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Phys 2310 Wed. Sept. 13, 2017Today’s Topics

– Chapter 15: The Superposition of Waves• Interference

– Methods for Adding Waves• Boundary Conditions

– Standing Waves & Normal Modes

– Supplementary Material: • Fourier Analysis

• Reading for Next Time

2

Homework this Week

SZ Chapter 15: #22, 32, 35, 39, 40, 42, 46, 47Due Monday Sept. 18

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SZ Chapter15: Superposition Principle• Concept of Superposition

– The addition of two waves is also a wave that satisfies the wave equation

– Result is the sum of the function (disturbance) at each point in space

• Superposition Examples– Consider two waves: A1 = 1.0, A2 =

0.9– Wave functions add according to

their amplitudes and phases– Note: (a) – addition (constructive

interference (b) - almost cancellation (destructive interference)

• Mathematical Techniques– Write some software that

evaluates waves at each point (x) and each time and adds them.

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SZ Chapter 15: Superposition Principle• Superposition Examples cont.

– Consider two waves: A1 = 1.0, A2 = 1.0

– Now waves have same amplitude and freq. but different phases

– Wave functions add according to their amplitudes and phases as before.• That is, we could

numerically compute the value of the wave at each value of kx and then add them together or we could make use of law of cosines since they have the same phase (next slide)

• We Need a Better Technique for Adding Amplitudes and Phases of Waves

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SZ Chapter 15: Graphical Addition of Waves

• Law of Cosines:– A2 = a1

2 +a22 +2a1a2cos(a1 – a2)

• Polar coordinates can be used to graphically add amplitudes and phases.– Think in terms of vectors

(phasors):A<f or sometimes A<qMultiple waves:a1<d1, a2<d2, etc.

– Constructive interference:a1 opposite a2

– Constructive interference:a1 parallel to a2– General interference:

• Compute components and add. Compute new amplitude and phase.

See diagram at right.

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Suppl: Group vs. Phase Velocity

• If two waves have different frequency the case is more complex

• For two harmonic waves with slightly different frequency:

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SZ Chapter 15: Addition with Same Frequency

• Algebraic Method– Recall that light waves are not affected when the beams cross.– The disturbance (E-field amplitude) just adds locally.

Consider two waves:E1 = E01 sin(ωt +α1) and E2 = E02 sin(ωt +α2 )where α = −(kx +ε) is the spatial portion of the phase.Adding the two waves (trig. identities):E = E01(sinωt cosα1 + cosωt sinα1)+E02 (sinωt cosα2 + cosωt sinα2 )and so when the spatial and time-dependent portions are separated:E = (E01 cosα1 +E02 cosα2 )sinωt + (E01 sinα1 +E02 sinα2 )cosωtNow note that the terms in () are constant in time. So we define:E0 cosα = E01 cosα1 +E02 cosα2 and E0 sinα = E01 sinα1 +E02 sinα2

Squaring and adding to get the intensity:E0

2 = E012 +E02

2 + 2E01E02 cos(α2 −α1) (note interference term)Dividing to get the phase:

tanα = E01 sinα1 +E02 sinα2

E01 cosα1 +E02 cosα2

thus:

E = E0 cosα sinωt +E0 sinα cosωtNote the interference term varies with OPL:δ = (kx1 +ε1)− (kx2 +ε2 ) or:

δ =2πλ0

n(x1 − x2 ) (where n(x1 − x2 ) is the OPL)

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SZ Chapter 15: Standing Waves

• Standing Waves: Special Case of Two Waves with the Same Frequency– Boundary Conditions: Wave is Fixed at Both Ends with Phase

Change Upon Reflection (see Figure 15.19)y1(x, t) = −Acos(kx +ωt) and y2 (x, t) = Acos(kx −ωt) so adding:y(x, y) = A[−cos(kx +ωt)+ cos(kx −ωt)] and using the trig. identity:cos(a± b) = cosacosb sinasinb we get:y(x, t) = (2Asinkx)sinωt (note the modulated amplitude)

Note that since both have the same frequency the result will also have the same frequency, only the amplitude changes with x position.

Nodes occur where the amplitude goes to 0. Requiring:

L = n λ2

for a string fixed at both ends (n is any integer). Solving for λ:

λn =2Ln

The corresponding frequencies (f = v / λ) are:

fn =v

2L with the wave velocity depending on mass/length and tension.

Thus the string can also vibrate in harmonics:yn (x, t) = AsinknxsinωtString instruments are designed to vibrate in the fundamental mode and the first few harmonics since pure tones sound like old-school video game music.

