of vibra-tions that occurs throughout the thickness ofthe material. The propagation characteristicsof Lamb waves depend on the density, elasticproperties, and structure of the material as wellas the thickness of the testpiece and the fre-quency. Their behavior, in general, resemblesthat observed in the transmission of electro-magnetic waves through waveguides.There are two basic forms of Lamb waves:

� Symmetrical, or dilatational� Asymmetrical, or bending

The form is determined by whether the particlemotion is symmetrical or asymmetrical withrespect to the neutral axis of the testpiece. Insymmetrical (dilatational) Lamb waves, thereis a compressional (longitudinal) particle dis-placement along the neutral axis of the plateand an elliptical particle displacement on eachsurface (Fig. 7a). In asymmetrical (bending)Lamb waves, there is a shear (transverse) par-ticle displacement along the neutral axis ofthe plate and an elliptical particle displacementon each surface (Fig. 7b). The ratio of themajor to minor axes of the ellipse is a functionof the material in which the wave is beingpropagated.Each class of Lamb waves is further subdi-

vided into several modes having different velo-cities. Theoretically, there are an infinitenumber of specific velocities at which Lambwaves can travel in a given material. Withina given plate, the specific velocities for Lambwaves are complex functions of plate thicknessand frequency. An example of a dispersioncurve is shown as Fig. 8 for the case of a1.0 mm thick aluminum plate, and modes areshown in a normalized thickness-frequencyscale with units of MHz-mm. These wavesare seeing increased use in inspection of plates,rods, and pipes. Many aspects of these wavesand their applications in NDT as well as Lamb

Fig. 4 Schematic of longitudinal ultrasonic waves.(a) Particle oscillation and resultant rarefaction

and compression. (b) Amplitude of particle displacementversus distance of wave travel. The wavelength, l, is thedistance corresponding to one complete cycle.

Fig. 5 Schematic of transverse (shear) waves. Thewavelength, l, is the distance correspondingto one complete cycle.

Fig. 6 Diagram of surface (Rayleigh) waves propagatingat the surface of a metal along a metal-air

interface. The wavelength, l, is the distance correspondingto one complete cycle.


wave generation and detection are discussedin the article “Guided Wave Testing” in thisVolume.

Major Variables in UltrasonicInspection

The major variables that must be consideredin ultrasonic inspection include both the char-acteristics of the ultrasonic waves used andthe characteristics of the parts being inspected.Equipment type and capability interact withthese variables; often, different types of equip-ment and wave types must be selected toaccomplish different inspection objectives(Ref1,5).The transductionmechanismemployedis also a key element in an inspection, not the leastwith regard to the wave field employed and ulti-mate sensitivity.The frequency of the ultrasonic waves used

affects inspection capability in several ways.Generally, a compromise must be madebetween favorable and adverse effects toachieve an optimal balance and to overcomethe limitations imposed by equipment and thematerial under test.Sensitivity, or the ability of an ultrasonic

inspection system to detect a very small dis-continuity, is generally increased by using rel-atively high frequencies (short wavelengths).Resolution, or the ability of the system to

give simultaneous, separate indications fromdiscontinuities that are close together both indepth below the front surface of the testpiece

and in lateral position, is directly proportionalto frequency bandwidth and inversely relatedto pulse length. Resolution generally improveswith an increase of frequency and bandwidththat accompanies a reduction in pulse length.Penetration, or the maximum depth (range)

in a material from which useful indicationscan be detected, is reduced by the use of higherfrequencies. This effect is most pronounced inthe inspection of metal that has coarse grainstructure or minute inhomogeneities, and incomposite materials because of the resultantattenuation and scattering of the ultrasonicwaves. Penetration limits due to attenuationare of less consequence in the inspection offine-grained, homogeneous metal and are moresignificant in a cast austenitic stainless steelwith large and complex-shaped grains.Beam spread, or the divergence from the

central beam axis for a single-element ultra-sonic transducer with a particular aperture, isalso affected by frequency. As frequencydecreases, the shape of an ultrasonic beamincreasingly departs from the ideal of zerobeam spread. This characteristic is observedat almost all frequencies used in inspection.Other factors, such as the transducer (searchunit) diameter and the use of focusing equip-ment, also affect beam spread. These issuesare discussed in greater detail in the section“Beam Spreading” in this article, and in thearticle “Ultrasonic Transduction (TransducerElements)” in this Volume.Sensitivity, resolution, penetration, and beam

spread are largely determined by the selectionof the transducer and are only slightly modifiedby changes in other test variables.Acoustic Impedance. When ultrasonic

waves traveling through one medium impingeon the boundary of a second medium at normalincidence, a portion of the incident acousticenergy is reflected back from the boundarywhile the remaining energy is transmitted intothe second medium. The characteristic that

determines the amount of reflection and trans-mission is the ratio of the acoustic impedanceof the two materials on either side of theboundary. If the acoustic impedances of thetwo materials are equal, there will be no reflec-tion. If the acoustic impedances differ greatly(as between a metal and air, for example),there will be virtually complete reflection. Thischaracteristic is used in the ultrasonic inspec-tion of metals to calculate the amounts ofenergy reflected and transmitted at impedancediscontinuities and to aid in the selection ofsuitable materials for the effective transfer ofacoustic energy between components in ultra-sonic inspection systems.The acoustic impedance for a longitudinal

wave, Zl, given in Pascal second per cubicmeter (Pa. s/m3) or the Rayl per square meter(Rayl/m2), which is defined as the product ofmaterial density, r, given in Kg per cubicmeter, and longitudinal wave velocity, Vl,given in meters per second, is:

Zl ¼ rVl (Eq 2)

The acoustic properties of several metalsand nonmetals are listed in Table 1. The acous-tic properties of materials are influenced byvariations in structure and metallurgical condi-tion. Therefore, for a given testpiece the prop-erties may differ somewhat from the valueslisted in Table 1.The percentage of incident energy reflected

from the interface between two materialsdepends on the ratio of acoustic impedances(Z2/Z1) and the angle of incidence. When theangle of incidence is 0� (normal incidence),the reflection coefficient, R, which is the ratioof reflected beam intensity, Ir, to incident beamintensity, Ii, is given by:

R ¼ Ir=Ii ¼ ½ðZ2 � Z1Þ=Z2 þ Z1Þ�2¼ ½ðr � 1Þ=ðr þ 1Þ�2 (Eq 3)

(a)

(b)

Fig. 7 Diagram of the basic patterns of (a) symmetrical(dilatational) and (b) asymmetrical (bending) Lamb

waves. The wavelength, l, is the distance corresponding toone complete cycle.

