Electronic supplement to Student Guide: Making Waves by Visualizing Fourier Transformation. Introduction

Transcription

1 Electroic supplemet to Studet Guide: Makig Waves by Visualizig Fourier Trasformatio By Robert Smalley, Jr. Itroductio The Fourier trasform itegrad visualizatio (FTIV) method itroduced here is based o a simple graphical presetatio of the Fourier trasform itegrad as a topographic surface. FTIV is useful for uderstadig the importat but ofte uderappreciated cotributio of the frequecy-domai represetatio s phase spectrum to the shape of the fuctio i the real-space. As the frequecy-domai is the atural space for describig waves, this approach ca also be used to develop isights ito complex wave behavior, such as the relatioship betwee travelig ad stadig waves ad the evolutio of the shape of a dispersig wave-trai. Figures -4 are the same as the figures i the prited versio. Figure 5 i this e- supplemet is a modified versio of Figure 5 i the prited versio. Figures 6 ad 7 i this e-supplemet are i additio to those i the prited versio. The equatios are umbered such that the whole umber equatios i red are the same as i the prited versio. Additioal equatios that are oly i this e-supplemet are umbered with decimal values that cout equatios betwee those i the prited versio. Backgroud Itroductory physics ad egieerig textbooks, which typically ecourage visualizatio, ofte start their discussio of Fourier aalysis with a figure showig how oe ca approximate a fuctio i some real-space by addig together siusoids i a appropriate frequecy-domai (Figure (A)). The itroductio of Fourier aalysis is also oftetimes associated with solutios to differetial equatios; especially the wave equatio for which siusoids are also solutios. I this case, a relatioship betwee the Fourier represetatio, which otherwise seems arbitrary, ad the physics of the problem is established, allowig oe to relate the two ad provide isights ito the physics. We will use examples with travelig ad stadig waves to show how a graphical presetatio ca illustrate fudametal ideas such as the shift theorem, superpositio ad statioary phase ad how they affect the time-domai shape. I the typical itroductio, the real-space is usually time or space (distace), ad we will use the terms time-domai or distace-domai as appropriate to refer to them. Figure (A) shows a time-domai fuctio, i this case a boxcar cetered at t = ad the first few terms of the Fourier series that approximates it

2 a u() t = + a ( t) = a a cos ω, ω ω, ad real (.) = (we will clarify the ω term below, for, ω ω cos ω t =, so the a term is just a DC shift ad treated separately). This is ot the most geeral (it ca oly represet symmetric fuctios for example) or elegat form of the Fourier series, but it allows a graphical illustratio that presets the fudametal ideas. The frequecy-domai represetatio of the boxcar is the set of weights, a, used i the Fourier series sum. Figure (A) also shows the time-domai approximatio obtaied from summig a larger umber of terms i the Fourier series. Note that the fial sum, u ( t), is i the timedomai oly, i.e. it is a fuctio of time oly, while the frequecy-domai represetatio is the series of weights, a, associated with frequecy oly. The fuctios, cos( ω t ), i Eq. (.) are kow as basis fuctios, ad they have both time ad frequecy i their argumet. We will use the term phase to refer to their argumets. The set of basis fuctios ca be used i a Fourier series to geerate ay other fuctio. The basis fuctios eed to have two importat properties. First they must be mutually orthogoal (perpedicular), which meas you caot make ay of the basis fuctios from a sum of the others. This is expressed mathematically as ( ωt) cos( ωmt) dt = δ ( m) = = = ad ( ) cos (.), th ω are the ad kow as the delta fuctio ad is defied as where cos( ω t ) ad cos( t m ) δ ( x) x = =, (.3) x th m basis fuctios ad ( m) δ is where x = m i this case. The set of basis fuctios must also have the property that they are complete. This meas that ay arbitrary fuctio, meetig some geeral coditios to esure the Fourier series coverges, ca be represeted usig oly this set of fuctios. While Figure (A) suggests the Fourier series represets a sigle boxcar, the Fourier series is actually periodic (more precisely, the Fourier series is oly applicable to periodic fuctios), with the period based o ω. If we plotted u ( t) for a rage of t larger tha that show i Figure (A) we would see this periodicity as a repetitio of the boxcar every mt, wheret = π ω is the period ad m is a iteger. Note that the fuctio u is cotiuous i the time-domai, but its represetatio i the frequecydomai is discrete. The geeral form of the Fourier series, which ca represet arbitrary periodic fuctios is, a u() t = + acos( ωt) + bsi ( ωt). (.4) =

3 This ca be writte as a ut () = + ccos( ω t+ φ). () = This form will facilitate makig the FTIV figures. The a, b ad c are all real with ad the phase value associated with the c = a + b (.) th or ω term is b φ = ta. (.) a The ω t + φ of the basis fuctio i Eq. (). Note the distictio betwee the phase argumet of the basis fuctio, the expressio ( ω t + φ ), ad the phase value, φ, which is part of the phase argumet. φ term provides a costat phase shift i the argumet ( ) The elegat form of the Fourier series is u( t) F i t e ω = =, (.3) which, with F i = c e φ, ca be writte i i t i( t ) u t c e φ e ω c e ω + = = φ (). (.4) = = (We will use lower case letters for real, ad upper case letters for complex values or fuctios.). The forms i Eq. (.4) ad Eq. (.3) are the forms that are usually preseted. They are difficult to use for graphical presetatios, however, as they have two parts (the weights for both the sies ad cosies, or the magitudes ad phase values for the complex expoetial) that caot be combied i a sigle illustrative figure. This is oe reaso why the boxcar was chose for Figure ; it is symmetric about zero makig φ =. We will fid the form i Eq. (), however, to be very useful i for makig the FTIV figures sice the phase value, φ, simply shifts or moves the positio of the cosie wave alog the t axis. This allows the full frequecy-domai represetatio to be show i terms of its amplitude ad phase i a clearly iterpretable form i the FTIV figures. The discrete sum of the Fourier series ca be geeralized to the cotiuous iverse Fourier trasform give by 3

