Chapter 4

Traveling Waves

4.1 Introduction

Todate,wehaveconsideredoscillations, i.e., periodic, often harmonic, variations of a physical characteristic of a system. The system at one time is indistinguishable from the system observed at a later time if the time difference is an integral number of temporal periods. To maintain oscillatory behavior, the energy of the oscillator must remain within the system, i.e., there can be no losses of energy. We will now extend this picture to oscillations that travel from the source and thus transport energy away. Energy must be continually added to the system to maintain the oscillation and the transported energy can do work on other systems at a distance. Our first task is to mathematically describe a traveling harmonic wave, i.e., denote a y [t] that travels through space. A harmonic oscillation y(t)=A0 cos(ω0t), can be converted into a traveling wave by making the phase a function of both x and t in a very particular way. Consider the general case of an oscillatory function of space and time:

y [z,t]=A0 cos [Φ [z,t]] = A0 cos [z,t] . We want this oscillation to move through space, e.g., toward positive z.Inotherwords,ifapoint of constant phase onthewave(e.g., a peak of the cosine created at a particular time τ)isatapoint x0 in space at a time t0, the same point of constant phase must move to z1 >z0 at time t1 >t0.

“Snapshots” of sinusoidal wave at two different times t0 and t1 >t0, showing motion of the peak originally at the origin at t0. Thewaveistravelingtowardsz =+ at velocity vφ. The phase of the first wave at the origin is 0 radians, but that of the∞ second is negative. Since the wave at location z1 and time t1 has the same phase as the wave at location z0 and time

27 28 CHAPTER 4. TRAVELING WAVES

t0,wecansaythat:

Φ [z0,t0]=Φ [z1,t1]= cos [z0,t0]=cos[z1,t1]= y [z0,t0]=y [z1,t1] . ⇒ ⇒ In addition, for the wave to maintain its shape, the phase Φ [x, t] must be a linear function of x and t; otherwise the wave would compress or stretch out at different locations in space or time. Therefore:

Φ [z,t]=αz + βt

= αz0 + βt0 = αz1 + βt1. ⇒

As discussed, if t1 >t0 = z1 >z0 (i.e., wave moves toward z =+ ), then α and β must have opposite algebraic signs: ⇒ ∞ Φ [z,t]= α z β t | | − | | By dimensional analysis, we know that α z β t has “dimensions” of angle [radians]. We have | | − | | already identified β = ω0, the angular frequency of the oscillation. Similarly, if [z]=mm must have dimensions of radians/mm, i.e., α tells how many radians of oscillation exist per unit length — the angular spatial frequency of the wave, commonly denoted by k:

y+ [z,t]=A0 cos [kz ω0t] traveling harmonic wave toward z =+ − − ∞ By identical analysis, we can derive the equation for a harmonic wave moving toward x = −∞

y [z,t]=A0 cos [kz + ω0t] traveling harmonic wave toward z = − − −∞ The waves are functions of both space and time, i.e., three dimensions [z,y,t] are needed to portray them. Generally we display y either as a function of z or fixed t, or as a function of t for fixed z:

4.1.1 2-D Plot of 1-D Traveling Wave The 1-D traveling wave is a function of two variables: the position z and the time t,andsomay be graphed on axes with these labels. An example is shown in the figure, where z is plotted on the horizontal axis and t on the vertical axis. In this case, the point at the origin at t =0has a phase of 0 radians. That point moves in the positive z direction with increasing time, and so is a wave of the form y [z,t]=cos[k0z ω0t] − The points with the same phase of 0 radians at later times are positioned along the line shown. The ∆z velocity of this point of constant phase is ∆t , and thus is the reciprocal of the slope of this line. 4.2. NOTATION AND DIMENSIONS FOR WAVES IN A MEDIUM 29 4.2 Notation and Dimensions for Waves in a Medium

Trigonometric Notation:

y [z,t] y0 = A cos Φ [z,t] = A cos(kz ωt + φ ) − { } ± 0 Complex Notation: iΦ[z,t] i(kz ωt+φ ) y [z,t]=Ae =Re Ae ± 0 n o y = position of the characteristic of the medium, e.g., [y]=angle, voltage, ... ; y0 = equilibrium value of the characteristic; A = amplitude of the wave, i.e., maximum displacement from equilibrium, [A]=[y]; z,t = spatial and temporal coordinates, [z]=length (e.g., mm), [t]= s; 1 2π T = period of the wave, [T ]=s,T = ν = ω ; λ = wavelength, [λ]= mm 2π radians ω = angular temporal frequency of the wave, ω = T , [ω]= s ; 2π radians k = angular spatial frequency of the wave, k = λ , [k]= mm ; cycles ω ν = temporal frequency of the wave, [ν]= s = Hertz [Hz], ν = 2π ; Φ = phase angle of the wave, [Φ]=radians, (in this case, Φ is linear in time and space);

