DiractionatUltrasonicWavesFebruary7,2005DiractionatUltrasonicWavesContents1 Introduction 32 ProductionofUltrasound 33 PropagationofSoundWaves 83.1 Solution of the Wave Equation for Periodical Excitation . . . 114 DiractionofLightatUltrasoundWaves 125 Intensitydistributionoftheinterferencepattern 166 Apparatus 226.1 Diraction from Sonic-Waves . . . . . . . . . . . . . . . . . . 226.2 The Striae method . . . . . . . . . . . . . . . . . . . . . . . . 227 Problems 237.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.2 General Problems . . . . . . . . . . . . . . . . . . . . . . . . . 232DiractionatUltrasonicWaves1 IntroductionThetermultrasound refers tomechanical oscillations, whosefrequenciesvary between 16 kHz, the upper audio limit of the human ear, and 1010Hz.A human ear can only hear sound in the range of approx. 16 to 16,000 Hz.Oscillationsintheregionbelow16Hzarereferredtoasinfrasound(e.g.earthquakes)andoscillationswithfrequenciesabove1010Hzarereferredtoashyper-sound. Thisupperfrequency-limitationfortheultrasoundisdeterminedbytheatomiccongurationofmatter. Thewavelengthoftheultrasoundinthisregionisintheorderofasmall multipleofthelatticeconstant a ( 103a); inthenext regionof hyper-sound, inwhichallthethermal movements of atoms andmolecules occur, strongquantum-mechanical eects have to be taken into account (phonon1theory). Belowtheseveryhighfrequencies of 1010Hzthematerial canbetreatedas acontinuumi.e. thelawsofclassical acousticsapplytoit, whichoriginallyonly dealt with the problem of the sound within the hearing range.Therearefewphysical applicationsforultrasound, themostimportantlythedeterminationofelasticityconstantsfrommeasurementsofthespeedof sound. Frommeasurements of thespeedof soundandabsorbability,thestructural natureofthematerial canbedeterminedwithinthescopeofmicroscopictheories. Todaytheapplicationsofultrasoundhavegrownoutofthenarrowareaofphysics; tobementionedaboveall isthesonar(found mainly on ships), the non destructive testing of material and medicaldiagnosticsinthehumanbody. Anultrasoundmicroscopeisalsounderdevelopment nowadays. Apart from these passive applications there is alsoanumber of activeapplications wheretheoscillationenergyis usedforperforming work processese.g. cleaning (ultrasonic baths), welding plasticsand treatment of ceramic materials.2 ProductionofUltrasoundThe oscillating systems used for the production of ultrasound waves shouldbe suitable for working with high frequencies. This means that all oscillatingsystems with spring and mass separated, which are used for the productionof sound in the hearing range, can not be used to produce ultrasound sincewecannotincreasetheireigenfrequenciesaboveacertainvalue. Instead,1soundparticles3DiractionatUltrasonicWavesFrequencyWave lengthA few inter-atomicintervals10-110111010109108107106105104103102101100InfrasoundAcousticrangeUltrasoundHypersoundIn solid s m c / 4000 m cm 1 . 0 25 In air s m c / 330 =mm m 20 20 Earthquake wavesm 100 For example: the wave lengthof green light is m 5 . 0 Figure 1: Sonic regions and typical wave lengthintheultrasoundrange, continuawhichareabletooscillateareusede.g.cavitieslledwithgasorliquidandsolidbodiesintheformof platesorbars. Inthesesystemstheelasticityofthematerial playstheroleofthespring, and the density together with the geometric properties play the roleofthemassi.e.thespringandmassaredistributedcontinuouslyovertheoscillator.Onecansimulateultrasonicoscillationwithfrequenciesuptoabout100kHzwithpurelymechanicoscillators. Examplesofmechanical oscillatorsare gas and liquid lled whistles, which work on the same principle as a setofwindinstruments. Fortheproductionofnon-sinusoidaloscillationsthehole-siren can be used.4DiractionatUltrasonicWavesOf far grater importance