Chaper 4 Traveling Waves 4.1 Inroducion To dae, we have considered oscillaions, i.e., periodic, ofen harmonic, variaions of a physical characerisic of a sysem. The sysem a one ime is indisinguishable from he sysem observed a a laer ime if he ime difference is an inegral number of emporal periods. To mainain oscillaory behavior, he energy of he oscillaor mus remain wihin he sysem, i.e., here can be no losses of energy. We will now exend his picure o oscillaions ha ravel from he source and hus ranspor energy away. Energy mus be coninually added o he sysem o mainain he oscillaion and he ranspored energy can do wor on oher sysems a a disance. Our firs as is o mahemaically describe a raveling harmonic wave, i.e., denoe a y [] ha ravels hrough space. A harmonic oscillaion y() =A 0 cos(ω 0 ), can be convered ino a raveling wave by maing he phase a funcion of boh x and in a very paricular way. Consider he general case of an oscillaory funcion of space and ime: y [z, ] =A 0 cos [Φ [z,]] = A 0 cos [z,]. We wan his oscillaion o move hrough space, e.g., oward posiive z. In oher words, if a poin of consan phase onhewave(e.g., a pea of he cosine creaed a a paricular ime τ) isaapoin x 0 in space a a ime 0, he same poin of consan phase mus move o z 1 >z 0 a ime 1 > 0. Snapshos of sinusoidal wave a wo differen imes 0 and 1 > 0, showing moion of he pea originally a he origin a 0. Thewaveisravelingowardsz =+ a velociy v φ. The phase of he firs wave a he origin is 0 radians, bu ha of he second is negaive. Since he wave a locaion z 1 and ime 1 has he same phase as he wave a locaion z 0 and ime 7
8 CHAPTER 4. TRAVELING WAVES 0,wecansayha: Φ [z 0, 0 ]=Φ [z 1, 1 ]= cos [z 0, 0 ]=cos[z 1, 1 ]= y [z 0, 0 ]=y [z 1, 1 ]. In addiion, for he wave o mainain is shape, he phase Φ [x, ] mus be a linear funcion of x and ; oherwise he wave would compress or srech ou a differen locaions in space or ime. Therefore: Φ [z,] = αz + β = αz 0 + β 0 = αz 1 + β 1. As discussed, if 1 > 0 = z 1 >z 0 (i.e., wave moves oward z =+ ), hen α and β mus have opposie algebraic signs: Φ [z,] = α z β By dimensional analysis, we now ha α z β has dimensions of angle [radians]. We have already idenified β = ω 0, he angular frequency of he oscillaion. Similarly, if [z] =mm mus have dimensions of radians/mm, i.e., α ells how many radians of oscillaion exis per uni lengh he angular spaial frequency of he wave, commonly denoed by : y + [z, ] =A 0 cos [z ω 0 ] raveling harmonic wave oward z =+ By idenical analysis, we can derive he equaion for a harmonic wave moving oward x = y [z, ] =A 0 cos [z + ω 0 ] raveling harmonic wave oward z = The waves are funcions of boh space and ime, i.e., hree dimensions [z,y,] are needed o porray hem. Generally we display y eiher as a funcion of z or fixed, or as a funcion of for fixed z: 4.1.1 -D Plo of 1-D Traveling Wave The 1-D raveling wave is a funcion of wo variables: he posiion z and he ime, andsomay be graphed on axes wih hese labels. An example is shown in he figure, where z is ploed on he horizonal axis and on he verical axis. In his case, he poin a he origin a =0has a phase of 0 radians. Tha poin moves in he posiive z direcion wih increasing ime, and so is a wave of he form y [z,] =cos[ 0 z ω 0 ] The poins wih he same phase of 0 radians a laer imes are posiioned along he line shown. The velociy of his poin of consan phase is z, and hus is he reciprocal of he slope of his line.
