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Traveling Waves 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].
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