4 where the weightig fuctio () ( ) i ω ω t u t = F e dω, (.5) ( ω) ( ω) ( ) i F a e φ ω = (.6) is kow as the Fourier trasform ad is the frequecy-domai represetatio of ( t) F ( ω ) is a cotiuous fuctio of ω oly, ad is complex with a magitude, ( ) kow as the amplitude spectrum, ad phase, ( ) fuctios u() t ad F ( ω) are kow as a Fourier trasform pair, expressed as ( ) F( ω) u t u. a ω, φ ω, kow as the phase spectrum. The, (.7) ad represet the same fuctio specified i the two domais. While there are some additioal coditios o the time-domai fuctio to esure that the Fourier trasform itegral is well behaved, the Fourier trasform is restricted to absolutely itegrable, operiodic fuctios. Followig Eq. (), it is more useful for FTIV to write the iverse Fourier trasform as ( ) ( ) = ( ) os + ( ) u t c ω c ωt φ ω dω, () i ( ) where the phase spectrum compoet, e φ ω, of the Fourier trasform is combied with i t the basis fuctio compoet, e ω. Give the weights for the Fourier series, it is easy to show how the summatio begis to look like the boxcar after summig just a few terms, ad the typical metal pictures of the Fourier series ad the iverse Fourier trasform are based o the idea of summatio. How does oe obtai the Fourier trasform weights, a, b, or F ( ω ) for the Fourier series, or the fuctio, F ( ω ), for the iverse Fourier trasform? Usig the form of the Fourier series i Eq. (), ad igorig the a term (which will go away upo itegratio for m ), cosider () cos( ω m ) u t t dt = (.) acos( ωt) bsi ( ωt) cos( ωmt) dt aδ ( m) + =. = We arrive at this result by chagig the order of itegratio ad summatio ad usig the orthogoality of the basis fuctios give i Eqs. (.) ad (.3). I the stadard form for the Fourier series weights we have 4

5 ad for the more elegat form, a = u() t cos ( ω t) dt π, (.) b = u() t si ( ω t) dt π, (.3) iω t F( ω ) = u() t e dt π. (.4) We ca iterpret the weights by turig to the mathematical cocept of correlatio, which, i this cotext, provides a measure of the similarity of two fuctios. t h t is The geeral expressio for the correlatio betwee the fuctios g ( ) ad ( ) () () g t h t dt. (.5) If the two fuctios are the same, the correlatio will be a maximum. If they are egatives of oe aother, the correlatio will be the same magitude but egative. For fuctios that are ot the same, the correlatio varies betwee these two limits ad has a miimum magitude of zero for two fuctios that are completely ucorrelated (two differet Fourier basis fuctios for example). The expressios i Eq. (.-.4) are correlatios that quatify how similar the time-domai fuctio, u ( t), is to each of the basis fuctios. As with the Fourier series, we will fid it helpful to use the form c = u() t cos ( ωt φ) dt π +, (.6) where c is the Fourier series weight ad φ is the phase value i Eq. (.), ad these terms give the amplitude ad phase spectra respectively. This is a form of the correlatio fuctio () = ( ) ( + ) q t g τ h τ t dτ, (.7) which fids the correlatio betwee the fuctios as a fuctio of a shift, t. The value of c ad φ are foud from the maximum of the correlatio fuctio as a fuctio of φ c u t t dt φ< π π. (.8) (, φ ) = max () cos ( ω + φ) Coceptually, the c ad φ are foud together by searchig over values of φ, which simply shifts the cosie basis fuctio back ad forth alog the time axis, to fid the maximum value of the itegral (correlatio) for c. Due to the periodicity of the cosie 5

6 fuctio, we oly have to examie phase shifts over a rage of π (for periodic sigals the resultig correlatio is also periodic, ad there is o sigle shift for maximum correlatio). I practice, we ca use the stadard forms i Eqs. (.) ad (.3) to calculate the weights, ad the use Eqs. (.) ad (.) to calculate the c ad φ. We could also evaluate the correlatio fuctio (which ca be doe easily i the frequecy-domai) ad choose its maximum value ad the associated shift. The weights i the frequecy-domai are therefore proportioal to how correlated or similar the time-domai fuctio ad the shifted basis fuctios are. We ow have two values associated with each frequecy, ω ; the weight, also called the magitude, c, ad a phase shift, φ, that are easily iterpreted both physically ad i the figures of the FTIV method. To fid the Fourier trasform, F ( ω ), the expressio for the Fourier series weights i Eq. (.6), ca be geeralized to which is also a correlatio betwee u ( t) iωt F( ω) = u() t e dω π, (.9) i t ad, although it is difficult to visualize ad is ot usually preseted as such. Followig the treatmet of the Fourier series, with i ( ) F ω = c ω e φ ω, we ca write ( ) ( ) e ω c( ω) = u( t) cos( ωt+ φ( ω) ) dt, (3) c ω is a where, as is the case with the Fourier series, φ ( ω ) is determied such that ( ) maximum. Note that the amig of the various Fourier etities is ot cosistet, ad there are several arbitrary choices i the defiitios. The Fourier series ad iverse Fourier trasform produce the time-domai represetatio from the frequecy-domai represetatio, while the Fourier trasform, or forward Fourier trasform, produces the frequecy-domai represetatio from the time-domai represetatio. There is o geeral ame for the expressio to geerate the weights of the Fourier series, Eqs. (.-.4,.6), or is there a iverse Fourier series. The sigs of the expoetial factors i Eqs. (.5) ad (.9) must be the egatives of oe aother, but the selectio of the sig is otherwise arbitrary. The π scalig factor may also be placed i the defiitio of the iverse Fourier trasform Eq. (.5) rather tha Eq. (.9), placed symmetrically o both the forward ad iverse trasforms as π, or doe away with altogether by a variable substitutio f = πω. Mathematicias, physical scietists, ad egieers each prefer a differet covetio. I geeral, i this tutorial presetatio, we will cocetrate o the fuctioal forms ad graphics ad ot show the scalig factors. Fourier Trasform Itegrad Visualizatio It is difficult, ufortuately, to illustrate may iterestig properties of the 6

