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The waveequationBy ENDERS ROBINSON and DEAN CLARK

T e foundation o f seismology s the theory of wave motion,a complicatedconcept hat is still - after centuriesof experi-ments and speculationsby many of the very greatest cientists- an are a of active research n many disciplines.Even simpleforms of wave motion are difficult to describe verbally; but,ironically, the simplest type of wave is rem arkably easy todescribe and subsequentlyanalyze) mathem atically.This is one of tho seareaswhere, n the wordsof Nob el PrizephysicistStevenWeinberg,mathematicshas a spooky corre-lation to the physicalworld. Although som enaturally occurr ingcrystals aveperfectgeometric hapes,ight triangles re a purelymathem atical concept.They exist outside our ordinary experi-enceof the physicalworld. Have you ever ound a perfectly ight-triangular rock, or blade of g rassor leaf in your back yard oron a field trip? Yet we remem ber ro m elementary rigonometry(the mathem atical analysis of the prope rtiesof triangles) thatthe graph of th e sine function - nothing more than the ratioof two sidesof a right triangle - perfectly represents ertainperiodicmotions, suchas he (small) oscillations f a pendulum.This type of sinusoidalmotion is calledsimpleharmonicmotion.The p ure sine curve, u = sin x, is quite restricted.The valueof u can never be greater than 1 or less han -1 and x m usttraverse distanceof 2t radians before one cycleof motion iscompleted.These imitations are, howe ver, ot serious.The sinefunction is easily tailored to rep resentany regularly repeatingmotion no matter what its height/depth (or amplitude), its fre-quency of oscillation, or its value when it cr oss es startingpoint (often the x = 0 line). Such an all-purpose sine functioncan be written, supp osingu to be the disturbancecausedby themotion. asu = A sin 27r ?- - cx T >wherex is distance nd I is time Five graphsof u vs.x are shownin Figure 1. The number A (chosen o be positive) representsthe amplitude; he distance etween onsecutive restss X (calledthe w avelength); he quantity T is the period or the time it takesthe wave o com pleteone cycle.The crestof the wave movesadistance X in time T Since X is a distance and T a tim e th equotient X/T equals the w aves velocity - almost always ex-

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pressedsimply as V. Therefore, itk 1~ 1 xt.dan d I I\ allowed totary, a wave crest sweepspast the fixed poml with a propaga-tion velocity given by 1:In seism ic work we usually plot wave motion as a fun ctionof time for fixed value of x and varying values of 1. This resultsin sine curves with a vertical (see Figure 2) orientation, ratherthan horizontal as in Figure 1.There are other useful ways n which we can write a sine func-tion to representwave motion. Instead of wavelengthand period,we can use wavenumber (k) and frequency (w) where

and then the sinusoidal wave may be expressedu = A sin (kx - wt)which representsa simple harmonic progressivewave. We canalso write this curve asu = A sin k(x - vt)because v = h/T = w/k.

The quantity w (w hich is expressed n un its of rad ians per sec-ond) representsangu lar frequen cy. It is related to c yclical fre-quencyf(expressed in Hertz) by the equation w = 27rf. Likewise

37 -_-_ - - - - - - -4 q---- - - - - -Figure 1 . Sinusoidal wave motion as a function of distance x.I I I I I I I 1 I *xI b-4

Figure 2. Sinusoidal wave motion as a function o f time t.

Figure 3. Illustration of f(x - vt).the quantity k (expressed n radians per meter) is the angularwavenumber, related to cyclical wavenumber x by the equ ationk = Zax.

We can generalize h is result without much difficulty. W e notethat the quantity x - vt reproduces tself when t becomes + tand x becomes x + vt' becausex + vt ~ vf t+ 1 =x - vt.Therefore any function of x - vt can be said to representa wave.An effective way to illustrate this is to imagine a ta ut stringlying on the x axis. If the string is displaced in any w ay perpen-dicular to the x axis, then the shape of the res ulting curve canbe written u = f(x). If the displacements alter in such a waythat the pulse travels with velocity v in the positive x directionwithout change of shape, the equation rep resenting the pulseat any time t will still be u = f(x) . . . . . provided that we movethe origin a distance vt in the positive direction (see Figure 3).In reference o the old origin, the equa tion of the pulse will havex replaced by x - vt or u = f(x-vt).