than mechanical oscillators are the electromechan-ical oscillators. Asthenameimplieselectricenergyisconvertedintome-chanical oscillation energy. In this group we can ndPiezoelectric converters.Magnetostrictive converters.Electrodynamic converters.Electrostatic converters.For the production of higher frequencies magnetostrictive and piezoelectricconverters are the most important.Magnetostrictiveconverters operate according to the Magnetostrictioneect. IfabarfromaFerromagneticmaterial,mostlyfromNi,ismagne-tized, itslengthlwillvaryslightlybyl, becausethemagneticmomentisalignedinthedirectionoftheeldandthusaectsthedeformationofthe lattice in the crystal. For aN-bar the relatively big changellamountsto 2.5 105inamagneticeldof 1Tesla(=14 105m/A). ApplyingoutsideforcestoanalreadymagnetizedN-barwillchangethemagnetiza-tionofthebar, whichinturnwillcauseavoltagesurgeinthebar. Thisvoltagesurgecanbemeasuredwhenaninductioncoiliswrappedaroundthe bar. Ultrasound can thus beproduced and measured using thesecon-vertersandgenerators. Magnetostrictiveoscillatorsaremostlysuitabletothe production of intense sound levels, up to 200 kHz.The piezoelectric converter is todays most frequently used sonic gener-atoranddetector. Comparedtothepreviouslydiscussedtechnologies,farhigher frequencies can be achieved (in the MHz range).The piezoelectric (or pressure-electric) eect was discovered in 1880 by theCurie brothers. With some crystals,when subjected to pressure or tensilestress in special crystallographic directions, electrical charges are realest oncertain crystal surfaces. The produced charges are proportional to the pres-sure or the stretch applied. The sign of the charges changes if for examplea compression alters into a dilation.The reversed piezoelectric eect was detected soon after in 1881. The samegroup of crystals , when put between two electrodes with a potential dier-ence, reacts by deformation. The direct piezoelectric eect is used for detec-5DiractionatUltrasonicWavestion of ultrasound waves and the reversed piezoelectric eect (Electrostric-tion) is used for their production. When placing an alternating voltage ontwo condenser plates, between which the crystal is located, the crystal willoscillate according to the frequency of the alternating voltage. The lengthvariationof thecrystal isproportional tothepiezoelectricmoduledandtheelectrical tensionputonit. Becausepiezoelectriccrystalsarealwaysanisotropic, d is a tensor and l also depends on the direction of the appliedelectric eld relative to the crystal axes. Putting a eld in parallel to a mainaxis we obtainl = djjU1(for quartz at low frequenciesd11 = 2.3 1012m/V ).For the receptor, the situation is analogous. The amplitude of the producedchange of pressure (by the sonic wave) is proportional to the tension on thecondenserplates. However, theproportionalityconstantsaredierentinthis case. The received voltageU2 isU2 = hii lwhereh is the deformation constant (for quartz with low frequenciesh11 =4.9 109V/m). For the same length variation, the voltagesU1andU2thusdier. The proportionU2U1for identical l is described by the square of thecoupling coecientkkii = _dii hiiIn general the coupling coecient at low frequencies is smaller than 1. How-ever, with resonance support and low damping the coupling coecient canbecome nearly 1.All crystals that show the piezoelectric eect have several similar qualities,theyisolatewellandhaveoneormorepolaraxes. A180turnofapolaraxis does not result in the same state. The piezoelectric eect now appearsin the directions of the polar axes. Crystals on which the piezoelectric eectcanbeobservedareforexample: lithiumsulfate,tourmaline,zincblende,Seignette-salt and tartaric acid.In a quartz crystal, which we want to study as an