7 relatioship betwee the frequecy-domai ad time-domai behaviors, especially those related to the phase spectrum, usig figures such as Figure (A). The vertical axis i the bottom portio of Figure (A), represets frequecy as oe goes betwee the labeled, horizotal frequecy axes. For the plot of each weighted siusoidal compoet, show i black, however, the idividual vertical axes represet amplitude for that compoet. The chage i amplitude ad sig of the siusoidal terms is due to the Fourier weightig for a boxcar, which is show i Figure (B) for both the cotiuous Fourier trasform (gree curve) ad the Fourier series (gree circles). I Figure (B), the values show for the Fourier trasform ad Fourier series have bee scaled to make the two plots overlay oe aother, illustratig they are the same fuctio (almost: the Fourier series is periodic ad the Fourier trasform is a sigle boxcar). A umber of blue circles are also show i the lower part of Figure (A) that defie lies i (, ) t ω where the argumet to the siusoid, ad, therefore, the value of the siusoid, is costat. The lies coectig these poits are kow as lies of costat phase. The argumet to the siusoidal basis fuctio, the phase, is the same alog each of them. We will retur to these lies later ad see that they are importat i uderstadig the shape of the time-domai fuctio. FTIV ca be used to provide a view of the frequecy ad time-domai Fourier pairs that more clearly illustrates the relatioship betwee the frequecy ad time-domai represetatios of a fuctio. I Figure (A), we show the basis fuctio compoet of the itegrad for both the forward ad iverse Fourier trasforms as a surface i ( t, ω ) space that is formed by drapig a altitude color-codig over a illumiated 3-D view of the topography, (, ω) cos( ωt) z t =. (3.) The applicatio of lightig ad shadig helps the brai visualize the surface. Sums over frequecy, i.e., sums alog lies parallel to the frequecy axis i Figure (B), max ( m) cos( ω m) u t = c t, (3.) = geerate the iverse Fourier trasform ad produce the time-domai fuctio also show i Figure (B). Note that we caot do cotiuous math o the computer ad, therefore, simulate the cotiuum by takig dese, discrete sampligs i both ω ad t. Itegratio over ω is replaced by summig over this dese samplig. The additioal discretizatio i time will itroduce additioal properties ad limitatios to the applicatio of Fourier aalysis, specifically a maximum frequecy related to the time step ad a periodicity i the frequecy-domai with respect to this maximum frequecy. While the differeces betwee cotiuous math ad discrete math are iterestig ad importat, ad have to be take ito accout whe producig the figures, they will complicate the curret discussio, ad we will ot discuss them further. We are, therefore, simulatig the iverse Fourier trasform to geerate the images i the figures usig Eq. (3.) ad ot usig the Fourier series, Eq. (). We will switch freely betwee the two forms sums or 7

8 itegratio as eeded. Eq. (3.) was used to produce the time-domai fuctio show alog the time axis of Figure (B) (the time-domai fuctio is rescaled for the graphics presetatio). While this method of calculatig the iverse Fourier trasform is computatioally iefficiet, it gives a very clear view of the process., calculated usig Eq. (3.) ad show i Figure (B), ad use it to calculate its Fourier trasform, we obtai the same set of weights that we started with, as expected ad show i Figure (C). As before, to produce the graphic display we are actually calculatig If we take the approximatio for the boxcar, u ( t) m c u t t (3.3) max ( ω) = ( m) cos ( ω m + φ) m= o the computer where t, ω, ad φ are discretized. The topographic surface i Figure (A) is composed of just the basis fuctio factors of the Fourier trasform itegrad. The time- ad frequecy-domai represetatios are show o the appropriate axes. To go from oe represetatio to the other, we multiply alog oe directio ad sum alog the other as show by the flow arrows at the top right of Figure (A). We ca see that the summatio ad correlatio views ca be applied to trasformig i either directio. The process, but ot the FTIV fields (compare Figures (B) ad (C)), is the same. Our atural bias from existig i the real-space domai is what determies which view is the most useful as a explaatory tool for coceptualizig the forward ad iverse trasforms. Oe ca also see i Figure (C) that the boxcar correlates well with the lower frequecy compoets of the basis set, particularly those whose period is greater tha the real-space width of the boxcar. The patter of relative icreases ad decreases i correlatio for the higher frequecy compoets of the basis set cotribute to makig the corers ad sides of the boxcar sharp ad vertical. To better uderstad how the phase spectrum i the frequecy-domai affects the shape of the time-domai fuctio, we will cosider a fuctio i which the frequecydomai weights are costat, c( ω ) =. Sice the amplitudes are all equal, the shape of the time-domai fuctio will ow be completely determied by the phase spectrum, or how the various siusoidal compoets lie up. We will start with the simplest phase spectrum specificatio i which the phase is a costat idepedet of frequecy ad is equal to at t =, i.e. all the siusoidal waves lie up at t =, show i Figure 3(A) (which shows the positive frequecy portio of Figure (A)). This phase spectrum produces a arrow spike i the time-domai located at t = which is a delta fuctio, Figure 3(B). I this case of uiform weights ad φ =, it is easy to see the lies of costat phase i the basis fuctio argumet of the Fourier trasform itegrad, as they are also cotour lies of costat amplitude i the Fourier trasform itegrad (Figure 3(A)). This oe-to-oe relatioship betwee the behavior of the phase argumet ad the value of the Fourier trasform itegrad, 8

9 (, ω) cos( ωt φ) z t = +, (3.4) for the delta fuctio, that if oe is a costat so is the other, holds alog ay sigle cotour of z( t, ω ). The relatioship does ot hold betwee two distict cotours i the Fourier trasform itegrad havig the same value, i which case the phase argumets will differ by π. We will, therefore, equate sigle cotours i the FTIV plot for the delta fuctio with costat value for the phase argumet i the remaider of this presetatio. This oe-to-oe relatioship betwee the Fourier trasform itegrad values ad the uderlyig phase argumet is ot true i geeral as ca be see i Figure (B) for the boxcar. We will ow use Figure 3 to illustrate a key relatioship betwee the phase argumet ad the time-domai shape of a fuctio. I Figure 3(C) ruig sums, or itegratios, over frequecy are show alog two vertical profiles i the FTIV image. I this case, we kow the expressio of the fuctio alog the vertical itegratio paths i the Fourier trasform itegrad, cos( ω t), ad ote two thigs: that the itegral of this fuctio is o-zero oly at t =, where the phase argumet is a costat (zero), ad that at all other times the itegral is zero (small), ad the phase argumet varies liearly with ω. At t = all the Fourier trasform itegrad terms, ad their phase argumets, are costat, as idicated by the vertical white ridge through the FTIV image. The ruig sum at t = is a liearly icreasig fuctio of the frequecy. At t =. 49, the phase argumet varies liearly, ad the values alog a vertical profile oscillate harmoically, as idicated by the oscillatig gray scale alog a vertical profile i the FTIV image i Figure 3(C). The ruig sum or itegral over frequecy for this oscillatig fuctio is also oscillatig ad remais small. I geeral the itegral of a oscillatory fuctio over a large umber of cycles is zero. These two observatios are the key to the priciple of statioary phase, which says that for fuctios that are oscillatory over most of their rage, the oly o-zero cotributios to the itegral of the fuctio come from the regios where the fuctio is o-oscillatory (e.g. Udías, 999). We will examie the priciple of statioary phase i more detail i the discussio of dispersio. I the followig discussio we will refer to the patter i Figure 3(C) as a skirt patter, with each lie of costat phase argumet beig a skirt. We will get costructive iterferece whe the skirts are vertical ad destructive iterferece whe they are slopig as we cross them alog vertical summatio or itegratio paths. Note that our sum over a fiite frequecy rage, plotted i Figure 3(B), actually produces a approximatio to the delta fuctio i the time-domai ω ( ) ( ω t) ( ω t) si ( t) cos ωtdω = sic ω, (3.5) ω which is kow as the sic fuctio. If we had take a exact time-domai boxcar i Figure (C) istead of the approximatio obtaied from Figure (B), we would ot have 9