We can therefore state that this is the general equation of awave with constant shape traveling in the positive direction withvelocity v. Furthermore, every wave of this type must be exp ressi-ble in this form. Similarly, a wavegoing in the opp osite direction(i.e. negative x) is represented by a function u = g(x + vt).T he first great insight into the mathem atical analysisof waveswas made by the legendary Greek m athematician an d religiousleader Pythagoras who is believed to have died abo ut 500 BC.He discovered hat the pitch of a sound from a plucked stringdepends upon the strings length, and that harmo nious soundsare given off by strings whose lengths are in the ratio of wholenumbe rs. Howe ver, significant additional progresswas impossi-ble until the invention of calculus, more than 200 0 years later,permitted the English mathe matician Brook l&ylor to make thefirst productive attempt a t the quantification of w ave motion.Consider a s tretched string with initial shape (x). Accordingto ba sic differential calcu lus, he s lope of the tangent line at anypoint rep resents he rate of changeof the functionf with respectto x. This rate of change is the first derivative of fwith respecttox; in turn, the rate of chan geof the slope (or secondderivativeoff with respect to x) represents he curvature of the function.Now consider the motion of an y particular point on thestring. We have seen that the traveling wave can be representedas (x - vt) which at the point x = 0 becomesmerely (-vt). Thatpoint is moving up and down approxim ately at right ang les tothe x a xis. Back to basic calculus. The points up-and-downvelocity is given by the first derivative offwith respect o t an dthe points acceleration is given by the second derivative offwith respect to t.When the string is in its equilibrium position (horizontal alongthe x axis), there is no net ve rtical force acting on a ny point o nthe string. However, when the string is curved, the tensionin the string exerts a restoring force which attempts to move itback to its equilibrium p osition. The m ore the curvature, the

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Small ~wvaturemeanssmall restoring force LargerYafre meanslargeesfarlngorceFigure 4. Force is proportional to curvature of string.

I +Nearly tratght,mepathmeansmall acceleration

,Strongly cu rved tome pathmeant large acceleralion

Figure 5. Acceleration proportional to the amount of bend inthe time path of a point on the string.greater this restoring force (Figure 4). Taylor noted this andreasoned that the restoring force is proportional to curva tureand, as a student of Sir Isaac Newton, he knew that force isproportional to acceleration (Figure 5). Th us he wrote theequationcurvature = (Y accelerationwhere 01 s a constant of proportionality. Taylor could not fullydevelop the properties of this equ ation be cause he had noknowledg e of partial derivatives. But after they were invented,his speculation was confirmed. It also turned out th at the con-stant of proportionality was 1/v2. In modern mathe maticalnotation, this equation is writtenah I a%-=-2a2 lJ* at

and it is known as the one-dimensional wave equation. W hengeneralized to three dim ensions, it gov erns the m elodies ofPythago ras, he propagation of seism icenergy hrough the earth,and all other w ave motion. The three dimensional wave equationisa54 a% ak 1 a%--+ x+ az =v at8x2If we now go back to one of our o riginal equations for simpleharmon ic motion- u = A sin (kx- wt) - and take secondpar-tial derivativeswith resp ect o x and t, and then substitute hesesecond derivatives into the one-dimensional w ave equation, w e

obtain-Ak2 sin(kx- tit) = $ (- Ati2) sin (kx - wf).After canceling comm on factors, we discover thatk2 = -,f_which is called the dispersion equation for the one-dime nsionalwave equation. It relates wavenumber and frequency. Whenextended to three dimensions, this equation becomesk: + k: + kl = $where k = Jk: + k: + kf is the wavenumberand k,, k,,, k, arethe wavenumbercompone nts in the three coordinate directions.

1 .,he \+l~~cquatlor I< arrequation m spaceand time coordinates1V,K; nd 1) whereag he related dispersion equation is an equa-flkjn in waven umbe r and tfrequencycoordinates (k,, k,, k,, and2). C;eophysicists sually want to look at the data in the familiar\pace-time display: but it is often advanta geous to transformthedata m to the wavenumber-frequencydomain for computerprocc~slng.The basis or the transformation from one domainto the other is the math ematical operation know n as the Fouriertransform.(One of the best treatments of the Fou rier transform, as itrelates o exploration geophy sics, s given on pages 10.6and 10.7of Roy Lindseths book Digital Proces singofGeophysicalata:A Review Figures 6, 7, 8, and 9 are adapted from this book.)The Fourier transform allows us to separate a seismic traceinto its individual frequency compone nts. If many trace s werethus broken down , we could createseismicsections omposedof traces of o nly on e, identical, frequency. Let us now ass umethat we have done this and ex amine three situations, in each ofwhich we will let frequency (w) be constant,As everygeophysicist now s, a wavefront s a curve of c onstantphase. f one follows the same crestof wavemotion, the line con-necting these points makes up a wa vefront. In each of the threefollowing exa mples, the wavefront will be a straight line but ineach case with a different angle to the horizontal. When awavefront strikes a horizontal line, the movem ent in the h ori-zontal direction gives rise to the ap parent horizontal velocity.If the wavefront s perfectly horizontal, then all points on it strikethe horizontal surfacesimultaneouslyso the apparent horizontalvelocity s infinite. A geophysical xam plewould be a deep reflec-tion app roaching he surface vertically from d epth. On the otherhand, if the wavefront makes an angle to the horizontal, thenthe apparent horizontal velocity would be finite.In the first case,our section s flat, i.e. the wavefront parallelsthe .u-axis no dip is present), which m eans each trace is identi-ca l ( Figure 6 ). As a result, any waveprofile in the x directionwould be a constant. If we meas ure he amp litudes of the wavealong any horizontal time line and then plot them against valuesof .Y (see the top of Figure), ou r graph w ould be a p erfectlystraight horizontal line. Since there are no oscillations. he wave-

iFigure 6.