example, we have threepolar axes. Quartz has the chemical formulaSiO2and it forms hexagonalcrystals. EverySi-atom has four positive elementary charges and everyO-atom two negative elementary charges. Figure 2(a) shows a structural cellof quartz.6DiractionatUltrasonicWaves(a)(b) Si +O -O -O -Si +Si +X 1 X 3 X 2 ----+ -- +++++ Si +O -O - O -Si + Si +Staggerd point ofthe + and -charges Dipol momentFigure 2: A structural cell of quartzX1, X2 and X3 are the polar axes. If we now apply, for instance, pressure indirection of the X1 axis , we reach the situation depicted in gure 2(b) wherewe see the resulting shift of the atoms and charges on the surface. Qualita-tively the same shift can be achieved, if a thrust is applied perpendicularlyto theX1 axis (transverse piezoelectric eect).Apartfromthisgroupofpiezoelectricsinglecrystalsthereisalsoaseriesof ferroelectricsubstances. Withthesessubstancesthedipolemomentisnot only produced by applying pressure or tension, but the electric dipolesalready exist within the crystal unit cell, similar to the magnetic momentsexistine.g. Fe. Forthisreasonferroelectricmaterialshaveaveryhighdielectric constant. Applying an electric eld at a high temperature alignsthese electrical dipole moments in the same way that a magnetic eld alignsthemagneticdipolemomentsinFe. Whenaferromagneticmaterial inan electric eld is cooled down below a certain temperature, known as theCurrie-temperature, the aligned electric dipole moments freeze. The out-come is a permanent macroscopic electric dipole. This modication remainstoalargeextent, aslongasthetemperatureofthesampledoesnotriseabove the Curie-temperature. Above the curie temperature the polarizationdisappearsirreversibly, i.e. thesamplemustbepolarizedagainusinganelectric eld.In contrast totheexample of aquartz piezo-crystal,ferromagnetic sound-converters are not necessarily single crystals. Poly-crystaline materials aresucient, which can be produced in an inexpensive way and in various forms7DiractionatUltrasonicWaves(plates, tubes, hollowspheres)bymeansofsintering. Examplesforthesematerialsare: leadzirconatetitanate(PZT), bariumtitanate, leadmetaniobium and lithium niobium.In this experiment a PZT oscillator is used. PZT is a mixture of PbZrO3 andPbTiO3. Thepiezoelectricconstantdependsonthemixingproportionofthe two materials and can vary within certain limits. The Curie-temperaturefor the PZT is about 250C.3 PropagationofSoundWavesIfpressureortension2isappliedtoamechanicalcontinuume.g.abaroflengthl and a cross sectionFit changes its shape ( by l and F).For a small perturbation l from the rest position we can use Hookes lawto describe the deformationllll=1E KFwhereEis the elasticity module,Kthe applied force andKFis the tension.ForliquidsorgasesthecompressibilityH isusedinsteadof1E. Insys-tems with static tension the isotermic compressibility is dominating, whilein high-frequency dynamic-tension systems the adiabatic compressibility isdominating,since there is not enough timefor the heat exchange betweenthe system and its surroundings.K) (1x K ) (2x Kwithoutstrainwith strain) (2x KRx xx x2=x1+xFigure 3:2inthedirectionofthelength8DiractionatUltrasonicWavesLets now look at thedynamic equilibriumof a masselement dFdx in aone dimentional bar or liquid, (x, t) represents a deection at location x attimet.A forceK(x1) applied the locationx1 therefore causes a length variation of(x1) (x2) in the volume element according to(x1 + x) (x1)x=xx=x1=1E K(x1)dF(1)In an equilibrium, a static (= time independent) forceK(x1) must be bal-ancedbyareactionforceKR(x2) = K(x2)i.e. K(x1) K(x2) = 0. Onthe other hand, with dynamic forces we must take the inertia of the volumeelementT= dFdx2t2into consideration.K(x1) K(x2) =Kx dx = 2t2 dFdx (2)By