10 obtaied the same sic fuctio we started with i the frequecy-domai (we would still have obtaied a sic fuctio, but of a differet width). Eq. (3.5) itroduces a fudametal property relatig the frequecy ad real-space extets of a fuctio. The more localized a fuctio is i oe domai, the more spread out it is i the other. For ω small, i.e. arrowbad i the frequecy-domai, sic( ω t) will be wide i the timedomai, while for ω large, i.e. broadbad i the frequecy-domai, it will be arrow i the time-domai. The arrowest fuctio i the frequecy-domai is a sigle frequecy, a delta fuctio, which is a ifiite extet siusoid i the time-domai. Similarly, as sic ω i the time-domai producig the delta fuctio (Figure 3(B)). Roughly speakig, the product of the widths of the fuctio i the two domais is a costat. This is a geeral property of the relatioship betwee ay Fourier trasform pairs. It appears i the wave particle duality cocept of quatum mechaics as the Heiseberg Ucertaity Priciple, where the probability desity fuctios for both positio ad mometum, ad time ad eergy, are Fourier trasform pairs (Gubbis, 4). We will see various forms of this itegral i the discussios that follow. ω i the frequecy-domai the width of ( t) Examples Travelig waves to stadig waves ad back. To illustrate the features of the FTIV presetatio, we will cosider two examples. I the first example, we will look at two waves travelig i opposite directios o a strig, illustratig the superpositio of waves ad the relatioship betwee travelig waves ad stadig waves. I the secod example, we will look at a dispersig wave-trai ad relate the behavior i the frequecy-domai to the real-space wave shape ad behavior. I order to cocetrate o the relatioships betwee the frequecy ad real-space represetatios i both cases, we will take the iitial disturbace to be a delta fuctio at x = ad t = (Figure 3(A) ad (B)). It is easy to show that fuctios of the form ( x vt) u ± (3.6) i the real-space are travelig wave solutios to the wave equatio u x = v u. (3.7) t This is kow as D Alembert s solutio, ad it represets the movemet of a arbitrary shaped wave of costat shape at a velocity, v. Cosider solutios to the wave equatio of the form ( ωt) cos kx ±, (3.8) which represet siusoidal waves travelig to the left or right, respectively, at velocity

11 v ω = (3.9) k i both the distace-domai ad the FTIV space. The argumet has bee chaged to use the waveumber, π k =, (3.) λ i radias/distace uit ad the agular frequecy, v ω = π f = π, (3.) λ i radias/time uit. With this chage, the argumet is i uits of radias, rather tha distace, ad is related to the wavelegth, λ, ad agular frequecy, ω, of the wave, rather tha the velocity. This chage ties the argumet to the FTIV image, where we ca see the wavelegth ad frequecy, but ot the velocity. The wave equatio is liear, ad by the priciple of superpositio, which allows us to combie solutios to the wave equatio to produce ew solutios, we ca add ay umber of solutios together to make a ew solutio. Now cosider a solutio to the wave equatio that is the superpositio of two siusoidal waves that are the same except for their directio of travel (, ) cos( ω ) cos( ω ) u x t = A kx+ t + A kx t. (4) (, ω) cos( ω ) cos( ω ) u k = A kx+ t + A kx t. (4.) Mathematically either ( x, t ) or (, ) wrote both above, but our bias of livig i the (, ) k ω ca be the idepedet variables, which is why we x t domai leads us to ormally select these as the idepedet variables. Usig simple trig idetities Eq. (4) ca be rewritte as ( ω ) ( ) u = Acos t cos kx, (4.) a siusoidal wave that is fixed i space, ( ) harmoically i time, cos( ω t) cos kx, whose amplitude is modulated. Waves of this type are kow as stadig or statioary waves. There are a large umber of web pages, may of which iclude aimatio, that show how the combiatio of travelig waves produces stadig waves, ad we do ot show that here. The spatial part of Eq. (4.) is zero, or has odes, at kx = π for =,,... For a strig of legth L with both eds fixed, the eds must be o odes. Lettig the eds of the strig be at x = ±L /, we fid that for all wavelegths λ = L/ for =,,... both eds of the strig are odes. Whe pairig the waves as we did here, you see oly the stadig wave whe lookig at the sum of the waves. Nothig seems to be goig aywhere, as it is i the math of the matched travelig waves, but you do ot see the travelig waves idividually. Note that we have switched the realspace axis from time to distace. The wave i real-space ow correspods to a sapshot,