4.. NOTATION AND DIMENSIONS FOR WAVES IN A MEDIUM 9 4. Noaion and Dimensions for Waves in a Medium Trigonomeric Noaion: y [z,] y 0 = A cos {Φ [z, ]} = A cos(z ± ω + φ 0 ) Complex Noaion: y [z, ] =Ae iφ[z,] =Re nae i(z±ω+φ )o 0 y = posiion of he characerisic of he medium, e.g., [y] = angle, volage,... ; y 0 = equilibrium value of he characerisic; A = ampliude of he wave, i.e., maximum displacemen from equilibrium, [A] =[y]; z, = spaial and emporal coordinaes, [z] = lengh (e.g., mm), [] = s; T = period of he wave, [T ]=s,t = 1 ν = π ω ; λ = wavelengh, [λ] = mm ω = angular emporal frequency of he wave, ω = π radians T, [ω] = s ; = angular spaial frequency of he wave, = π radians λ, [] = mm ; ν = emporal frequency of he wave, [ν] = cycles s = Herz [Hz], ν = ω π ; Φ = phase angle of he wave, [Φ] =radians, (in his case, Φ is linear in ime and space); φ 0 = iniial phase of he wave, i.e., phase angle @ =0,z =0, [φ 0 ]=[Φ] =radians. σ = wavenumber, σ = 1 λ, he number of wavelenghs per uni lengh, [σ] = mm 1. Relaions beween he phase and he emporal frequencies ω = Φ ν = ω π = 1 Φ π 4.3 Velociy of Traveling Waves The phase velociy v φ of a wave is he speed of ravel of a poin of consan phase. A definiion for phase velociy can be derived by dimensional analysis: [v φ ]= mmper s; [ω] =radians per s; []= radians per mm: h ω i = = radians per second radians per mm radian- mm = = mm radian- s s Slighly more rigorously, we can find he phase velociy of a wave by aing derivaives of he equaion for he wave: y [z, ] = A cos [z ω + φ 0 ], y = ( ω)a sin [z ω + φ 0 ]=+Aω sin [z ω + φ 0 ], y = ()A sin [z ω + φ z 0 ]= A sin [z ω + φ 0 ] ³ y v φ = z = = ω = ω, ³ y z or by considering he poin of consan phase b radians: µ b z ω = b = z = + ω ³ ω = b0 + b 0 b is a new consan Consider he posiions z 1 and z of he same poin of consan phase a differen imes 1 and :
30 CHAPTER 4. TRAVELING WAVES ³ ω z 1 = b 0 + 1 ³ ω z = b 0 + = z 1 z = z = v φ x = ω = v φ. ³ ω ³ ω ( 1 )= 4.4 Superposiion of Traveling Waves Consider he superposiion of wo raveling waves wih he same ampliude, differen phase velociies, and differen frequencies: y 1 [z, ] = A cos [ z ω 1 ] y [z, ] = A cos [ z ω ]. We can use he same derivaion developed for oscillaions by defining a new frequency for boh: Ω 1 z Ω z ω 1 ω y [z, ] = y 1 [z,]+y [z,] =A {cos [ z ω 1 ]+cos[ z ω ]} ½ µ µ ¾ 1 z z = A cos ω 1 +cos ω = A {cos [Ω 1 ]+cos[ω ]} µ µ Ω1 + Ω Ω1 Ω = A cos cos jus as before.by evaluaing he sum and difference frequencies, we obain: µ Ω1 + Ω = µ 1 z ω 1 + z where avg +,ω avg ω 1 + ω µ ω = 1 + z µ ω1 + ω avg z ω avg µ Ω1 Ω = µ 1 z ω 1 z where mod,ω mod ω 1 ω + ω = z ω 1 ω mod z w mod
4.5. STANDING WAVES 31 4.5 Sanding Waves Consider he superposiion of wo waves wih he same ampliude A 0, emporal frequency ν 0,and wavelengh λ 0, bu ha are raveling in opposie direcions: f 1 [z,]+f [z,] = A 0 cos [ 0 z ω 0 ]+A 0 cos [ 0 z + ω 0 ] 0 z ω 0 = A 0 cos + 0z + ω 0 0 z ω 0 cos 0z + ω 0 0 z + 0 z = A 0 cos + ω 0 + ω 0 0 z 0 z cos + ω 0 ω 0 = A 0 cos [ 0 z] cos [ ω 0 ] = A 0 cos [ 0 z] cos [ω 0 ], because cos [ θ] =+cos[+θ] = A 0 cos π zλ0 cos [πν 0 ] This is he produc of a spaial wave wih wavelengh λ 0 and a emporal oscillaion wih frequency ν 0. Sanding waves produced by he sum of waves raveling in opposie direcions, shown as funcions of he spaial coordinae a five differen imes. The sum is a spaial wave whose ampliude oscillaes. 4.6 Anharmonic Traveling Waves, Dispersion Thus far he only raveling waves we have considered have been harmonic, i.e., consising of a single sinusoidal frequency. From he principle of Fourier analysis, an anharmonic raveling wave can be decomposed ino a sum of raveling harmonic wave componens, i.e., waves of generally differing