Fig. 8 Example of dispersion curves, calculated for aluminum 1.0 mm thick, with symmetric and antisymmetricmodes, shown in normalized units (MHz-mm). A0 and S0 are fundamental modes; index indicatesincreasing higher order. Calculated by N. Pei, Iowa State University


Where Z1 is the acoustic impedance ofmedium 1, Z2 is the acoustic impedance ofmedium 2, and r equals Z2/Z1 and is theimpedance ratio, or mismatch factor. WithT designating the transmission coefficient,

R + T = 100%, because all the energy is eitherreflected or transmitted, and T is simplyobtained from this relationship.The transmission coefficient, T, also can be

calculated as the ratio of the intensity of the

transmitted beam, It, to that of the incidentbeam, Ii, from:

T ¼ It=Ii ¼ 4Z2Z1=ðZ2 þ Z1Þ2 ¼ 4r=ðr þ 1Þ2(Eq 4)

When a longitudinal ultrasonic wave inwater (medium 1) is incident at right anglesto the surface of an aluminum alloy 1100 test-piece (medium 2), the percentages of acousticenergy reflected and transmitted are calculated(based on data from Table 1) as:

Impedance ratio ðrÞ ¼ Z2Z1 ¼ 1:72=0:149 ¼ 11:54

Reflection coefficient ðRÞ ¼ ½ðr � 1Þ=ðr þ 1Þ�2¼ ð10:54=12:54Þ 2¼ 0:71 ¼ 71%

Transmission coefficient ðTÞ ¼ 1� R ¼ 0:29 ¼ 29%

The same values are obtained for R and Twhen medium 1 is the aluminum alloy andmedium 2 is water.For normal incidence the energy transmitted

as a function of acoustic impedance ratio fortwo semi-infinite media is shown in Fig. 9.Angle of Incidence. Only when an ultra-

sonic wave is incident at right angles on aninterface between two materials (normal inci-dence; i.e., angle of incidence = 0�) do trans-mission and reflection occur at the interfacewithout any change in beam direction. At anyother angle of incidence, the phenomena ofmode conversion (a change in the nature ofthe wave motion) and refraction (a change indirection of wave propagation) must be consid-ered. These phenomena may affect the entirebeam or only a portion of the beam, and thesum total of the changes that occur at the inter-face depends on the angle of incidence and thevelocity of the ultrasonic waves on either sideof the point of impingement on the interface.All possible ultrasonic waves leaving this pointare shown for an incident longitudinal ultra-sonic wave in Fig. 10. In many cases, not allof the waves shown in Fig. 10 will be pro-duced in any specific instance of obliqueimpingement of an ultrasonic wave on theinterface between two materials. The wavesthat propagate in a given instance depend onthe ability of a particular wave mode to existin a given material, the angle of incidence ofthe initial beam, and the velocities of the wavemodes in both materials.The general law that describes wave behav-

ior at an interface is known as Snell’s law.Although originally derived for light waves,Snell’s law applies to acoustic (elastic) waves,including ultrasound, and also to many othertypes of waves. According to Snell’s law, theratio of the sine of the angle of incidence tothe sine of the angle of reflection or refractionequals the ratio of the corresponding wavevelocities. Snell’s law applies even if modeconversion takes place. Mathematically,Snell’s law can be expressed as:

sina= sinb ¼ V1=V2 (Eq 5)

Table 1 Acoustic properties of several metals and nonmetals

Material Density (r), kg/m3

Sonic velocities, m/s

Acoustic impedance (Z1)(d), MRaylVl(a) Vt(b) Vs(c)

Ferrous metals

Carbon steel, annealed 7850 5940 3240 3000 46.6Alloy steelAnnealed 7860 5950 3260 3000 46.8Hardened 7800 5900 3230 . . . 46.0

Cast iron 6950–7350 3500–5600 2200–3200 . . . 25–4052100 steelAnnealed 7830 5990 3270 . . . 46.9Hardened 7800 5890 3200 . . . 46.0

D6 tool steelAnnealed 7700 6140 3310 . . . 47.0Hardened 7700 6010 3220 . . . 46.0

Stainless steelsType 302 7900 5660 3120 3120 44.7Type 304L 7900 5640 3070 . . . 44.6Type 347 7910 5740 3100 2800 45.4Type 410 7670 5390 2990 2160 41.3Type 430 7700 6010 3360 . . . 46.3

Nonferrous metals

Aluminum 1100-O 2710 6350 3100 2900 17.2Aluminum alloy 2117-T4 2800 6250 3100 2790 17.5Beryllium 1850 12800 8710 7870 23.7Copper 110 8900 4700 2260 1930 41.8Copper alloys260 (cartridge brass, 70%) 8530 3830 2050 1860 32.7464 to 467 (naval brass) 8410 4430 2120 1950 37.3510 (phosphor bronze, 5% A) 8860 3530 2230 2010 31.2752 (nickel silver 65-18) 8750 4620 2320 1690 40.4

LeadPure 11340 2160 700 640 24.5Hard (94Pb-6Sb) 10880 2160 800 730 23.5

Magnesium alloy M1A 1760 5740 3100 2870 10.1Mercury, liquid 13550 1450 . . . . . . 19.5Molybdenum 10200 6250 3350 3110 63.8NickelPure 8800 5630 2960 2640 4.95Inconel 8500 5820 3020 2790 4.95Inconel X-750 8300 5940 3120 . . . 4.93Monel 8830 5350 2720 2460 4.72

Titanium, commercially pure 4500 6100 3120 2790 2.75Tungsten 19250 5180 2870 2650 9.98

Nonmetals

Air(e) 1.29 331 . . . . . . 0.0004Ethylene glycol 1110 1660 . . . . . . 1.8GlassPlate 2500 5770 3430 3140 14.4Pyrex 2230 5570 3440 3130 12.4

Glycerin 1260 1920 . . . . . . 2.4OilMachine (SAE 20) 870 1740 . . . . . . 1.50Transformer 920 1380 . . . . . . 1.27

Paraffin wax 900 2200 . . . . . . 2.0PlasticsMethylmethacrylate (Lucite,Plexiglas)

1180 2670 1120 1130 3.2

Polyamide (nylon) 1000–1200 1800–2200 . . . . . . 1.8–2.7Polytetrafluoroethylene (Teflon) 2200 1350 . . . . . . 3.0

Quartz, natural 2650 5730 . . . . . . 15.2Rubber, vulcanized 1100–1600 2300 . . . . . . 2.5–3.7Tungsten carbide 10000–15000 6660 3980 . . . 67.0–99.0WaterLiquid(f) 1000 1490 . . . . . . 1.49Ice(g) 900 3980 1990 . . . 3.6