12 at a fixed time, of the wavefield i space, similar to a sapshot of the waves o the surface of a pod after havig throw i a pebble. With this chage i axis, frequecy will mea spatial frequecy i cycles/distace uit rather tha cycles/time uit, ad we will specify frequecy usig spatial frequecy, f k or waveumber, k. We could also produce a real-space time series at a fixed positio, but the wave sapshot is easier to visualize. The time series gives the time history of the motio at a fixed positio, the displacemet history of a cork floatig i the pod ito which we threw our pebble, ad the fial result is a seismogram. The time ad distace-domai figures are idistiguishable i the sese that oe caot tell, without the labelig of the axes, if oe is lookig at a sapshot of the wave i space or a time history (seismogram) of the wave at a fixed positio. We will ow combie the priciple of superpositio together with Fourier aalysis, which says we ca make arbitrary fuctios from weighted sums of sies ad cosies. For simplicity, cosider the delta fuctio agai. Start with two of the delta fuctios whose FTIV images are show i Figure 3(A). Let each oe propagate at the same velocity, v, with oe wave goig to the left ad the other oe to the right (Figures 4(A) ad (B)). As all compoets travel at the same velocity, after a time t the delta fuctios move without a chage of shape to ew positios x = ± vt. This is just D Alembert s solutio. We ca use superpositio to add the idividual travelig wavefields i Figures 4(A)-ii ad (B)-ii directly to obtai the combied travelig wavefield i Figures 4(C)-ii where the waves pass through oe aother. Returig to the two delta fuctios, sice the order i which we do the additios/itegratios does ot affect the result, we ca first combie each pair of Fourier compoets at a give frequecy to make a stadig wave at that frequecy ad the itegrate, or sum, these stadig waves over frequecy. Combiig the FTIV fields i Figures 4(A)-i ad (B)-i gives the FTIV field show i Figure 4(C)-i, where oe ca see patters remiiscet of, but ot the same as, the patters i the FTIV image for the idividual delta fuctios. There is aother view of the FTIV field show i Figure 4(C)-i obtaied by examiig the two parts of the product, cos( ω t) ad cos( kx ), i the stadig wave view, show o the right side i Figure 4(C)-i. The cos ( kx) part is the same as the delta fuctio i Figures 3(A) ad b. At t =, cos( ω t) =, ad the stadig wave view has a delta fuctio at x =. At some later time, t = t, ad rememberig that the ω for each Fourier compoet is a liear fuctio of k (Eq. 3.9, we will reserve writig ω ( k ) for the case of dispersive waves where the relatioship betwee ω ad k is ot liear), the iverse Fourier trasform at each positio is (, ) = cos( ω ) cos( ) = δ ( ± ω ) u x t t kx dk kx t (5) by the orthogoality of the cosie fuctios i the itegrad, producig the travelig wave delta fuctios i Figure 4(C)-ii at positios

13 x ωt k =± =± vt, with L L x. (5.) At time cos ω t term effectively selects two vertical profiles at ± x i the FTIV image of the delta fuctio at x = that have the same agular frequecy as the modulatio i time, as is show graphically i the right had side of the FTIV image show i Figure 4(C)-i. The itegrad at ± x is proportioal tocos, which is, ω. At all other locatios the itegrad is oscillatig, ad, as we have see before, itegrates to zero. t, therefore, the ( ) Aother feature of the stadig wave viewpoit is that it ca geerate a more complete costructio of the wavefield. The travelig ad stadig wave views discussed above correspod to two, of several, geeral approaches to geeratig sythetic seismograms: ray tracig ad modal aalysis. Ray tracig has the advatage that each pulse or phase (a ew use of the term phase meaig the arrival of a specific pulse, such as the P phase, o the seismogram) i the sythetic seismogram is idetifiable ad has a kow ray path (this correspods to the process illustrated i Figure 4(C)-ii). The disadvatage is that we oly get the phases, or ray paths, we specify. That is oe of the reasos we ca idetify them, ad dispersio (which we will do ext) caot be hadled easily. I the modal aalysis method, we get the complete seismogram, ad it is easy to iclude dispersio, but we caot easily idetify the phases, ray paths, etc. (the process illustrated i Figure 4(C)-i). I the modal view, if we let t icrease beyod the time for the delta fuctio to travel to the ed of the strig, it will correctly geerate the series of iverted, reflected waves produced each time the waves arrive at the fixed eds of the strig. This feature ca be explaied two ways. The first, which works for both the modal ad ray tracig views, is based o the ifiite extet of the harmoic waves combied with the atural periodicity imposed by the boudary coditios, i our case the fixed eds. I this view the two sets of mathematically defied periodic waves, extedig from to, oe goig to the right ad the other to the left, move ito ad out of the regio L x L where the strig exists. These two waves, as they pass through oe aother, always combie i such a way that at the eds of the strig they cacel to meet the boudary coditios there (fixed). The secod is that i the modulated mod π periodicity of the modulatio i time, stadig wave view, the ( ) cos( ωt) cos( ω( t mt) ) = +, (5.) provides the reflected waves, both iside ad outside the limits of the strig, so it is ot ecessary to cosider ifiite extet mathematically defied waves. I a travelig wave or ray tracig view with a D Alembert type solutio, the reflectios would have to be doe specifically at each ed of the strig. Both approaches have to take ito accout the physics ad boudary coditios but offer differet views of the solutio. The modal view for the strig is relatively easy to modify for other boudary coditios. For example, a strig fixed at oe ed ad free at the other correspods to takig the travelig 3

14 or modal views for the strig with both eds fixed ad cuttig it i the middle, at x = (just redraw Figure 4 with the limits at x = ad x = ±L / ), where the boudary coditio for the phase argumet of the cosie term at a free ed is φ = π,. Reflectios from the free ed are ot iverted, ad the maximum amplitude at the free ed is double the maximum amplitude of the wave. We also see i Figures 4(A)-i ad (A)-ii that movig, or shiftig, the delta fuctio i space (or time) is the same as a phase shift alog the space (or time) axis of the cos( ω t + φ ) terms of the Fourier trasform itegrad (step i Figure ). Movig the delta fuctio to x = xδ, each frequecy compoet i the Fourier trasform itegrad moves the same distace i space (or time), but a differet distace i phase of the argumet, i.e. a differet umber of wavelegths. This is a graphical view of the Fourier shift theorem F ikxδ ikxδ ( f ( x x )) = e F( k) = e F( f ( x) ). (5.3) δ Shiftig a wave i space (or time), therefore, does ot chage its shape or affect the weights of the Fourier compoets, it oly chages the phase spectrum of the ikxδ compoets liearly with frequecy through the e term. This is the frequecydomai expressio of traslatioal ivariace i the time- ad distace-domais. There is also a breakdow i the symmetry betwee the forward ad iverse Fourier trasforms here as the phase shift for both the forward ad iverse Fourier Trasform is always parallel to the real axis (step i Figure ). There is o equivalet to traslatioal ivariace i the frequecy-domai, where the origi is uique. Dispersio: Next cosider the case of a travelig wave i a medium i which the velocity of wave propagatio is a fuctio of frequecy, i.e. a wave travelig i a dispersive medium. If each frequecy compoet propagates at its ow velocity, v( f k ) (or v( k ) ), two thigs happe as the wave propagates for some time t. First, as before, the positio of the wave moves to a ew locatio. Secod, ad more iterestigly, is that the shape of the wave chages. This does ot correspod to a D Alembert type solutio. The FTIV image ca be used to illustrate several aspects of the evolutio of the wave shape as a fuctio of time, ad the developmet of this method, applied to seismic surface wave aalysis, was first preseted i Brue, et al. (96) ad Nafe ad Brue (96). We will ow examie this case, focusig o the relatioship betwee the Fourier trasform space ad real-space shapes, illustrated i Figure 5, i more detail. Detailed mathematical presetatios ad isights ito the physics ca be foud i Brue et al. (96) ad Nafe ad Brue (96) ad i most itermediate ad advaced seismology textbooks (Båth, 968; Acheback, 973; Udías, 999; Aki ad Richards; ; Pujol, 3). Startig as a delta fuctio show i Figures 3(A) ad (B), after some time t, the idividual harmoic waves will each travel at their ow velocity some distace, 4