3 CHAPTER 4. TRAVELING WAVES ampliudes over a discree se of frequencies: X X y [z, ] = y n = A n cos [ n z ω n + φ n ], n=1 n=1 where A n, n,andω n are he ampliude, angular spaial frequency, and angular spaial frequency of he n h wave. Therefore, we can define he phase velociy of he n h wave as: (v φ ) n = ω n. n Now suppose ha a paricular anharmonic oscillaion is composed of wo harmonic componens y [x, ] =y 1 (x, y)+y [x, ]. If he wo componens have he same phase velociy, (v φ ) 1 =(v φ ),hen poins of consan phase on he wo waves move wih he same speed and mainain he same relaive phase. The shape of he resulan wave is invarian over ime. Such a wave is called nondispersive, because poins of consan phase on he componens do no separae over ime. Wha if he phase velociies are differen, i.e., if (v φ ) 1 6=(v φ )? In his case, poins of consan phase on he wo waves will move a differen velociies, and herefore he disance beween poins of consan phase will change as a funcion of posiion or ime. Therefore he shape of he superposiion wave will change as a funcion of ime; hese waves are dispersive. Noe ha he dispersion is a characerisic of he medium wihin which he waves ravel, and no of he waves hemselves. I is he medium ha deermines he velociies and hus wheher he waves ravel ogeher or if hey disperse wih ime and space. 4.7 Average Velociy and Modulaion (Group) Velociy We added wo raveling waves of differen frequencies and obained he same resul we saw when adding wo oscillaions: he sum of wo harmonic waves yields he produc of wo harmonic waves wih modulaion and average spaial and emporal frequencies. Using he new erms: avg, mod,ω avg, and ω mod,wecandefine he phase velociies of he average and modulaion waves: v avg ω ω1 +ω avg = 1+ = ω 1 + ω avg 1 + v mod ω mod mod = ω1 ω = ω 1 ω Thesewovelociieshavehesamemeaningashephasevelociyofhesinglewave,i.e., i is he velociy of a poin of consan phase of he average raveling wave frequency or of he modulaion wave frequency, or beas wave. The modulaion velociy is also commonly called he group velociy. 4.7.1 Example: Nondispersive Waves (v φ ) 1 =(v φ ) In a nondispersive medium, he phase velociy is consan over frequency (or wavelengh), i.e., (v φ ) 1 = ω 1 =(v φ ) = ω. Noe ha ω 1 6= ω and 6= only he raios are equal. Now find expressions for v mod and v avg. v avg = ω ω 1+ω avg = + = ω 1 + ω ω 1 ³1+ ω ω 1 = avg 1 + ³1+ Since ω 1 = ω for nondispersive waves = ω ω 1 = and: v avg = ω 1 1+ 1+ = ω 1 = v 1 = v = v avg. 1
4.8. DISPERSION RELATION FOR NONDISPERSIVE TRAVELING WAVES 33 Similarly for he velociy of he modulaion wave: v mod = ω mod mod = ω 1 ω Since ω 1 = ω for nondispersive waves, hen ω ω 1 = Noe also ha ω mod = ω 1 ω = ω 1 ω ω 1 ³1 ω ω 1 = ³1 and: v mod = ω 1 1 = ω 1 = v 1 = v = v mod = v avg. 1 1 = ω = dω = ω mod In a nondispersive medium, all waves (all spaial and emporal frequencies and all modulaion and average waves) ravel a he same velociy. 4.8 Dispersion Relaion for Nondispersive Traveling Waves Waves are nondispersive in some imporan physical cases: e.g., ligh propagaion in a vacuum and audible sound in air. Since ω =v φ, we can easily express he emporal angular frequency ω in erms of he angular wavenumber : ω = ω() =(v φ ) where v φ is consan, so ha ω. The expression of ω in erms of is called a dispersion relaion. We can plo ω [] vs., giving a sraigh line in he nondispersive case. Dispersion Relaion for Nondispersive Waves, Two ypes of wave wih differen velociies (v φ ) 1 > (v φ ). 4.9 Dispersive Traveling Waves The more general, more common, and more imporan case is ha of dispersive waves. Here, he phase velociy v φ = ω is no consan; v φ varies wih frequency. This is he normal sae of affairs