φ0 = initial phase of the wave, i.e., phase angle @ t =0,z =0, [φ0]=[Φ]=radians. 1 1 σ = wavenumber, σ = λ , the number of wavelengths per unit length, [σ]= mm− . Relations between the phase and the temporal frequencies

∂Φ ω = − ∂t ω 1 ∂Φ ν = = 2π −2π ∂t 4.3 Velocity of Traveling Waves

The phase velocity vφ of a wave is the speed of travel of a point of constant phase. A definition for phase velocity can be derived by dimensional analysis: [vφ]= mmper s; [ω]=radians per s;[k]= radians per mm: ω radians per second radian- mm mm = = = = ⇒ k radians per mm radian- s s h i Slightly more rigorously, we can find the phase velocity of a wave by taking derivatives of the equation for the wave:

y [z,t]=A cos [kz ωt + φ ] , − 0 ∂y = ( ω)A sin [kz ωt + φ ]=+Aω sin [kz ωt + φ ] , ∂t − − · − 0 · − 0 ∂y = (k)A sin [kz ωt + φ ]= Ak sin [kz ωt + φ ] ∂z − · − 0 − · − 0 ∂y ∂z ∂t ω ω vφ = = = = , ∂t ³ ∂y ´ − k k ¯ ¯ ∂z ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ³ ´ ¯ ¯ or by considering the point¯ of constant¯ phase b radians:

b ω ω b kz ωt = b = z = + = b0 + tb0 is a new constant − ⇒ k kt k ≡ k µ ¶ ³ ´ Consider the positions z1 and z2 of the same point of constant phase at different times t1 and t2: 30 CHAPTER 4. TRAVELING WAVES

ω z = b + t 1 0 k 1 ³ω ´ z = b + t 2 0 k 2 ³ ´ ω ω = z1 z2 = ∆z = (t1 t2)= ∆t ⇒ − k − k ∆x ω ³ ´ ³ ´ vφ = = vφ. ≡ ∆t k

4.4 Superposition of Traveling Waves

Consider the superposition of two traveling waves with the same amplitude, different phase velocities, and different frequencies:

y1 [z,t]=A cos [k1z ω1t] − y2 [z,t]=A cos [k2z ω2t] . −

We can use the same derivation developed for oscillations by defining a new frequency for both:

k1z Ω1 ω1 ≡ t − k2z Ω2 ω2 ≡ t −

y [z,t]=y1 [z,t]+y2 [z,t]=A cos [k1z ω1t]+cos[k2z ω2t] { − − } k1z k2z = A cos ω1 t +cos ω2 t t − t − ½ ∙µ ¶ ¸ ∙µ ¶ ¸¾ = A cos [Ω1t]+cos[Ω2t] { } Ω1 + Ω2 Ω1 Ω2 =2A cos t cos − t 2 2 ∙µ ¶ ¸ ∙µ ¶ ¸ just as before.By evaluating the sum and difference frequencies, we obtain:

Ω1 + Ω2 k1z k2z t k1 + k2 ω1 + ω2 t = ω1 + ω2 = z t kavgz ωavgt 2 t − t − 2 2 − 2 ≡ − µ ¶ µ ¶ µ ¶ µ ¶ k1 + k2 ω1 + ω2 where kavg ,ωavg ≡ 2 ≡ 2

Ω1 Ω2 k1z k2z t k1 k2 ω1 ω2 − t = ω1 + ω2 = − z − t kmodz wmodt 2 t − − t 2 2 − 2 ≡ − µ ¶ µ ¶ k1 k2 ω1 ω2 where kmod − ,ωmod − ≡ 2 ≡ 2 4.5. STANDING WAVES 31 4.5 Standing Waves

Consider the superposition of two waves with the same amplitude A0, temporal frequency ν0,and wavelength λ0, but that are traveling in opposite directions:

f1 [z,t]+f2 [z,t]=A0 cos [k0z ω0t]+A0 cos [k0z + ω0t] − k0z ω0t k0z + ω0t k0z ω0t k0z + ω0t =2A0 cos − + cos − 2 2 · 2 − 2 ∙ ¸ ∙ ¸ k0z + k0z ω0t + ω0t k0z k0z ω0t ω0t =2A0 cos + − cos − + − − 2 2 · 2 2 ∙ ¸ ∙ ¸ =2A0 cos [k0z] cos [ ω0t] · − =2A0 cos [k0z] cos [ω0t] , because cos [ θ]=+cos[+θ] · − z =2A0 cos 2π cos [2πν0t] λ · ∙ 0 ¸

This is the product of a spatial wave with wavelength λ0 and a temporal oscillation with frequency ν0.