15 ( ) x = v k t, (5.4) producig the FTIV image ad real-space plots, show i Figures 5(A) ad (B). The frequecy ad real-space domais are ow related by u( x, t) = cos( kx ω ( k) t) dk (5.5) (Brue et al., 96, Nafe ad Brue, 96) where ( k) kv( k) ω =. (5.6) The vertical lie of costat phase (of the argumet) origially at x = ad t = i Figure 3(A) gets sheared ito a ew shape at t show by the cya lie i Figure 5(A) (step i Figure ). By rescalig the x axis i Figure 5(A) by t, the axis ca be chaged to uits of velocity, ad we ca use the same figure to plot velocity as a fuctio of frequecy ad a velocity axis is draw across the top of Figure 5(A). The cya lie is, therefore, also a plot the of the velocity at which the waves i the FTIV field move as a fuctio of frequecy, v( k ). The velocity distributio used here was costructed to be differetiable ad produce dispersio similar to that for surface waves i the earth, where the loger wavelegth, lower frequecy waves propagate faster. I this example, the velocity distributio has costat velocity segmets for the lowest ad highest frequecies, travelig at speeds v ad v, respectively, ad varies smoothly ad mootoically betwee them. As time icreases, part of the vertical skirt at x = ad t = becomes slopig, ad we fid a ew collectio of poits, ( x, k ), i the FTIV image show by the mageta lie where the iitially slopig skirts deform such that they ow have a fold at which poit they go through vertical. We will ow examie the developmet of the real-space wavefield i Figure 5(B) from the iitial shape. Compared to the delta fuctio, the travelig wavefield has some wider, more complicated shape that is cofied approximately to the regio vt 3 < x< vt. Note that the x axis i Figures 5(A) ad (B) is shifted, or paed, to follow the ceter of the dispersig wave-trai. The observed width of the wavefield is wider tha what oe would expect if the width was based o the fast ad slow limits of the velocity distributio, vt < x< vt. From where does this low velocity limit, v 3, which is ot a physical velocity of the medium, arise? The aswer ca be illustrated through aalysis of the skirt patter. To assist i the discussio of the details of the dispersed wave-trai, we will divide both axes of the plot of the ( x, k ) space ito a umber of regios. The divisios alog the frequecy axis are idicated by the horizotal lies α through δ i Figure 5(A) ad divide the mageta curve, which we will discuss shortly, ito five braches labeled I through V. Usig the lies v, v, ad v 3 of the velocity axis of Figure 5(A) for referece, the real-space axis ca also be divided ito the regio to the left of v, where 5

16 the skirts all slope to the left, a arrow regio about v ; the regio betwee the lies for v ad v, where the skirts are folded over a sigle time; a arrow regio about v, a regio betwee v ad v 3, where the skirts are folded over twice with the folds facig opposite directios; a arrow regio about v 3, where the slope of the skirts goes through vertical without a fold; ad fially a regio to the right of v 3 where the skirts all slope to the right. We will call the regios VL, V, V, V, V3, V33, ad V3R respectively. Returig to the FTIV image of the dispersed wave i Figure 5(A), we observe that i regios VL ad V3R the skirts, or lies of costat phase (of the argumet), are all slopig outward, similar to the skirts i Figure 3(A), ad itegratio alog vertical paths there will be zero. A example of this is show by the oscillatory itegrad ad small ruig sum (itegratio) i Figure 5(C)-i. The wavefield at time t will, therefore, ot be foud i regios VL or V3R, outside the approximate limits V ad V33. We will ext look at the regio V, where the skirt patter is vertical for < k < α, where all the Fourier compoets arrive travelig at a costat velocity, v, ad the slopes for α < k. Rememberig that regios of the itegrad where the skirts slope do ot cotribute to the itegral, we ca limit our itegrad to the rage < k < α. We might expect these waves to combie ito a wave of some shape that also travels at v ad we will see that this is the case. Cosider the itegratio over a fiite rage of cotiuous frequecies, all travelig at the same velocity, v, where ( x t) k ( ) + δ k, = cos( ω ) u x t kx t dk, (5.7) k δ k u, is the wavefield i the distace-domai. This itegral looks very similar to Eq. (3.5) ad evaluatig it (Udías, 999) gives ( X ) si u( x, t) cos k x t sic X cos k x t X ( ω ) = ( ) ( ω ), ( ) which is the product of a carrier, cos( kx ω t) X x v t δ k, (5.8), where ω = kv, with a sic modulatio fuctio. (We are agai cocetratig o the pricipal properties of the solutio ad are cotiuig to igore the scalig see the texts referred to earlier for full solutios). The carrier is simply a travelig siusoid with waveumber k, ad frequecy ω. I the argumet of the sic fuctio, the ( x v t) term shifts the locatio of the sic fuctio by vt, while the δ k term cotrols the width. The relatioship betwee δ k ad k determies the shape of u ( x, t). For k small (log wavelegth) ad δ k large the sic fuctio is arrower tha the wavelegth of the travelig siusoidal wave, ad the resultig shape looks like a sic pulse, as ca be see i the leadig wiggle of the wavefield ear x 7. 5 ad labeled ii, i Figure 5(B). For k large (short wavelegth) ad δ k small the sic fuctio is wider tha the wavelegth of the travelig siusoidal wave ad the resultig wave shape looks like a sic modulated siusoidal wave group or packet. This ca be see i regio V where the skirt patter is also vertical i the high frequecy rage, δ < k < k, ad all the Fourier compoets travel at a max 6