Figure 7.

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length is infinite a nd the correspondtng w,l\cnt:lnt)r.l (h) I, ~et,~,This w ave motion is representedon the uavenumber-flequeni:!doma in by the single point (k,o) = (0,~) because k z 0 andd represents our constant frequen t!N xt consider a section of traces with the \ame constant fre-quency (w) but whe re the wave front has a \ma ll amou nt otdip( Figure 7 ). The motion along the horizontal time line w*ouldlook qu ite different from that of the flat-wavefront section. Eachof the vertical traces s now slightly out of phasewith its adjoin-ing traces. As a result, motion in the x-direction is not constan t,but will have a low -wavenum ber sinusoidal character. The cor-responding low wavenumber is paired with the frequency andbecomes a single point (k,w) on the graph .If the amou nt of w avefront dip is increasedand the frequencyremains the same as n the preceding wo cases, he w avenumberincreases (Figure 8 ). E vidently, for any given frequency , thewavenu mber is related directly to the amount of dip of thewavefront.Our x-t graphs represent time sections; that is they are plotsof time vs. distance.Since the ratio of distance o time producesvelocity, it is seen that the slope of the wavefront line (time/distance) is the reciprocal of velocity. Thus any sinuso idal wavemotion which has the same wavefront dip will have the sameapparent horizontal velocity.In a fourth example, the wavefront dip is held constant forany chosen wavenumber k). Since we are holding the wavefrontdip con stant, it follows tha t we are holding the velocity (v) con-stant. As we know , the frequency (w) is specified by the disper-sion equation, w = kv. Now let us look at the plot of LLvs .k for a constant velocity ( Figure 9 ). The dispersionequationsays hat this plot is a straight line with slope v. That is if wechange W, we see that the wavenu mber (k) falls on a straightline as shown in the figure.This allows us to do some startling things in data proc essing.All the wavefronts with a given velocity are spread all ove r theroutine seismic ime section. In other words, each wavefront hasa different intercept with the horizontal (x) axis. However, if w etransform this data into the frequency-wavenumber omain, all

+,Figure 8.

Figure 9.

(a )Seismicectionwithhigh-velocity (b) Fourier ransformwith the tworeflections nd low-velocity round dispersion ines.roll. BI

Zeroingoutgroundmlldispersion ine

(d) Seismic section without theground1011. (c) Fourier translorm with theground-roll ispersion ine zeroedout.Figure 10. Pie-slice filtering.of the wavefronts are transformed into a single dispersion linewhich goes through the origin in the w-k plane. Thus if we wantto wipe out all the waveswith this apparent velocity, all we haveto do is zero-out the appro priate dispersion ine. This is the basisof velocity filtering, w hich is also called pie sliceor fan filtering.

In F igure IO, we see four diag rams (a), (b), (c), and(d) whichare connected in a clockwise fashion by operations A, B, an dC. Diagram (a) showsa seismicsection with two reflected eventsand three ground-roll events. These events criss-crossand thatwould m ake separation difficult in the x vs. I form. Howev er,let us now exploit the fact that the reflected events have highapparen t horizontal velocity, while the ground-roll eventshavelow velocity. With (a) as input, the Fourier transform A givesus (b) as outT ut. In (b ) the two reflected events appear as onedispersion line through the o rigin, whereas the three ground-roll events appea r as a separate dispersion line throu gh theorigin. The fact that dispersion ines alwaysgo through the originmeans that the two types of eventshave been separated.That is,diagram (b) has the appearanceof a fan, or a sliced-uppie, whereone pie slice contains the reflected events and another pie slicethe ground-roll eve nts. In operation B, we let the computer eatup (i.e. erase) the pie slice containing the gro und-roll disper-sion line, thereby giving diagram (c). W ith (c) as input, theinverse Fourier transform operator C gives us (d) as output.As expected, the unwanted ground-roll events do not appearin diagram (d), and we are left with only the des ired reflections.T e use of frequency-wavenumberanalysis s one of the mostpowerful tools of seismic data processing. As we have seen,measurement f wavenumber s a function of frequencyprovidesa reliable means of separating nd measuring he variousvelocitycomponents on a seismicsection. Other im portant seismicpro-cessingoperations (including dip moveo ut as well as migration)can make use of w-k analysis. All these processingmethods aretied physically to the d ispersion equation, wh ich in turn followsfrom Taylors inspired insight (in 17 15) spatial curvatu re isproportional to tempo ral acceleration that led to the originalformulation of the wave equation. &

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