(a) Longitudinal (compression) waves. (b) Transverse (shear) waves. (c) Surface waves. (d) For longitudinal waves Z1 = rV1. (e) At standardtemperature and pressure. (f) At 4 �C (39 �F). (g) At 0 �C (32 �F)


where a is the angle of incidence, b is theangle of reflection or refraction, and V1 andV2 are the respective velocities of the incidentand reflected or refracted waves. Both a andb are measured from a line normal to theinterface.The general relationship applying to reflec-

tion and refraction, taking into account allpossible effects of mode conversion for anincident longitudinal ultrasonic wave, asshown in Fig. 10, is given as:

sina1=Vl 1ð Þ ¼ sina01=Vl 1ð Þ ¼ sina0t=Vt 1ð Þsinb1=Vlð2Þ ¼ sinbt=Vt 2ð Þ

(Eq 6)

where al is the angle of incidence for incidentlongitudinal wave in material 1, a0l is the angleof reflection for reflected longitudinal wave inmaterial 1 = al, a0t is the angle of reflectionfor reflected transverse wave in material 1, blis the angle of refraction for refracted

longitudinal wave in material 2, bt is the angleof refraction for refracted transverse wave inmaterial 2, Vl(1) is the velocity of incident lon-gitudinal wave in material 1 = velocity ofreflected longitudinal wave in material 1, Vt(1)

is the velocity of reflected transverse wave inmaterial 1, Vl(2) is the velocity of refracted lon-gitudinal wave in material 2, and Vt(2) is thevelocity of refracted transverse wave in mate-rial 2.For quantities that are shown in Fig. 10 but

do not appear in Eq 6, bs is the angle of refrac-tion for refracted surface (Rayleigh) wave inmaterial 2 = 90�, and Vs(2) is the velocity ofrefracted surface (Rayleigh) wave in material2. Equation 6 can apply to similar relationshipsfor an incident transverse (instead of longitudi-nal) wave by substituting the term sin at/Vt(1)

for the first term, sin al/Vl(1). Correspondingly,in Fig. 10, the incident longitudinal wave atangle al (with velocity Vl(1) in material 1)

would be replaced by an incident transverseangle at equal to a0t (with velocity Vt(1)).Critical Angles. If the angle of incidence

(al, Fig. 10) is small, ultrasound waves propa-gating in a given medium may undergo modeconversion at a boundary, resulting in thesimultaneous propagation of longitudinal andtransverse (shear) waves in a second medium.If the angle is increased, the direction of therefracted longitudinal wave will approach theplane of the boundary (b1 ! 90�). At somespecific value of a1, bl will exactly equal 90�,above which the refracted longitudinal wavewill no longer propagate in the material, leav-ing only a refracted (mode-converted) shearwave to propagate in the second medium.This value of a1 is known as the first criticalangle. If a1is increased beyond the first criticalangle, the direction of the refracted shearwave will approach the plane of the boundary(bt ! 90�). At a second specific value of a1,bt will exactly equal 90�, above which therefracted transverse wave no longer will prop-agate in the material. This second value of al iscalled the second critical angle.Critical angles are of special importance in

ultrasonic inspection. Values of al betweenthe first and second critical angles are requiredfor most angle-beam inspections. Surface waveinspection is accomplished by adjusting theincident angle of a contact-type search unit sothat it is a few tenths of a degree greater thanthe second critical angle. At this value, therefracted shear wave in the bulk material isreplaced by a Rayleigh wave traveling alongthe surface of the testpiece. As mentioned pre-viously in this article, Rayleigh waves can beeffectively sustained only when the mediumon one side of the interface (in this case, thesurface of the testpiece) is a gas. Conse-quently, surface wave inspection is primarilyused with contact methods.In ordinary angle-beam inspection, it is

usually desirable to have only a shear wavepropagating in the test material. Because longi-tudinal waves and shear waves propagate atdifferent speeds, echo signals will be receivedat different times, depending on which typeof wave produced the echo. When both typesare present in the test material, confusing echopatterns may be shown on the display device,which can lead to erroneous interpretations oftestpiece quality. Frequently, it is desirable toproduce shear waves in a material at an angleof 45� to the surface. In most materials, inci-dent angles for mode conversion to a 45� shearwave lie between the first and second criticalangles. Typical values of al for all three ofthese—first critical angle, second criticalangle, and incident angle for mode conversionto 45� shear waves—are listed in Table 2 forvarious metals.Beam Intensity. The intensity of an ultra-

sonic beam is related to the amplitude of parti-cle vibrations. Acoustic pressure is the termmost often used to denote the amplitude ofalternating stresses exerted on a material by a

10

0.1

0.01

0.0010.001 0.01 0.1

Ene

rgy

tran

smis

sion

coe

ffici

ent,

T

1

Semi-infinitemedium 1

Semi-infinitemedium 2

Impedance ratio z2/z1

10 100 1000

Reflected energy

Incidentenergy

Transmittedenergy

T = 4z2/z1 T = 4z1/z2

z1 z2

Fig. 9 Acoustic energy transmission across an interface between two semi-infinite media, calculated at normalincidence. Source: Ref 9. Reprinted with permission from Institute of Electrical and Electronics Engineers(IEEE), # Copyright 1965

Material 1

Vl(1)

Vt(2)

Vs(2)

Vl(2)

Vt(1)

Vl(1)

α1

βs

β1

βt

α′1

α′tInterface

Material 2

Fig. 10 Diagram showing relationship (by vectors) of all possible reflected and refracted waves to an incidentlongitudinal wave of velocity Vl(1) impinging on an interface at angle al relative to normal to theinterface. See text for explanation of vectors.


propagating ultrasonic wave. Acoustic pres-sure is directly proportional to the product ofacoustic impedance and amplitude of particlemotion. The acoustic pressure exerted by agiven particle varies in the same directionand with the same frequency as the positionof that particle changes with time. Acousticpressure is the most important property of anultrasonic wave, and its square determines theamount of energy (acoustic power) in thewave. It should be noted that acoustic pressureis not the intensity of the ultrasonic beam.Intensity, which is the energy transmittedthrough a unit cross-sectional area of the beam,is proportional to the square of acousticpressure.Although piezoelectric transducer elements

sense acoustic pressure, ultrasonic systems donot measure acoustic pressure directly. How-ever, receiver-amplifier circuits of most ultra-sonic instruments are designed to produce anoutput voltage proportional to the square ofthe input voltage from the transducer. There-fore, the signal amplitude of ultrasound thatis typically displayed in a commercial ultra-sound NDT unit is a value proportional to thetrue intensity of the reflected ultrasound.The law of reflection and refraction described