Standing waves produced by the sum of waves traveling in opposite directions, shown as functions of the spatial coordinate at five different times. The sum is a spatial wave whose amplitude oscillates.

4.6 Anharmonic Traveling Waves, Dispersion

Thus far the only traveling waves we have considered have been harmonic, i.e., consisting of a single sinusoidal frequency. From the principle of Fourier analysis, an anharmonic traveling wave can be decomposed into a sum of traveling harmonic wave components, i.e., waves of generally differing 32 CHAPTER 4. TRAVELING WAVES amplitudes over a discrete set of frequencies:

∞ ∞ y [z,t]= yn = An cos [knz ωnt + φn] , n=1 n=1 − X X where An,kn,andωn are the amplitude, angular spatial frequency, and angular spatial frequency of the nth wave. Therefore, we can define the phase velocity of the nth wave as:

ωn (vφ)n = . kn Now suppose that a particular anharmonic oscillation is composed of two harmonic components y [x, t]=y1(x, y)+y2 [x, t]. If the two components have the same phase velocity, (vφ)1 =(vφ)2,then points of constant phase on the two waves move with the same speed and maintain the same relative phase. The shape of the resultant wave is invariant over time. Such a wave is called nondispersive, because points of constant phase on the components do not separate over time.

What if the phase velocities are different, i.e., if (vφ)1 =(vφ)2? In this case, points of constant phase on the two waves will move at different velocities, and6 therefore the distance between points of constant phase will change as a function of position or time. Therefore the shape of the superposition wave will change as a function of time; these waves are dispersive. Note that the dispersion is a characteristic of the medium within which the waves travel, and not of the waves themselves. It is the medium that determines the velocities and thus whether the waves travel together or if they disperse with time and space.

4.7 Average Velocity and Modulation (Group) Velocity

We added two traveling waves of different frequencies and obtained the same result we saw when adding two oscillations: the sum of two harmonic waves yields the product of two harmonic waves with modulation and average spatial and temporal frequencies. Using the new terms: kavg,kmod,ωavg, and ωmod,wecandefine the phase velocities of the average and modulation waves:

ω1+ω2 ωavg 2 ω1 + ω2 vavg = = ≡ k k1+k2 k + k avg ¡ 2 ¢ 1 2 ω1 ω2 ωmod ¡ −2 ¢ ω1 ω2 vmod = k k = − ≡ kmod 2 k1 k2 ¡ −2 ¢ − Thesetwovelocitieshavethesamemeaningasthephasevelocityofthesinglewave,¡ ¢ i.e., it is the velocity of a point of constant phase of the average traveling wave frequency or of the modulation wave frequency, or beats wave. The modulation velocity is also commonly called the group velocity.

4.7.1 Example: Nondispersive Waves (vφ)1 =(vφ)2 In a nondispersive medium, the phase velocity is constant over frequency (or wavelength), i.e.,

ω1 ω2 (vφ)1 = =(vφ)2 = . k1 k2

Note that ω1 = ω2 and k1 = k2 — only the ratios are equal. Now find expressions for vmod and vavg. 6 6

ω2 ω1+ω2 ω1 1+ ωavg 2 ω1 + ω2 ω1 vavg = = = = k1+k2 k k + k ³ k2 ´ avg 2 1 2 k1 1+ k1 ³ ´ Since ω1 = ω2 for nondispersive waves = ω2 = k2 and: k1 k2 ⇒ ω1 k1 1+ k2 ω1 k1 ω1 vavg = k = = v1 = v2 = vavg . k1 1+ 2 k1 k1 4.8. DISPERSION RELATION FOR NONDISPERSIVE TRAVELING WAVES 33

Similarly for the velocity of the modulation wave:

ω2 ω1 ω2 ω1 1 ωmod −2 ω1 ω2 − ω1 vmod = = = − = k1 k2 kmod k1 k2 ³ k2 ´ −2 k1 1 − − k1 ³ ´ Since ω1 = ω2 for nondispersive waves, then ω2 = k2 and: k1 k2 ω1 k1

ω 1 k2 ω 1 − k1 1 vmod = k = = v1 = v2 = vmod = vavg . k1 1 2 k1 − k1

ω1 ω2 ∆ω dω Note also that ωmod = − = = = ωmod k1 k2 ∆k dk − ⇒ In a nondispersive medium, all waves (all spatial and temporal frequencies and all modulation and average waves) travel at the same velocity.