dierentiating (1) and placing the result in (2) we obtain2t2 c2s2x2= 0, cs =E=_1Had (3)f x x 0 x 1 c s t 1 t 0 =0 t 1 >0 c s Figure 4: f(x1, t1) = f(x0, t0) = f(x1cst1)This wave equation is valid for sound waves with small amplitudes(x, t).The most general solution of this equation has the form9DiractionatUltrasonicWaves(x, t) = f(x cst) +g(x +cst) (4)Herefandg represent elastic deviations of an arbitrary form, which prop-agate at the speed of soundcs. The wavefpropagates in the direction ofthe positivex axis whileg propagates along the direction of the negativexaxis , see gure 4.Accordingtoour assumptions waveequation(3) applies tolongitudinalwaves( cs)inliquidsorgas. Transversewaves( cs)ontheotherhand, cannotbecarriedforwardbyliquidsorgas. Theequationappliesalso to expansion waves in thin bars ( = diameter of the bar). Theseexpansion waves (Fig.5) are formed due to the lateral contraction during thelength variation of the bar.Longitudinal waveExpansion waveTransverse waveFigure 5: Dierent kinds of wavesExpansion waves are actually a mixture (linear combination) of longitudinalandtransversal waves. Whenusinganinnitelyextendedmedium, theselateral contractions cannot be formed. In this case more complicated elasto-dynamics equations are required instead of the simple wave equation. Then,the speed of light depends on the lateral contraction coecient for both lon-gitudinal and transverse waves .The wave velocity, in media with geometricaldimensions of the order of a wavelength, depend on these dimensions or tothe wave length respectively.10DiractionatUltrasonicWaves3.1 SolutionoftheWaveEquationforPeriodicalExcitationSince we try to excite the piezoelectric converter using sines oscillations offrequency , we expect the liquid in which the crystal is placed to have alsoperiodical waves. Hence we make the ansatz(x, t) = 1ei(tKx)+2ei(t+Kx)(5)whereK =cs=2 =wave number.1is the amplitude of the outgoing wave from the converter and2is theamplitudeofthewavereectedfroml (seegure6).12isthereectionfactor.x = 0 x = l xAei tFigure 6: standing waveThe source of the piezoelectric excitation is located atx = 0, thus(0, t) = Aeit(6)We will assume total reection from the barrier at locationl, this would bethe case for an innitely sti barrier.hence (l, t) = 0 (7)11DiractionatUltrasonicWavesFromthetwoboundaryconditions aboveandequation(5) weobtainastanding wave(x, t) = A sin(K(l x))sin(Kl) eit(8)If we would have no reection at all,i.e. perfect transmission or absorptionatx =l,wewouldget2= 0andwouldhaveonlytravelingwaves. Therealityinourexperimentissomewhereinthebetween. Consequentlyweshould formulate the boundary conditions dierently. If 1(x, t) is a wave ina adjacent medium, we require that atx =lthe deections and forces areequal.so (l, t) = 1(l, t), Ex= E11x(9)4 DiractionofLightatUltrasoundWavesIn1932DebyeandSearsdiscoveredintheUSAandLucasandBiquarddiscoveredinFrancethattransparentmediadiractlightwhenanultra-sound wave is sent trough them. This eect is a consequence of a periodicalvariationoftherefractiveindex,whichinturnisaconsequenceofalocalperiodical pressure change caused by the ultrasound wave. Figure 7 showstheexperimentalsetupwhichallowstheobservationofdiractionoflightby an ultrasonic wave.A thin slit,lit up by the lampLa serves as a source of light. The lensL1is placed in the distance of its focal length from the gap and thus producesabroadbeamof parallel light. Thelightthenpenetratesatransparentmedium (gas, liquid or solid) in which an ultrasonic wave transducerQ, lo-cated perpendicularly to the direction of the incidence light beam, produceselastic waves. For experiments with liquids or gases a container with plane-parallel glass walls is required. The second lensL2 projects a real imageS