in Eq 5 or 6 gives information regarding only thedirection of propagation of reflected andrefracted waves and says nothing about theacoustic pressure in the reflected or refractedwaves. When ultrasonic waves are reflected orrefracted, the energy in the incident wave is par-titioned among the various reflected andrefracted waves. The relationship among acous-tic energies in the resultant waves is complexand depends both on the angle of incidence andon the acoustic properties of the matter on oppo-site sides of the interface.The variation of acoustic pressure (not

energy) with angle of reflection or refraction(a0l, bl, or bt, defined in Fig. 10) that resultswhen an incident longitudinal wave in waterhaving an acoustic pressure of 1.0 arbitraryunit impinges on the surface of an aluminumtestpiece is shown as Fig. 11. At normal

incidence (al = a0l = bl = 0), acoustic energyis partitioned between a reflected longitudinalwave in water and a refracted (transmitted)longitudinal wave in aluminum. Because ofdifferent acoustic impedances, this partitioninduces acoustic pressures of approximately0.8 arbitrary units in the reflected wave inwater and approximately 1.9 units in the trans-mitted wave in aluminum. Although it mayseem anomalous that the transmitted wavehas a higher acoustic pressure than the incidentwave, it must be recognized that it is acousticenergy, not acoustic pressure, that is parti-tioned and conserved.In Fig. 11, as the incident angle, a1, is

increased, there is a slight drop in the acousticpressure of the reflected wave, a correspondingslight rise in the acoustic pressure of therefracted longitudinal wave, and a sharper risein the acoustic pressure of the refracted trans-verse wave. At the first critical angle for thewater-aluminum interface (a1 = 13.6, b1 = 90,and bt = 29.2), the acoustic pressure of the lon-gitudinal waves reaches a peak, and therefracted waves go rapidly to zero (point A0,Fig. 11). Between the first and second criticalangles, the acoustic pressure in the reflectedlongitudinal wave in water varies as shownbetween points A and B in Fig. 11. Therefracted longitudinal wave in aluminummeanwhile has disappeared. Beyond the sec-ond critical angle (al = 28.8�), the transversewave in aluminum disappears, and there istotal reflection at the interface with no partitionof energy and no variation in acoustic pressure,as shown to the right of point B in Fig. 11.Curves similar to those in Fig. 11 can be

constructed for the reverse instance of incidentlongitudinal waves in aluminum impinging onan aluminum-water interface, for incidenttransverse waves in aluminum, and for othercombinations of wave types and materials.Details of this procedure are available in vari-ous texts (Ref 1, 5). These curves are impor-tant because they indicate the angles ofincidence at which energy transfer across theboundary is most effective. For example, at

an aluminum-water interface, peak transmis-sion of acoustic pressure for a returning trans-verse wave echo occurs in the sector fromapproximately 16 to 22� in the water relativeto a line normal to the interface. Consequently,35 to 51� angle beams in aluminum are themost efficient in transmitting detectable echoesacross the front surface during immersioninspection and therefore can resolve smallerdiscontinuities than beams directed at otherangles in the aluminum. The partition ofacoustic energy at a water-steel interface isillustrated with Fig. 12, which shows thatshear waves, with no longitudinal component,in steel can be produced between approxi-mately 15 and 28� angle of incidence. Anexample showing the utilization of thesemode-conversion phenomena in shear wavetransducers is given in Fig. 13, which displaysthe pressure ratios and critical angles forPlexiglas (Perspex) on aluminum.

Attenuation of Ultrasonic Beams

The intensity of an ultrasonic beam that issensed by a receiving transducer is consider-ably less than the intensity of that initiallytransmitted. The factors primarily responsiblefor the loss in beam intensity can be classifiedas transmission losses, interference effects, andbeam spreading.Causes of transmission losses include

absorption, scattering, and acoustic impedancemismatch effects at interfaces. Interferenceeffects include diffraction and other effectsthat create wave fringes, phase shift, or fre-quency shift. Beam spreading involves mainlya transition from plane waves to either spheri-cal or cylindrical waves, depending on theshape of the transducer-element face. The

Table 2 Critical angles for immersion and contact testing, and incident angle for 45�shear wave transmission, in various metals

Metal

First critical angle,degrees(a), for:

Second critical angle,degrees(a), for:

45� shear wave incident angle,degrees(a), for:

Immersiontesting(b)

Contacttesting(c)

Immersiontesting(b)

Contacttesting(c)

Immersiontesting(b)

Contacttesting(c)

Steel 14.5 26.5 27.5 55 19 35.5Cast iron 15–25 28–50 . . . . . . . . . . . .

Type 302 stainless steel 15 28 29 59 19.5 37Type 410 stainless steel 11.5 21 30 63 20.5 39Aluminum alloy 2117-T4 13.5 25 29 59.5 20 37.5Beryllium 6.5 12 10 18 7 12.5Copper alloy 260 (cartridgebrass, 70%)

23 44 46.5 . . . 31 67

Inconel 11 20 30 62 20.5 38.5Magnesium alloy M1A 15 27.5 29 59.5 20 37.5Monel 16.5 30 33 79 23 44Titanium 14 26 29 59 20 37

(a) Measured from a direction normal to surface of test material. (b) In water at 4 �C (39 �F). (c) Using angle block (wedge) made of acrylic plastic

Refracted longitudinalwave in aluminum

Reflected longitudinalwave in water

Refracted transversewave in aluminum

A′0

Angle of reflection or refraction, degrees

5

4

3

Aco

ustic

pre

ssur

e, a

rbitr

ary

units

2

1

010 20 30 40 50 60 70 80 90

Total reflectionA B

Fig. 11 Variation of acoustic pressure with angle ofreflection or refraction during immersion

ultrasonic inspection of aluminum. The acousticpressure of the incident wave equals 1.0 arbitrary unit.Points A and A0 correspond to the first critical angle,and point B to the second critical angle, for this system.