4.8 Dispersion Relation for Nondispersive Traveling Waves

Waves are nondispersive in some important physical cases: e.g., light propagation in a vacuum and ω audible sound in air. Since k =vφ, we can easily express the temporal angular frequency ω in terms of the angular wavenumber k:

ω = ω(k)=(vφ) k where vφ is constant, so that ω k. · ∝ The expression of ω in terms of k is called a dispersion relation. We can plot ω [k] vs. k, giving a straight line in the nondispersive case.

Dispersion Relation for Nondispersive Waves, Two types of wave with different velocities

(vφ)1 > (vφ)2.

4.9 Dispersive Traveling Waves

The more general, more common, and more important case is that of dispersive waves. Here, the ω phase velocity vφ = k is not constant; vφ varies with frequency. This is the normal state of affairs 34 CHAPTER 4. TRAVELING WAVES for light traveling in a medium such as glass. The common specification of the phase velocity of light in medium is the refractive index n: c n = vφ where vφ is the phase velocity of light in the medium. In a dispersive medium, we can interpret group velocity in another way:

ω(k)=k vφ · dω d = vmod = = (k vφ) ⇒ dk dk · dk dvφ dvφ = vφ + k = vφ + k . dk · dk dk µ ¶ µ ¶ µ ¶

In other words, the group velocity is the sum of the phase velocity vφ and a term proportional to dvφ dk , which is the change in phase velocity with wavenumber:

dvφ > 0= vmod > vφ dk ⇒ dvφ < 0= vmod < vφ. dk ⇒ As the phase velocity varies, the refractive index varies inversely (faster velocity = smaller index). Variation of the refractive index implies a change in the refractive angle of light entering⇒ or exiting the medium (via Snell’s law). Variation of refractive index with wavelength implies that different frequencies will refract at different angles. This is the principle of the dispersing prism.

4.9.1 Example: Dispersive Traveling Waves Consider a medium with dispersion relation of the form of a power law:

ω(k)=αk where is a real number. The average and modulation velocities are:

ω α(k) v = = = αk 1 avg k k − dω d 1 vmod = = αk = αk − = vavg. dk dk · ¡ ¢ ¡ ¢ So if >1,thenvmod >vavg,andif<1,vmod

Phase and modulation (group) velocities on the dispersion plot ω [k]. The phase velocity at ω1 wavenumber k1 is , while the velocity of the modulation wave is the slope of the dispersion curve k1 dω evaluated at k1, vmod = . dk k=k1 In a medium with normal dispersion, the refractive index nincreaseswith frequency ν(ω) and de- ¯ dn creases with wavelength λ. Therefore n decreases as the wavenumber¯ k increases, i.e., dk > 0.Thus in real media, the average waves travel faster than the modulation.

Refractive index n vs. wavelength λ for several media, demonstrating the decrease in index (and thus increase in phase velocity) of light with increasing wavelength.

4.9.2 Propagation of the Superposition of Two Waves in Media with Nor- mal and Anomalous Dispersion

Recall that an anharmonic, though periodic, oscillation can be expressed as a sum of harmonic terms of different frequencies, i.e., as a Fourier series. We can therefore find the effect of dispersion on an anharmonic traveling wave by decomposing it into its Fourier series of harmonic terms and propagating each separately at its own velocity. The resultant is found by resumming the resulting 36 CHAPTER 4. TRAVELING WAVES components. For example, if:

A1 A1 f [z,t]=A1 sin [k1z ω1t]+ sin [k2z 3ω1t]+ sin [k3z 5ω1t] − 3 − 5 −

As we’ve already seen, f [z,t] is the sum of the first three terms of a square wave. In the nondispersive case, k2 =3k1 and k3 =5k1,andv1 = v2 = v3. The wave at the source is shown below:

In dispersive media, v1 = v2 = v3, and the relative phase of the three components will vary as the wave travels through space.6 Therefore6 the resultant wave will become increasingly distorted.

Normal Dispersion

“Snapshots” of sums of two traveling waves with different frequencies at five different times in a medium with normal dispersion, so the wave with longer wavelength travels faster than that with shorter wavelength. The “modulation” wave moves more slowly than the “average” wave. 4.9. DISPERSIVE TRAVELING WAVES 37

Anomalous Dispersion

“Snapshots” of sums in a medium with anomalous dispersion, so the wave with shorter wavelength travels faster than that with longer wavelength and the “modulation” wave moves faster than the “average” wave. 38 CHAPTER 4. TRAVELING WAVES