17 costat velocity, v. Here we fid a sic modulated wave packet of high frequecy siusoidal waves at locatio iv, ear x 3. 5, i Figure 5(B). I regios V ad V the sic modulatio evelopes move at the same velocity as the compoet waves (parts I ad V of the velocity plot). I Figures 5(C)-ii ad (C)-iv the itegrad ad the ruig sums (itegratios) are plotted, ad we ca see that the cotributios to the real-space wavefield come from the regios i gray, where the velocity, itegrad ad itegrad phase are all costat. I both V ad V, the sic modulatio restricts the resultig real-space wavefield to a pulse or wave packet over a fiite regio of space movig at v or v respectively, ad we ca see that this regio cotais the eergy or iformatio carried by the wave. I this case, the real-space wave packet or pulse travels at the same velocity as the compoet waves i the FTIV image (cya curve). We will see shortly, that this is ot always the case, ad we will eed to distiguish betwee two velocities. We will ow look at regios V ad V3 where the skirts have oe or two idividual folds. I this regio, we fid that the wiggles i the wavefield do ot travel at the velocity show by the cya curve but travel at a velocity defied by sectios II-IV of the mageta curve i Figure 5(A). We will ow itroduce the term group velocity for the velocity at which waves travel i the real-space with their eergy ad iformatio (mageta curve) ad the term phase velocity for the velocity of the idividual compoet waves ad the velocity at which they travel i the FTIV field (cya curve). Note that a sigle frequecy wave would travel at the phase velocity. We will ow ivestigate these velocities, especially the group velocity, more thoroughly. Before discussig Figure 5 i more detail, we will first examie the basic ideas of phase ad group velocity by lookig at the simplest case i which the two velocities are differet; the superpositio of two siusoidal waves at slightly differet frequecies or wavelegths, travelig at slightly differet phase velocities. For a smoothly varyig velocity, start with a referece wave travelig at some waveumber k ad velocity v( k ) ad costruct two ew waves at waveumbers ( k δ k) ±, (5.9) travelig at velocities with frequecies ( ) dv k v( k± δ k) = v( k) ± δ k, (5.) dk ( )( ) ω ± δω = v k± δk k± δk, (5.) uder the coditio that δ k << k, δω << ω. (5.) 7

18 Combiig these two waves we obtai cos( ( δ ) ( ω δω) ) cos( ( δ ) ( ω δω) ) F = A k+ k x + t + A k k x t, (5.3) which, as before, ca be rewritte as ( ω ) ( δ δω ) F = Acos kx t cos k x t. (5.4) This could be broke dow further ito fuctios of ( kx ), ( ω t), ( δ kx) ad ( t) δω but doig so does ot provide additioal physical isight. I the real-space domai this is the product of two travelig waves, oe travelig at a velocity ( ) v k ad the other travelig at a velocity, δω ( k) δ k ( k ) ω =, (5.5) k ( ) dv k = v( k) + k = U( k). (5.6) dk We recogize the first velocity, Eq. (5.5), as the phase velocity of the referece wave. The secod velocity is ew ad does ot correspod to the velocities of the waves used i the superpositio, but to the velocity of the secod travelig wave factor i Eq. (5.4), ad this velocity is called the group velocity. If all the waves travel at the same phase velocity, dv( k ) dk =, ad U( k) v( k) =, there is o eed to distiguish betwee phase ad group velocities, or carry aroud the k depedece, although usig the term group velocity esures we are talkig about the velocity of eergy propagatio. The stadard presetatio of the relatioship betwee the phase ad group velocity is show i Figure 6(A), while a 3D surface view is show i Figure 6(B). I this simple case, oe ca determie the phase velocity by measurig the wave at two or more positios ad fidig the velocity at which the wiggles travel (the slated dashed red lie i Figure 6(A) follows the crest of a wiggle). Oe has to be careful to ot skip wiggles, by a iteger umber of wavelegths, which ca easily happe if the measuremet poits are too far apart as the wiggles are idistiguishable. The group velocity, show by the slatig gree lie i Figure 6(A), is determied by measurig the wave evelope at two or more positios to fid the velocity of the evelope. This may be difficult as the evelope is ot well defied whe there are a small umber of wiggles i the evelope. Determiig group velocity also suffers from the wiggle skippig problem, as the groups are also idistiguishable, but with respect to the evelope which skips i steps of half its wavelegth. If oe sits o the evelope of a group of waves, oe travels at the group velocity ad the wave crests ru by udereath at the phase velocity (the waves i the group ca ru forward, backward or, if v = U, be statioary). As we saw above, eergy is carried by the wave at the group velocity as this is the 8

19 velocity at which the waves arrive, or are observed, i the real-space. This is ot obvious with the ifiite extet waves i Figure 6, but oe ca get a idea of how this works if oe surfs a crest of the waveform (the oe o the left marked i red i Figure 6(B) for example). Oe moves at the phase velocity, but the wave rus out i the trough ad you get dumped i the water ad have to wait for aother crest from behid to surf forward agai (the crest o the right i Figure 6(B) marked i red for example). This ew crest also rus out whe it arrives at the trough, ad you have to catch a later crest agai. You are basically limited to catchig the crests i oe modulatio package of the evelope. Your log term average forward speed, therefore, is the speed at which the evelope travels, which is the group velocity. This aalogy does ot work whe the phase velocity is less tha the group velocity. I this case, eergy still travels at the group velocity, which ca be see if the surfboard is log eough that it is surfig the evelope, rather tha the idividual wave crests, ad agai moves forward at the group velocity. As i the case of the travelig-stadig wave combiatio, there are may aimatios o the web of this effect. We ca ow apply these ideas to the dispersig wave-trai i Figure 5, which illustrates graphically the observatio that phase velocity is a easy cocept to uderstad, but a hard quatity to measure as othig is physically travelig at the phase velocity, ad that group velocity is a hard cocept to uderstad, but a easy quatity to measure, as this is the velocity at which the physical wavefield travels. If oe jumps o a crest of the real-space wave, oe fids the crest to have some wavelegth ad be travelig at some velocity. As oe rides alog with that crest, however, both the wavelegth ad the velocity chage with time. Both chages are directly related to the evolutio of the fold i the skirt patter i FTIV space with time. We will see that at ay time, ridig o the crest of the wave at x, with waveumber k, i real-space is equivalet to ridig at the same positio, x, o the crest of the Fourier compoet that has the same frequecy i FTIV space. As time progresses, however, the skirt chages shape ad moves through FTIV space samplig other frequecies ad phase velocities. If oe wats to follow a idividual crest i the real-space domai, oe has to chage frequecies ad, therefore, velocities, i FTIV space by followig the mageta group velocity curve i Figure 5(A). The distictio betwee the phase ad group velocities is clear i the FTIV space as is the relatioship of the evolutio of the skirt patter i FTIV space to the velocity ad shape of the waves i the real-space. To illumiate the relatioship betwee the patters see i the FTIV space ad the shape of the real-space fuctio we will ow retur to the discussio of the priciple of statioary phase. I its geeral form, the priciple of statioary phase, first outlied by Cauchy (Erdelyi, 955) ad fully developed by Stokes ad Kelvi (Båth, 968), cosiders itegrads which are the product of two fuctios, a slowly varyig fuctio, iψ( k, x) ϕ k, ad a oscillatory fuctio, e, ( ) b ( ) ϕ ( ) ( k, x) iψ f x k e dk a = (5.7) 9