wave physics that completely describe thesethree effects are discussed in various texts(Ref 1, 5).Acoustic impedance effects (see the sec-

tion “Acoustic Impedance” in this article) canbe used to calculate the amount of energy thatis reflected during the ultrasonic inspection ofa testpiece immersed in water. This reductionin intensity occurs primarily because of energypartition when waves are only partly reflected,for example, at the aluminum-water or steel-water interfaces. Additional losses wouldoccur due to absorption and scattering of theultrasonic waves, as discussed in the sections“Absorption” and “Scattering” in this article.Similarly, energy loss can be calculated for

a discontinuity that constitutes an ideal reflect-ing surface, such as a delamination that is nor-mal to the beam path that interposes a metal-air interface larger than the ultrasound beam.For example, in the straight-beam inspectionof an aluminum alloy 1100 plate containing adelamination, the final returning beam, afterpartial reflection at the front surface of theplate and total reflection from the delamina-tion, would have a maximum intensity 8% ofthe incident beam. By comparison, only 6%was found for the returning beam from theplate that did not contain a delamination. Sim-ilar calculations of the energy losses caused byimpedance effects at metal-water interfaces forthe ultrasonic immersion inspection of severalof the metals listed in Table 3 yield the follow-ing back-reflection intensities, which are

expressed as a percentage of the intensity ofthe incident beam.The loss (knockdown) for graphite epoxy

composite materials is even larger than formetals with fiber attenuation and lower mate-rial density. The loss in intensity of returningultrasonic beams is one basis for characterizingflaws in testpieces. As indicated previously,acoustic impedance losses can severely dimin-ish the intensity of an ultrasonic beam.Because only a small fraction of the area ofan ultrasound beam is reflected from small dis-continuities, it is obvious that ultrasonic instru-ments must be extremely sensitive to smallvariations in intensity if small discontinuitiesare to be detected. The ultrasound intensity ofcontact techniques is usually greater than thatof immersion techniques; that is, smaller dis-continuities will result in higher amplitude sig-nals. Two factors are mainly responsible forthis difference:

� The back surface of the testpiece is a metal-airinterface, which can be considered a totalreflector.Comparedtoametal-water interface,this results in an approximately 30% increasein back-reflection intensity at the receivingsearch unit for an aluminum testpiece coupledto the search unit through a layer of water.

� If a couplant whose acoustic impedancemore nearly matches that of the testpieceis substituted for the water, more energy istransmitted across the interface for boththe incident and returning beams. For most

applications, any couplant with an acousticimpedance higher than that of water is pre-ferred. Several of these are listed in thenonmetals group in Table 1. In addition tothe liquid couplants listed in Table 1, sev-eral semisolid or solid couplants (includingwallpaper paste, certain greases, honey,elastomers, and some adhesives) havehigher acoustic impedances than water.

Absorption of ultrasonic energy occursmainly by the conversion of mechanicalenergy into heat. Elastic motion within a sub-stance as an ultrasonic wave propagatesthrough it alternately heats the substance dur-ing compression and cools it during rarefac-tion. Because heat flows so much moreslowly than an ultrasonic wave, thermal lossesare incurred, and this progressively reducesenergy in the propagating wave. A related ther-mal loss occurs in polycrystalline materials; a

0 10 20 30 40 50 60 70 90°

50 60 70 90°403020100

1.0

0.9

0.8

r = 1.00 g/cm3

V1 = 1490 m/s LiquidSolid

a R

S

S

S

L

Lbt

bt

r = 7.7 g/cm3

V1 = 5900 m/sVt = 3230 m/s

Water

Steel

Ene

rgy

ratio

RR

0.2

0.1

00 5 10 15

Angle of incidence, a, deg20 25 30 35

10

Loss

, dB

7

13

20

βL1

β1

Fig. 12 Partition of acoustic energy at a water-steel interface. The reflectioncoefficient, R, is equal to 1 – (L + S), where L is the transmission

coefficient of the longitudinal wave and S is the transmission coefficient of thetransverse (or shear) wave.

Critical Angle, L

Crit

ical

Ang

le, S

25.6°

62.4

°

θJ θf

θTS

θTS

θr

0TL

0RN

0f

01

0.8

0.6

0.4

0.2

1020

30

40

50

60

70

80

90

1020

30

40

50

60

70

80

90

80

60

70

50

40

30

20100

1

0.8

0.8

Plexiglass

Aluminum

1

0.6

0.6

0.4

0.4

0.2

0.2 0

Longitudinal Shear

Fig. 13 Pressure ratios for refracted shear (transverse) wave for Plexiglas and steelinterface. Calculated by M. Baquera, Iowa State University

Table 3 Back-reflected intensity at metal-water interfaces

MaterialBack-reflection intensity,

% of incident beam intensity

Magnesium alloyM1A

11.0

Titanium 3.0Type 302 stainlesssteel

1.4

Carbon steel 1.3Inconel 0.7Tungsten 0.3


thermoelastic loss arises from heat flow awayfrom grains that have received more compres-sion or expansion in the course of wave motionthan did adjacent grains. For most polycrystal-line materials, this effect is most pronouncedat the low end of the ultrasonic frequencyspectrum (Ref 1).Vibrational stress in ferromagnetic and fer-

roelectric materials generated by the passageof an acoustic wave can cause motion ofdomain walls or rotation of domain directions.These effects may cause domains to bestrengthened in directions parallel, antiparallel,or perpendicular to the direction of stress.Energy losses in ferromagnetic and ferroelec-tric materials also may be caused by a micro-hysteresis effect, in which domain wallmotion or domain rotation lags behind thevibrational stress to produce a hysteresis loop.In addition to the types of losses discussed

previously, other types exist that have not beenaccounted for quantitatively. For example, ithas been suggested that some losses are causedby elastic-hysteresis effects due to cyclic dis-placements of dislocations in grains or grainboundaries of metals.Absorption can be thought of as a braking

action on the motion of oscillating particles.This braking action is more pronounced whenoscillations are more rapid, that is, at high fre-quencies. For most materials, absorption lossesincrease directly with frequency.Scattering of an ultrasonic wave occurs

because most materials are not homogeneous.Crystal discontinuities, such as grain bound-aries, twin boundaries, composite materials,and minute nonmetallic inclusions, deflectsmall amounts of ultrasonic energy out of themain ultrasonic beam. In addition, especiallyin mixed microstructures or anisotropic materi-als, mode conversion at crystallite boundariestends to occur because of slight differences inacoustic velocity and acoustic impedanceacross the boundaries.Scattering in metals is highly dependent on

the relation of crystallite size (mainly grainsize) to ultrasonic wavelength. When grainsize is less than 0.01 times the wavelength,scatter is negligible. Scattering effects varyapproximately with the third power of grainsize, and when the grain size is 0.1 times thewavelength or larger, excessive scatteringmay make it nearly impossible to conductvalid ultrasonic inspections.In some cases, determination of the degree

of scattering can be used as a basis for accep-tance or rejection of parts. Some cast ironscan be inspected for the size and distributionof graphite flakes. Similarly, the size and dis-tribution of grain size, microscopic voids insome powder metallurgy parts, or of strength-eners in some fiber-reinforced or dispersion-strengthened materials, can be evaluated bymeasuring attenuation (scattering) of an ultra-sonic beam (Ref 10–12). This topic is dis-cussed further in the articles “NondestructiveEvaluation of Pressed and Sintered Powder