of the gap S on a screen. If the ultrasound wave is excited, several orders ofthe spectrum of the lamp La can be seen on both sides of S

. By introducinga lterFinto the path of the light beam we produce monochromatic lightand obtain only one interference strip of each order nearS

.To understand this phenomenon we must assume that the local periodical12DiractionatUltrasonicWavesQ LaLSL 1 L 2 Figure 7: The experimental setup which allowed the observation of dirac-tion of light by an ultrasonic wave.pressurechangesoftheelasticwavecreatelocal changesintherefractionindexof themedium. Thesurfacesof equal phaseS(x, y, z)of thelightwavearethennolongerplains(S(x, y, z)=k0xforlightpropagationinthex-direction.) but they become a sine function with the same period asthe ultrasonic wave. This arises from the fact that in an area with a higherrefractive indexn the light wave travel slower than in an area with a lowerrefractive index (c =c0n ).When we describe an elastic wave by the local density(y, t) instead of bythe deection(y, t) we can easily convince our selfs that we obtain(y, t) = 0 +0(y, t)y(10)For the refractive indexn(y, t) we expect accordinglyn(y, t) = n0 + n(y, t) (11)The relation between the refractive index, the dielectric constant and thedensityisdescribedbythewell-knownClausius-Mossottiequationfromthe theory of the dielectric constants.1 1 + 2=130NLM = const. (12)13DiractionatUltrasonicWavesWhereNListheLoschmidtnumber, Mthemolecularweightandthepolarisability of the atoms or molecules. The polarisability can be regardedas independent of density. At optical frequencies in a non-magnetic medium( = 1) we have = n2. We therefore obtain from equation (12) =3( + 2)( 1)(13)combining (10) and (11) with = 2n n we ndn(y, t) =(n20 + 2)(n201)6n0(y, t)y(14)The light waveE(x, y, z, t) (or the eld vectorsD,HandB) obeys a waveequation in the same way the ultrasonic wave does. The wave equations forlight can be obtained from Maxwells equations,E = Bt(15)H =Dt(16) D = 0 (17)B = 0H (18)D = 0E (19)The exact wave equation for an inhomogeneous medium =(r, t) is verycomplicated. For a homogeneous medium =const. we can simply obtainthewaveequationbyderivingequation(16) bytimeandbyusingalsoequations (18) and (15).14DiractionatUltrasonicWaves2

