Chapter 7: Superposition


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1 Chapter 7: Superposition Introduction The wave equation is linear, that is if ψ 1 (x, t) ψ 2 (x, t) satisfy the wave equation, then so does ψ(x, t) = ψ 1 (x, t) + ψ 2 (x, t). This suggests the Principle of Superposition : the resultant disturbance at any point in a medium is the algebraic sum of the separate constituent waves. The Addition of Waves of the Same Frequency A solution of the wave equation can be written as E(x, t) = E sin[ωt (kx + ɛ)], in which E is the amplitude of the harmonic disturbance propogating along the x axis. This is Suppose there are two such waves, E(x, t) = E sin[ωt + α(x, ɛ]. E 1 = E 1 sin(ωt + α 1 ), E 2 = E 2 sin(ωt + α 2 ), each with the same frequency speed, coexisting in space. Then E = E 1 + E 2 is such that E = E sin(ωt + α), E 2 = E2 1 + E E 1E 2 cos(α 2 α 1 ), tanα = E 1sinα 1 + E 2 sinα2 E 1 cosα 1 + E 2 cosα 2. That is the composite wave is harmonic of the same frequency as the constituents, although its amplitude phase are different. This phase difference between these two waves may arise from a difference in path lengths is δ, δ = 2π λ (x 1 x 2 ) + (ɛ 1 ɛ 2 ). 1
2 If the wave emitters are initially in phase, then ɛ 1 = ɛ 2, because n = c/v = λ /λ δ = 2π λ n(x 1 x 2 ). The quantity n(x 1 x 2 ) is known as the optical path difference or OPD = Λ. Thus δ = k Λ. Waves for which ɛ 1 epsilon 2 is constant are said to be coherent. When the OPD is equal to, ±2π, ±4π,..., the resultant amplitude is a maximum because of constructive interference. When the OP D is equal to ±π, ±3π,..., the resultant amplitude is a minimum because of destructive interference. Superposition of Many Waves Consider the addition of N waves such that Then E 2 = E = i=1 i=1 E i cos(α i ± ωt). E = E cos(α ± ωt), E 2 i + 2 tanα = N j i i=1 cos(α i α j ), i=n i=1 E isinα i i=n. i=1 E i cosα i Consider N atomic emitters compising an ordinary light source. Each photon emitted is associated with a short duration oscillatory wave pulse. That is each atom is an independent source of wave trains with each wave train lasting about 1 to 1 ns. After each wave train, the atom will emit a new wavetrain with a totally rom phase completely independent of the phase of the previous wavetrains of the phase of those emitters around it. The resultant flux density asrising from N independent sources having rom, rapidly varying phases is given by N times the flux density of any one sourca E 2 = NE2 1. If sources are coherent, then E 2 = N2 E 2 1. Coherent sources generally alter the spatial distribution of energy but not the total energy. 2
3 Sting Waves Previousy considered two harmonic waves propogating in the same direction  now consider two waves propogating in te same direction. COnsider E l = E l sin(kx + ωt + ɛ l ) E r = E rsin(kx ωt + ɛ r ). The composite disturbance is E = E l + E r, with some trignometric manipulation, E(x, t) = 2E l sinkxcosωt, without too much loss of generality, we have assumed E l = E r. This is the equation for a sting or stationary wave. Its profile does not move through space. At certain points, nodes, namely x =, λ/2, λ, 3λ/2,...the disturbance is zero at all times. Halfway between the nodes, that is, at x = λ/4, 3λ/4,..., the amplitude has a maximum value of ±2E ol : these are the anitnodes. The addition of Waves of Different Frequency Consider the addition of two waves travelling in the same direction but with different frequency. Let E 1 = E 1 cos(k 1 x ω 1 t), E 2 = E 2 cos(k 2 x ω 2 t), k 1 > k 2 ω 1 > ω 2. The net composite wave can be formulated as E = 2E 1 cos( 1 2 [(k 1 +k 2 )x (ω 1 +ω 2 )t]) cos 1 2 [(k 1 k 2 ) (ω 1 ω 2 )t]. Now define ω k as the average angular frequency average propogation number. Similarly the quantities ω m k m are the modulation frequency modulation propogation number respectively. That is ω = 1 2 (ω 1 + ω 2 ), ω m = 1 2 (ω 1 ω 2 ), Then k = 1 2 (k 1 + k 2 ), k m = 1 2 (k 1 k 2 ). E = 2E l cos(k m x ω m t)cos( kx ωt). 3
4 The total disturbance may be viewed as a travelling wave of frequency ω having a timevarying or modulated amplitude E (x, t) such that, E(x, t) = E (x, t)cos( k ωt), E (x, t) = 2E l cos(k m x ω m t). If ω 1 ω 2, then ω is much greater than ω m E (x, t) will change slowly. It can be shown that E 2 (x, t) = 2E2 1 [1 + cos(2k mx 2ω m t)]. Thus E 2(x, t) oscillates about a value of 2E2 1 with an angular frequency of (ω 1 ω 2 : the beat frequency. Thus E, the amplitude of the resulting wave varies at the modulation frequency but E 2 varies at twice that  the beat frequency. Group Velocity When a number of differentfrequency harmonic waves superimpose to form a composite disturbance, the resulting modulation envelope will travel at a speed different from that of the constituent waves. Group velocity verses phase velocity. The phase velocity in this situation is v = ω/ k, that is the phase velocity of the carrier wave. The group velocity relates to the rate at which the modulation envelope advances i.e. v g = ω m /k m. If v 1 = v 2 then v g = v. Phase velocity can be greater than c, but phase velocity doesnt carry information or energy. Group velocity can also be greater than c. In either case, information or energy cannot be transmitted at a speed greater than c. Anharmonic Periodic Waves Superposition of harmonic functions having different amplitudes frequencies produces an anharmonic wave. Fourier Series: A function f(x) can be synthesized by a sum of harmonic functions whose wavelengths are integral submultiples of λ (ie. λ, λ/2, λ/3...). 4
5 Any periodic function, f(x),, knowing f(x), f(x) = A 2 + A m cosmkx + B m sinmkx m=1 m=1 A m = 2 λ B m = 2 λ λ λ f(x)cosmkxdx, f(x)sinmkxdx. ω is the fundamental. Subsequent omegas like 2ω, 3ω,... are the harmonics. Fourier series so represented has the form f(x) = C + C 1 cos( 2π λ x + ɛ 1) + C 2 cos( 2π λ x + ɛ 2) +... Can extend this to nonperiodic waves abd write f(x) = 1 π [ A(k)coskxdk + A(k) = B(k) = f(x)coskxdx, f(x)sinkxdx. B(k)sinkxdk], A(k), B(k) are the contributions to angular spatial frequency between k, k + dk. The frequency bwidth is k or ω. 5
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