Metallurgy Parts” and “Nondestructive Evalu-ation of Additively Manufactured MetallicParts” in this Volume. This assessment ofmaterial state (material state awareness) is see-ing increased attention as fabricators seek touse designed microstructures. It is also becom-ing increasingly necessary in addition tomacrodefect detection (cracks) to addressmicrostructure characterization and detect met-allurgical anomalies such as hard a inclusionsin titanium disks and local zones where thereare variations in either microstructure, suchas grain size, or elastic properties.The significance of the various mechanisms

for scattering and absorption vary with fre-quency. An example of the range of frequency-dependent absorption and scattering effects forthe case of a 0.6mmmean grain diameter alumi-num is illustrated with the frequency depen-dence of attenuation from 0.1 MHz to 1 GHz asshown in Fig. 14. The dominant scatteringeffects are seen to be closely related to the wave-length-grain size ratio (Ref 1, 13).Diffraction. An acoustic beam propagating

in a homogeneous medium is coherent; thatis, all particles that lie along any given planeparallel to the wave front vibrate in identicalpatterns. When a wave front passes the edgeof a reflecting surface, the front bends aroundthe edge in a manner similar to that in whichlight bends around the edge of an opaqueobject. When the reflector is very small com-pared to the ultrasound beam, as is usual fora small crack, pore, or an inclusion, wavebending (forward scattering) around the edgesof the reflector produces an interference pat-tern in a zone immediately behind the reflectorbecause of phase differences among differentportions of the forward-scattered beam. Theinterference pattern consists of alternateregions of maximum and minimum intensity

that correspond to regions where interferingscattered waves are respectively in phase andout of phase (Ref 1, 5, 14, 15).Diffraction phenomena must be taken into

account during the development of ultrasonicinspection procedures. Unfortunately, onlyqualitative guidelines can be provided. Entry-surface roughness, the type of machined sur-face, and machining direction all can influenceinspection procedures. In addition, the rough-ness of a flaw surface affects its echo patternand must be considered (Ref 16).An ultrasonic beam striking a smooth inter-

face is reflected and refracted; but the fieldmaintains phase coherence, and beam behaviorcan be predicted analytically. A rough inter-face, however, modifies boundary conditions,and some of the beam energy is diffracted.Beyond the interface, a coherent wave mustre-form through phase reinforcement and can-cellation. The wave then continues to propa-gate as a modified wave.The influence on the beam depends on the

roughness, size, and contour of the modifyinginterface. For example, a plane wave strikinga diaphragm containing a single hole onewavelength in diameter will propagate as aspherical wave from a point (Huygens)source. The wave from a larger hole will re-form in accordance with the number of wave-lengths in the diameter. In ultrasonic inspec-tion, a 2.5 mm (100 min.) surface finish mayhave little influence at one inspection fre-quency and search-unit diameter, defined interms of roughness to wavelength ratio, butmay completely mask subsurface discontinu-ities at other inspection frequencies orsearch-unit diameters.

Diffracted signals have seen increased inter-est in terms of their use for defect sizing. Thistopic is considered in the articles “Ultrasonic

Fig. 14 Attenuation of longitudinal waves in polycrystalline aluminum. Source: Ref 13. Reprinted by permission ofthe publisher Taylor & Francis Ltd


Imaging and Sizing” and “Sizing with Time-of-Flight Diffraction” in this Volume (Ref 17).Near- and Far-Field Effects. The face of a

single-element ultrasonic transducer piezo-electric crystal does not vibrate uniformlyunder the influence of an impressed electricalvoltage. Rather, the crystal face vibrates in acomplex manner that can most easily bedescribed as a mosaic of tiny, individual crys-tals, each vibrating in the same direction butslightly out of phase with its neighbors. Eachelement in the mosaic acts like a point (Huy-gens) source and radiates a spherical waveoutward from the plane of the crystal face.Near the face of the crystal, the compositeultrasonic beam propagates chiefly as a planewave, although spherical waves emanatingfrom the periphery of the crystal face produceshort-range ultrasonic beams referred to asside lobes. Because of interference effects,as these spherical waves encounter oneanother in the region near the crystal face, aspatial pattern of acoustic pressure maximaand minima is set up in the composite ultra-sound beam. The region in which these max-ima and minima occur is known as the near-field (Fresnel field) of the ultrasound beam(Ref 1, 5, 18).Along the central axis of the composite

ultrasound beam, the series of acoustic pres-sure maxima and minima becomes broaderand more widely spaced as the distance fromthe crystal face, d, increases. Where d becomesequal to N (with N denoting the length of thenear-field), the acoustic pressure reaches afinal maximum and then decreases approxi-mately exponentially with increasing distance.Both a graphical and pictorial representationof the near- and far-field acoustic pressuresare shown in Fig. 15. The length of the near-field is determined by the size of the radiatingcrystal and the wavelength, l, of the ultrasonicwave. For a circular radiator of diameter D, thelength of the near-field can be calculated from:

N ¼ D2 � l2� �

4l(Eq 7)

When the wavelength is small with respect tothe crystal diameter, the near-field length canbe approximated by:

N ¼ D2

4l¼ A

pl(Eq 8)

where A is the area of the crystal face.At distances greater than N, known as the

far-field of the ultrasonic beam, there are nointerference effects. At distances from N toapproximately 3N from the face of a circularradiator, there is a gradual transition to aspherical wave front. At distances of more thanapproximately 3N, the ultrasonic beam from arectangular radiator more closely resembles acylindrical wave, with the wave front beingcurved about an axis parallel to the longdimension of the rectangle.

Near- and far-field effects also occur whenultrasonic waves are reflected from interfaces.The reasons are similar to those for near- andfar-field effects for transducer crystals. Thatis, reflecting interfaces do not vibrate uni-formly in response to the acoustic pressure ofan impinging ultrasound wave. Near-fieldlengths for circular reflecting interfaces canbe calculated from Eq 7 and 8. Table 4 listsnear-field lengths corresponding to severalcombinations of radiator diameter and ultra-sonic frequency. The values in Table 4 werecalculated from Eq 7 for circular radiators ina material having an ultrasonic velocity of6000 m/s (~4 miles/s) and closely approximateactual lengths of near-fields for longitudinalwaves in steel, aluminum alloys, and certainother materials. Values for radiators withdiameters of 25, 13, and 10 mm (1, ½, and3/8 in.) correspond to typical search-unit sizes,and values for radiators with diameters of 3 and1.5 mm (1/8 and 0.060 in.) correspond to typical“hole sizes” in standard reference blocks.Beam Spreading. In the far-field of an

ultrasonic beam, the wave front expands withdistance from a radiator. The angle of diver-gence from the central axis of the beam froma circular radiator is determined from ultra-sonic wavelength and radiator size as:

g ¼ 2 sin�1 0:5lD

� �(Eq 9)

where g is the angle of divergence in degrees,l is the ultrasonic wavelength, and D is thediameter of a circular radiator. Equation 9 isvalid only for small values of l/D, that is, onlywhen the beam angle is small.When the radiator is not circular, the angle

of divergence cannot be assessed accuratelyby applying Eq 9. For noncircular search units,beam spreading is most accurately foundexperimentally.