20 (Båth, 968). I additio to providig isight ito the propagatio of dispersed waves, the priciple of statioary phase provided a importat trick that facilitated evaluatio of the iverse Fourier trasform whe computers were people. These two poits were actually the motivatio for its developmet, although it ca also be derived by moder aalysis techiques. By usig the delta fuctio i the real-space domai, we oly have iψ( k, x) to cosider the oscillatory term, e, sice ϕ ( k ) = F( δ ) =. This greatly simplifies the discussio of statioary phase without losig the isight. We will ow apply our earlier observatio i Figure 3(C), that the oly o-zero cotributios to the iverse Fourier itegratio come from regios where the phase of the argumet of the itegrad is a costat, or equivaletly its derivative is zero. We have to remember that while idividual cotour lies of the image i Figure 5(A) have costat values of the fuctio they are ot cotours of the argumet F( k, x) = cos( kx ω ( k) t) (5.8) ( kx ω ( k ) t) (5.9) directly, but they ca be iterpreted as lies of costat phase of the argumet whe the Fourier weights are costat. O ay sigle cotour lie we will cosider f ( k, x ) = costat, or (, ) f k x dk x = (5.) to be equivalet to ( ) ( ) ω ( ) Ψ kx, = kx kt = costat, or Ψ ( kx, ) k x =, ad ( k) x dω = = U( k), (6) t dk x which says the cotributio to the wavefield from the Fourier compoet k arrives at x U k. at the group velocity, ( ) I regio V, after the large iitial arrival, we fid a exteded trai of siusoidal waves whose wavelegth (this is a wavefield sapshot) slowly decreases. Lookig at the itegrad ad ruig sum i Figure 5(C)-iii we ca see that the major cotributio to the wavefield i regio V comes from the small regio, idicated by the gray backgroud, which comes from the area aroud the fold i the skirt patter i the FTIV image i Figure 5(A), where the slope is vertical or the phase of the argumet i the iverse Fourier trasform itegrad is statioary. A frequecy, k, that has a statioary phase, i.e. a vertical skirt i Figure 5(A), defies a sigle poit ( x, k ) i the FTIV space. The mageta lie i Figure 5(A) coects the poits ( x, k ) i the FTIV that have statioary phase ad defies the group velocity, the velocity at which the observed

21 waves of each wavelegth, ad the eergy associated with them, travels. I regio V, oe brach of statioary phase is foud at log wavelegths ad produces a wavefield of siusoidal wiggles at a slowly varyig istataeous frequecy. I regio V3, we fid two braches of statioary phase: oe alog a cotiuatio of the mageta curve from regio V, ad a separate oe at shorter wavelegths. By superpositio, we add the results from each brach of statioary phase idividually as show i Figure 5(C)-v, where we agai see that the o-zero cotributios to the fial value at that positio comes from the two braches of statioary phase, show i gray. If we divide the frequecy axis betwee α ad δ ito two parts about the frequecy where the group velocity turs aroud ( f k ) ad itegrate each part idividually, the real-space dispersed wave-trai from the low frequecy part has icreasigly shorter wavelegths as oe goes from v to v3. This effect is clearly see i regio V, ad is called ormal dispersio. The sum for the higher frequecy part has icreasigly shorter wavelegths as oe goes from v 3 back to v ad this is called aomalous dispersio. I regio V3, the two patters are superimposed, makig them hard to see idividually. We also ote here that i order to use the priciple of statioary phase, which is a approximatio, we have to let the wave propagate for sufficiet time such that the itegrad is highly oscillatory over all its rage (skirts are sufficietly sheared) except for the immediate regio of the statioary phase. This is typically true for travel times that are much greater tha the periods of the frequecy compoets about the poit of statioary phase (Pujol, 3). Performig the iverse Fourier trasform i regios V ad V3 requires evaluatig Eq. (5.5). This itegral caot, i geeral, be evaluated i closed form due to the ω ( k ) term. While the details of the mathematics i this ext sectio ca get quite complicated, the basic method is straightforward. We have see that oly a small regio of the rage of itegratio about the poit of statioary phase cotributes to the value of the itegratio. We will, therefore, cocetrate o evaluatig the itegral i that regio ad the fial result will tur out to be relatively simple. For well behaved itegrads, the itegral i the regio of iterest ca be approximated usig a Taylor series expasio of the argumet to the itegrad about a poit of statioary phase, ( k, x ), to fid u ( x, t). From Eq. (6) we saw that the cotributio to the wavefield from the Fourier compoet k arrives at x at the group velocity, U ( k ). Keepig the lowest o-zero term of the Taylor series expasio of Ψ ( kx, ) about the poit of statioary phase, where ω =, ad usig the prime to represet ddk ( ) (, ) ω ( )( ) ω ( ) Ψ k x k x + k k k t (6.) Pluggig this ito Eq. (5.5), usig the expoetial form Eq. (5.7) to make the itegratio easier (we will take the real part at the ed), usig Eq. (6), otig that the itegral outside the eighborhood of k is ull so the limits ca be expaded to ±, doig a little algebra, ad agai igorig scalig costats our approximatio becomes