34 CHAPTER 4. TRAVELING WAVES for ligh raveling in a medium such as glass. The common specificaion of he phase velociy of ligh in medium is he refracive index n: n = c v φ where v φ is he phase velociy of ligh in he medium. In a dispersive medium, we can inerpre group velociy in anoher way: ω() = v φ = v mod = dω = d ( v φ) µ µ µ dvφ dvφ = v φ + = v φ +. In oher words, he group velociy is he sum of he phase velociy v φ and a erm proporional o, which is he change in phase velociy wih wavenumber: dv φ dv φ > 0= v mod > v φ dv φ < 0= v mod < v φ. As he phase velociy varies, he refracive index varies inversely (faser velociy = smaller index). Variaion of he refracive index implies a change in he refracive angle of ligh enering or exiing he medium (via Snell s law). Variaion of refracive index wih wavelengh implies ha differen frequencies will refrac a differen angles. This is he principle of he dispersing prism. 4.9.1 Example: Dispersive Traveling Waves Consider a medium wih dispersion relaion of he form of a power law: ω() =α where is a real number. The average and modulaion velociies are: v avg = ω = α( ) = α 1 v mod = dω = d α = α 1 = v avg. So if >1, henv mod >v avg,andif <1, v mod <v avg.thefirs relaion corresponds o anomalous dispersion and he second o normal dispersion. The dispersion relaion for normal dispersion is nonlinear and concave down, while ha for anomalous dispersion is nonlinear and concave up. Of course, for nondispersive waves he dispersion relaion is linear.
4.9. DISPERSIVE TRAVELING WAVES 35 Phase and modulaion (group) velociies on he dispersion plo ω []. The phase velociy a wavenumber is ω 1, while he velociy of he modulaion wave is he slope of he dispersion curve evaluaed a, v mod = dω. =1 In a medium wih normal dispersion, he refracive index nincreaseswih frequency ν(ω) and decreases wih wavelengh λ. Therefore n decreases as he wavenumber increases, i.e., dn > 0. Thus in real media, he average waves ravel faser han he modulaion. Refracive index n vs. wavelengh λ for several media, demonsraing he decrease in index (and hus increase in phase velociy) of ligh wih increasing wavelengh. 4.9. Propagaion of he Superposiion of Two Waves in Media wih Normal and Anomalous Dispersion Recall ha an anharmonic, hough periodic, oscillaion can be expressed as a sum of harmonic erms of differen frequencies, i.e., as a Fourier series. We can herefore find he effec of dispersion on an anharmonic raveling wave by decomposing i ino is Fourier series of harmonic erms and propagaing each separaely a is own velociy. The resulan is found by resumming he resuling
36 CHAPTER 4. TRAVELING WAVES componens. For example, if: f [z, ] =A 1 sin [ z ω 1 ]+ A 1 3 sin [ z 3ω 1 ]+ A 1 5 sin [ 3z 5ω 1 ] As we ve already seen, f [z, ] is he sum of he firs hree erms of a square wave. In he nondispersive case, =3 and 3 =5,andv 1 = v = v 3. The wave a he source is shown below: In dispersive media, v 1 6= v 6= v 3, and he relaive phase of he hree componens will vary as he wave ravels hrough space. Therefore he resulan wave will become increasingly disored. Normal Dispersion Snapshos of sums of wo raveling waves wih differen frequencies a five differen imes in a medium wih normal dispersion, so he wave wih longer wavelengh ravels faser han ha wih shorer wavelengh. The modulaion wave moves more slowly han he average wave.
4.9. DISPERSIVE TRAVELING WAVES 37 Anomalous Dispersion Snapshos of sums in a medium wih anomalous dispersion, so he wave wih shorer wavelengh ravels faser han ha wih longer wavelengh and he modulaion wave moves faser han he average wave.
38 CHAPTER 4. TRAVELING WAVES