Dt2= Ht= _10

Bt_ =10_

E_ = 10_

E (

E)_2

Et2100

E = 0. (20)Analogouslytothesoundwavetterm(00)1/2isequivalenttothepropagationspeedof thephasec. Thusinvacuumc0=100andinamediumc =c0n(n =). The abbreviation

Estands for {E1, E2, E3}.The separation anzatz

E = u(x, y, z)eitleads to a time independent waveequation.u +k20n2u = 0 (21)Whereko =c0=20. Simple solutions of this equation are,e.g. plain wavesforn = const.

E(x, y, z, t) =

Aeitkxk = k0nwhere the modulus of

A corresponds to the amplitude and the direction tothe polarization of the light beam.From analogy to the plain wave we obtainuj = Aj(x, y, z)eik0S(x,y,z)(22)From placing this result in the time independent wave equation (21) we see,that in the extreme case of geometrical opticsi.e.k0 , the equation issatised if_Sx_2+_Sy_2+_Sz_2= n2(23)15DiractionatUltrasonicWaves1AjAj S = 12 S (24)S(x, y, z) is called the eikonal. Surfaces that obey the equationS(x, y, z) =const. are constant phase surfaces of the light wave and grad(S) then indicatethe direction of the light wave.Undertheconditionthatthetemporalandlocalchangesoftherefractiveindexn(y, t) are small compared tothe temporal and local changes of thesourceof light, i.e. soundlightandsoundlightwecanuseanapproximationofaslowlyvaryingfunctionn(y, t)insteadoftheconstantrefractive indexExact solutions of theeikonal equation(23) for our problemn=n0 +n(y, t) might still be very dicult to calculate. However, the equation tellsushowonecanobtainanimageofthesurfacesofconstantphaseusingasimple graphical method. If we have one known surface of a constant phase(e.g. boundarycondition)wecanuseittoconstructagroupof parallelsurfaceswithconstantinnitesimalgaps |S|=n(y). Theconstructioncorresponds exactlytoHuygens principlewhereeachpoint of thewaveeldisastartingpointofaspherical wave. Drawsuchplanes, orcurvesof constantphaseforanultrasoundwave, foraxedpointintime, i.e.n = n0 + n0 sin(K) with n0< n0.5 Intensitydistributionof theinterferencepat-ternTo calculate the interference pattern on the screen we need to apply Huy-gens principle once more. We consider every point on the exit plainx = aaftertheultrasoundeldasastartingpointof aspherical wavewithanamplitude and phase of the light wave in this point, see gure 8. Since theexact solution of the eikonalSin the domain of the ultrasonic wave is verydicult, we would like to use an approximation in which the light rays passinparallelthroughtheultrasoundeldandaremodulatedonlyinthelo-cal and temporal phase. This corresponds to a linear approximation of theeikonal and amplitude.S = n(y, t) x and A = A0(25)16DiractionatUltrasonicWavesI III IIyx2l 2l Rr yx=0 x=aFigure 8: DiractionIn this approximation all the polarization directions are identical, so we canabandon the vector notation.In region I we have parallel light soEI = A0ei(tk0x)(26)in region II, from our assumptionEII = A0ei(tk0n(y,t)x)(27)weconsiderthewaveinregionIII asasuperpositionof spherical wavesEIIIEIII = EII 1Rei(tk0R) y (28)For a bigR or observations around small angles we haveR= r y sin() (29)17DiractionatUltrasonicWavesWe assume that the ultrasonic wave propagates in they-direction, thusn(y, t) = n0 + n0 sin(t Ky) (30)Together with (27), (28) and (29) we getEIII =A0rei(tk0n0ak0r)_+ l2l2ei(k0n0a sin(tKy)k0y sin()dyWecanexactlyexpandtheeisinfunctionintheintegrandintoaFourierseries. We knoweia sin()==+

=J(a)ei. (31)JaretheBessel functionsof integerorder. Afterexchangebetweentheintegral and the sum, the integral turns into=+

=_ l2l2(1)J(k0na)eitei(k0 sin()K)ydyThe integral is now simple to calculate and we obtain the following for thelight wave of region IIIEIII=A0lrei(kon0a+kr)=+

=(1)J(k0n0a) sin[l2(k0 sin() K)]l2(k0 sin() K)ei(+)t(32)Theintensityisproportionaltothetemporalaverageofthesquareoftheamplitude or in complex notationI =lim1_0(E E)dt (33)18DiractionatUltrasonicWavesCalculating the last equation with EIII, we obtain a double sum of the formI =lim

FF 1_0ei()tdt =

FFThe intensity is accordinglyI =A20l2r2+

=J2(k0n0a) _sin[l2(k0 sin() K)]l2(k0 sin() K)_2(34)Thefunctionsin2xx2(seegure9)ismaximal forx=0anddropsrapidlyfrom each side of x = 0. Thus we have strong intensity (interference fringes)around the anglesin our wave eld (r l).k0 sin K = 0 (35)For small angles we obtain = kk0= 0(36)Whereindicates the order of the diraction spectrum.0 1 sin 2 x x 2 +3 x +23+2 x -0.1 0.1 0.3 0.5 0.7 0.9 1.1 012345 J 2 2 J 2 (x) J 0 2 J 1 2 x Figure 9:We nd that the distance of the interference fringes from each other is thesameaswithadiractiongratingwithalatticeconstantofthelengthofthe ultrasonic wave.19DiractionatUltrasonicWavesThe intensity of the interference fringes of the-th order is proportional tothe square of the Bessel functionJI J2(k0n0a) (37)An appropriate choice of light frequency (k0), width a or intensity (n0) ofthe ultrasound wave can thus cause the extenuation of specic ordersNext we apply the Bessel functions sum rule+