Beam diameter also depends on the diame-ter of the radiator and the ultrasonic wave-length. The theoretical equation for –6 dBpulse-echo beam diameter is:

�6 dB beam diameter ¼ 1:032D

4� S (Eq 10)

Where S is the focusing factor and is � 1.Focusing the transducer (S > 1) produces asmaller beam. For a flat (nonfocused) trans-ducer (S = 1), the beam has a diameter of0.25 D at the near-field distance N, where Ndepends on the ultrasonic wavelength asdefined in Eq 7.The overall attenuation of an ultrasonic

wave in the far-field can be expressed as:

P ¼ P0 exp �aLð Þ (Eq 11)

Where P0 and P are the acoustic pressures atthe beginning and end, respectively, of a sec-tion of material having a length L and an atten-uation coefficient a. Attenuation coefficientsmost often are expressed in nepers per centi-meter or decibels per millimeter. Both nepersand decibels are units based on logarithms—nepers on natural logarithms (base e) anddecibels on common logarithms (base 10).Numerically, the value of a in decibels permillimeter (dB/mm) is equal to 0.868, thevalue in nepers per centimeter.A table of exact attenuation coefficients for

various materials, if such data could be deter-mined, would be of doubtful value. Ultrasonicinspection is a process subject to wide varia-tion in responses, and these variations arehighly dependent on structure and propertiesin each individual testpiece. Attenuationmainly determines the depth to which ultra-sonic inspection can be performed as well asthe signal amplitude from reflectors within atestpiece. Table 5 lists the types of materialsand approximate maximum inspection depth

Fig. 15 (a) Data curve showing variation of acoustic pressure with distance ratio for a circular search unit.Distance ratio is the distance from the crystal face, d, divided by the length of the near field, N.

(b) Image of a transducer pressure field, with Y1 associated with distance ratio of approximately ½ and Y0

associated with distance ratio of 1. Reprinted with permission of the American Society for Nondestructive Testing(ASNT) Inc. This reprint contains copyrighted property of ASNT and may not be duplicated or altered in any manner.


corresponding to low, medium, and high atten-uation coefficients. Inspection depth is alsoinfluenced by the decibel gain built into thereceiver-amplifier of an ultrasonic instrumentand by the ability of the instrument to discrim-inate between low-amplitude echoes and elec-tronic noise at high gain settings.

Common Implementations forUltrasonic Inspection

The two major methods that most commonlyimplement ultrasonic inspection with single-element transducers are the pulse-echo and thetransmission methods. The primary differencebetween these two methods is that the transmis-sion method involves two transducers andinvolves only the measurement of signal attenua-tion, while the pulse-echo method can be imple-mented with a single transducer and is used tomeasure both transit time and signal attenuation.The pulse-echo method also can be imple-

mented with a phased array, and it remainsone of the most widely used ultrasonic modal-ities, involving the detection of echoes pro-duced when an ultrasonic pulse is reflectedfrom a discontinuity or an interface in a test-piece. This method is used in flaw locationand thickness measurements. Flaw depth isdetermined from the time-of-flight betweenthe initial pulse and the echo produced by aflaw. Flaw depth also may be determined bythe relative transit time between the echo pro-duced by a flaw and the echo from the backsurface. Flaw sizes are estimated by comparingthe signal amplitudes of reflected ultrasoundfrom an interface (either within the testpieceor at the back surface) with the amplitude of

ultrasound reflected from a reference reflectorof known size or from the back surface of atestpiece having no flaws.The transmission method, either reflection

(pitch-catch) or through-transmission, involvesonly measurement of signal attenuation, andalso can be used in flaw detection. In thepulse-echo method, it is necessary that aninternal flaw reflect at least part of the ultra-sound energy onto a receiving transducer.However, echoes from flaws are not essentialto their detection. Merely the fact that theamplitude of the back-reflection from a test-piece is lower than that from an identicalworkpiece known to be free of flaws impliesthat the testpiece contains one or more flaws.The technique of detecting the presence offlaws by ultrasound attenuation is used intransmission methods as well as in the pulse-echo method. The main disadvantage of atten-uation methods is that flaw depth cannot bemeasured.The principles of each of these two inspec-

tion methods are discussed in the articles“Basic Inspection Methods (Pulse-Echo andTransmission Methods)” and “UltrasonicImaging and Sizing” in this Volume, whichalso present forms of data presentation, inter-pretation of data, and effects of operatingvariables, together with presentations of thevarious forms of implementation includinguse of phased arrays. Subsequent articles inthis Volume describe various components andsystems for ultrasonic inspection, referencestandards, and inspection procedures andapplications.The application of ultrasonic techniques also

involves other methods, such as acousticalmicroscopy, acoustical holography, the

frequency modulation technique, spectral analy-sis, and ultrasound conduction. The first of thesemethods is discussed in the article “AcousticMicroscopy” in this Volume. The second ofthese is covered in the article “Acoustical Holog-raphy” in this Volume. Three less commonlyusedmethods are briefly summarized as follows.The frequency modulation (FM) method,

which was the precursor of the pulse-echomethod, is another flaw-detection technique.In the FM method, the ultrasonic pulses aretransmitted in wave packets whose frequencyvaries linearly with time. The frequency varia-tion is repeated in successive wave packets sothat a plot of frequency versus time has asaw tooth pattern. There is a time delaybetween successive packets. Returning echoesare displayed on the readout device only ifthey have certain characteristics as determinedby the electronic circuitry in the instrument.Although not as widely used as the pulse-echomethod, the FM method has a lower signal-to-noise ratio and therefore somewhat greaterresolving power (Ref 19).Spectral analysis, which can be used with

both the through-transmission or pulse-echomethods, involves determination of the fre-quency spectrum of an ultrasonic wave afterit has propagated through a testpiece. The fre-quency spectrum can be determined either bytransmitting a pulse and using a fast Fouriertransform (FFT) to obtain the frequency spec-trum of the received signal or by sweepingthe transmission frequency in real-time andacquiring the response at each frequency. Theincreasing use of the pulse method is attributedto improvements in the speed of digital signalprocessing including using a FFT.Spectral analysis is used in transducer eva-

luations and may be useful in defect character-ization. However, because the spectralsignatures of defects are influenced by severalother factors (such as the spectrum of the inputpulse, coupling details, and signal attenuation),defect characterization primarily involves thequalitative interpretation of echoes in the timedomain.Spectral analysis also can be used to mea-

sure the thickness of thin-walled specimens.A short pulse of ultrasound is a form of coher-ent radiation; in a thin-walled specimen thatproduces front- and back-wall echoes, the tworeflected pulses show phase differences andcan interfere coherently. If the pulse containsa wide band of frequencies, interference max-ima and minima can occur at particular fre-quencies, and these can be related to thespecimen thickness. These approaches havebeen known by various terms including ultra-sonic spectroscopy (Ref 20, 21).Ultrasound conduction is used in flaw

detection by monitoring the intensity of arbi-trary waveforms at a given point on the test-piece. These waveforms transmit ultrasonicenergy, which is fed into the testpiece at someother point without the existence of a well-defined beam path between the two points.

Table 4 Near-field lengths for circular radiators in a material having an ultrasonicvelocity of 6000 m/s (4 miles/s)

Frequency, MHz

Wavelength

Near-field length for radiator with diameter of:

25 mm (1 in.) 13 mm (1

/

2 in.) 9.5 mm (3/8 in.) 3.2 mm (1/8 in.) 1.5 mm (0.060 in.)

mm in. cm in. cm in. cm in. cm in. cm in.

1.0 6.0 0.24 2.5 1.0 0.52 0.20 0.23 0.09 . . . . . . . . . . . .

2.0 3.0 0.12 5.3 2.1 1.3 0.50 0.68 0.27 0.009 0.0035 . . . . . .

5.0 1.2 0.04 13.4 5.3 3.3 1.3 1.9 0.75 0.18 0.07 0.02 0.00810.0 0.6 0.02 27 11 6.7 2.6 3.8 1.5 0.40 0.16 0.08 0.0315.0 0.4 0.015 40 16 10 4.0 5.7 2.2 0.62 0.24 0.14 0.05525.0 0.24 0.009 67 26 17 6.7 9.4 3.7 1.04 0.41 0.24 0.095

Table 5 Approximate attenuation coefficients and useful depths of inspection for variousmetallic and nonmetallic materials, using 2 MHz longitudinal waves at room temperature

Attenuation coefficient,dB/mm (dB/in.)

Useful depth ofinspection, m (ft) Type of material inspected

Low: 0.001–0.01(0.025–0.25)

1–10 (3–30) Cast metals: aluminum(a), magnesium(a). Wrought metals: steel, aluminum,magnesium, nickel, titanium, tungsten, uranium

Medium: 0.01–0.1(0.25–2.5)

0.1–1 (0.3–3) Cast metals(b): steel(c), high-strength cast iron, aluminum(d), magnesium(d).Wrought metals(b): copper, lead, zinc. Nonmetals: sintered carbides(b),some plastics(e), some rubbers(e)

High: >0.1 (>2.5) 0–0.1 (0–0.3)(f) Cast metals(b): steel(d), low-strength cast iron, copper, zinc. Nonmetals(e):porous ceramics, filled plastics, some rubbers

(a) Pure or slightly alloyed. (b) Attenuation mostly by scattering. (c) Plain carbon or slightly alloyed. (d) Highly alloyed. (e) Attenuation mostly byabsorption. (f) Excessive attenuation may preclude inspection.


This method now has evolved into at least twofamilies of measurement methods: The first isresonant ultrasound spectroscopy (RUS), ofwhich there are now commercial units beingsold and used to assess product quality(Ref 22), and diffuse field ultrasonics, whichis seeing use in assessing aging and degrada-tion in concrete structures and remains the sub-ject of interest as a structural health monitoring(SHM) approach (Ref 23). This topic is dis-cussed in detail in the article “NondestructiveEvaluation and Life Assessment” in thisVolume.

REFERENCES

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3. P.C. Miedlar, A.P. Berens, A. Gunderson,and J.P. Gallagher, “Analysis and SupportInitiative for Structural Technology(ASIST); Delivery Order 0016: USAFDamage Tolerant Design Handbook:Guidelines for the Analysis and Designof Damage Tolerant Aircraft Structures,”AFRL-VA-WP-TR-2003-3002, 2003

4. M. Farley, 40 years of progress in NDT –History as a Guide to the Future, Proc.40th Annual Review of Progress in

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6. I.A. Viktorov, Rayleigh and Lamb Waves,Springer, 1967

7. J.L. Rose, Ultrasonic Waves in SolidMedia, Cambridge University Press,1999

8. M. Redwood, Mechanical Waveguides,Pergamon Press, 1960

9. L.C. Lynnworth, Ultrasonic ImpedanceMatching from Solids to Gases, IEEE T.Son. Ultrason., Vol 12 (No. 2), June1965, p 37–48

10. K. Goebbles, Materials Characterizationfor Process Control and Product Confor-mity, CRC Press, 1993

11. F. Margetan, Bruce Thompson: Adventuresand Advances in Ultrasonic Backscatter,Review of Progress in Quantitative Nonde-structive Evaluation, Vol 31A, AIP Conf.Proc., Vol 1430, 2012, p 54–82

12. E.P. Papadakis, Scattering in Polycrystal-line Media, Methods in ExperimentalPhysics, Vol 19, Academic Press, 1981,p 237–298

13. R.T. Smith and R.W.B. Stephens, Effectsof Anisotropy on Ultrasonic Propagationin Solids, Progress in Applied MaterialsResearch, E.G. Stanford, J.H. Fearon, andW.J. McGonnagle, Ed., Vol 5, Gordonand Breach, London, 1964, p 39–64

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17. J.P. Charlesworth and J.A.G. Temple, Engi-neering Applications of Ultrasonic Time-of-Flight Diffraction, 2nd ed., Research Stud-ies Press, Baldock, UK, 2001

18. M.G. Silk, Ultrasonic Transducers forNondestructive Testing, Adam Hilger/IOP, 1984

19. F. Lam and J. Szilard, Pulse CompressionTechniques in Ultrasonic NondestructiveTesting, Ultrasonics, Vol 14 (No. 3),1976, p 111–114

20. D.W. Fitting and L. Adler, UltrasonicSpectral Analysis for Nondestructive Eval-uation, Plenum Press, 1981

21. A.F. Brown, Ultrasonic Spectroscopy,Ultrasonic Testing, J. Szilard, Ed., Wiley,1982, p 167–215

22. A. Migliori and J.L. Sarrao, ResonantUltrasound Spectroscopy, Wiley, 1997

23. R.L.Weaver andO.I. Lobkis, Diffuse Fieldsin Ultrasonics and Seismology,Geophysics,Vol 71 (No. 4), 2006, p S15–S19


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