=J2(x) = 1thismeansthatthesumofintensitiesoverallordersisaconstant, whichcorresponds to the intensity of the incident light.Anotherspecialfeatureresultingfromthemovementofourphasegratingcan be explained by wave equation (32) forEIII. The light observed in the-th order does not have the original frequencybut is shifted by-timesthe frequency of the ultrasonic wave, see gure 10. = +ultrasound light + 2 2 + = 2 = 2 = 1 = 0 = 1Figure 10:We can explain this frequency shift with the Doppler eect or with quantummechanics in an easy manner. In quantum mechanic the sound- and light-(electromagnetic-) elds are quantized, i.e. the energy of a eld of frequencycan only change between discrete values (). A sonic-eld of frequency20DiractionatUltrasonicWavescanthus bedescribedthroughquasi-particles (theso-calledphonons)whichhasanenergyof E= perparticle. Thediscreteenergychangeimpliesnowthatwecaneitherproduceorannihilate(emitorabsorb)anentire particle with an energy of in the sonic-eld. The same explanationapplies for the electromagnetic eld as well. In the later case the particlesarecalledphotons andhavetheenergy. Themomentum pof thesemass-less particles is p =

k (38)If we look at the diraction of light from the ultrasonic wave as scatteringof photons by phonons, the particles must fulll the energy and momentumconservationlaws. Hereweshouldpayattention, though, that for eachphoton only whole numbers () of phonons can be absorbed (+) or emitted(-).

k1 =

k0 + K momentum conservation (39)1 = 0 + energy conservation (40)k 1 k 0 k Figure 11:From the momentum conservation law we attain|k1|2+|k0|22|k0||k1| cos = 2K2(41)|k1| =1c0n=20n=wavenumber of theincident light inaliquidwithrefractive indexn. (0 undc0 are values in a vacuum)Here , whichmeans |k1| = |k0| andthusfromequation(41)weobtain21DiractionatUltrasonicWaves2 sin 2=Kk1 = 0nTheangleistheanglebetweenincidentandscatteredphotonsintheliquid. For perpendicular incidence of the photons on the liquid we observethe scattered photons in air,nL = 1, at the angle(refraction law).sinsin== nso = 0In the-th order of the diraction spectrum we see now the photons thatabsorbedphonons, oremittedphononsfor f4Figure 12: Striae method7 Problems7.1 Experimentala) Photographthediractionpatternof dierent ultrasonicfrequencies.Calculate the wave length of the elastic wave from the distance of the inter-ference stripes and determine the sound propagation velocity in xylene.b)Photographtheultrasonicwaveusingthestriaemethodandcalculatethe sound propagation velocity.7.2 GeneralProblems1. Calculatetheultrasonicwave (x, t) inliquidfor partial reectionand show, with the help of Eq. (9), that the reection factor for theamplitude on the boundary layer between two media, 1 and 2, is givenbyR =1c12c21c1 +2c2(where is density andc is the sound velocity) .2. Howbigisthemaximal change(=amplitude)n0oftherefractionindex of a traveling sine wave if the ultrasonic converter emits a power-density ofW23DiractionatUltrasonicWavesnxylene= 1.497 = 0.86grcm3W = 1Wattcm2Hint: calculatethepower-densityof thesonicwaveasafunctionofthe amplitudeA0(y, t) = A0 sin(t Ky)3. Use Huygens principle to draw surfaces of constant phase of the lightwave as it passes through the liquid (at a certain time,t = const).4. Which frequencies can be seen in the dierent orders of diraction forstanding waves?Hint: Treatthestandingwavesastwocounterpropagatingtravelingwavesandapplythesummarizedresultsfrom gure10. Assumethatthe diraction occurs rst at one of the counter propagating waves andthen after at the other one.5. An appropriate choice of light frequency or ultrasound intensity can,inthecaseoftravelingwaves, causealossofthezerothorderfromthe screen. Is this also possible for standing waves?24