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SZ Chapter 15: Addition with Same Frequency

• Standing Waves– Oscillating string constrained at both ends has A = 0 at endpoints.– Weiner showed that the E field at the surface of a mirror must be

zero, just like the E&M theory of a conductor.– Reflected waves are like standing waves with a phase change of p.

In order to satisfy this boundary condition consider both waves:E = EI +ER = E0 I[sin(kx +ωt)+ sin(kx −ωt)]And since:

sinα + sinβ = 2sin 12

(α +β)cos 12

(α +β) we have:

E(x, t) = 2E0 I sinkxcosωt (standing wave)Note that one part oscillates in time but the amplitude or "envelope" doesn't, hence the name. This results in nodes and anti-nodes.In addition, Weiner's experiment showed that reflected light can interfere.This technique can be used to measure the wavelength of light from the OPD.

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Supplementary: 2-d Harmonic Oscillator

• Now Consider a 2-d Harmonic Oscillator

x = A1 cos(ω1t +α1) and y = A2 cos(ω2t +α2 )Since the cos() functions vary between +1 and -1 the x displacement is confined within ±A1 and the y displacement is confined within ±A2. Thus any point:P(x, y, t) will be confined within the corresponding rectangle.

If ω1 ≠ω2 or ω1

ω2

≠ ratio of whole numbers then the pattern

will never repeat and eventually fill in the entire region. For the special case where ω1 =ω2 :x = A1 cos(ωt) and y = A2 cos(ωt +δ) consider a few cases:1:δ = 0. In this case: x = A1 cos(ωt) and y = A2 cos(ωt) and so:

y = A2

A1

x and so P traces out a diagonal line.

2 :δ = π / 2 and so now we have:x = A1 cos(ωt) and y = A2 cos(ωt +δ) = −A2 sin(ωt) and since:sin2ωt + cos2ωt =1 we must have:x2

A12 +

y2

A22 =1 which is the equation of an ellipse with axes along

the x and y axes. Note further that as t begins to increase x decreases while y goes negative. Thus the vector between the origin and P rotates clockwise.

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Supplementary: 2-d Harmonic Oscillator

• 2-d Harmonic Oscillator Continued

3: If ω1

ω2

is the ratio of whole numbers we see complex but repeating pattern.

Consider the case where ω2 = 2ω1. During one cycle of ω2 we go through only one half cycle of ω1. The result is the family of curves shown below depending on the phase offset between the two wavefunctions.

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SZ Chapter 15: Addition of Many Waves with Same Frequency

• Superposition of Many Waves– Now consider that case of many waves:

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Suppl.: Addition of Many Waves with Same Frequency

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Suppl.: Addition of Many Waves with Same Frequency

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SZ Chapter 15: Addition with Different Frequency

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French: Anharmonic Periodic Waves• Fourier Series

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French: Anharmonic Periodic Waves

• Fourier Series– Knowing a function, f(x), we can

compute the coefficients. The number of terms depends on the precision required.

– Note how only 6 terms (in sine) can reproduce the sawtooth function rather well since it is odd.

– Adding more terms is easy with a computer (even Excel).

– A few hundred terms may be necessary to accurately reproduce both the slopes and the discontinuities. The number required depends on the precision required.

– Note the decreasing amplitude of the higher-order terms in the frequency spectrum (typical).

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French: Anharmonic Periodic Waves

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French: Non-periodic Waves• Fourier Integrals

– If we accept that we can model any function with an infinite series of sines and cosines it seems reasonable to generalize Fourier series to Fourier Integrals. See figure 7.21 for how In this case we have the Fourier Transform:

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Supplement: Pulses and Wave Packets• Square Pulse

– Fourier transform of a square pulse is called the Sinc function

• Finite Wavetrain– A given Fourier component goes on

forever– A finite wavetrain requires additional

Fourier modes. • Think of the wave being modulated by a

square pulse.• Thus a finite wavetrain (pulse) can be

thought of as a wave packet.• A the wavetrain becomes long its

frequency spectrum shrinks and vice versa.

• Coherence Length– Any real E-M wave is not absolutely

monochomatic but has a natural width.– There must be a coherence length and a

coherence time as a result.• Namely: Dlc = cDtc

– Coherence length and time are a measure of space and time over which the wave has an approximately constant wavelength or frequency.

– Note that white light (large bandwidth) can then be understood as the superposition of large numbers of monochromatic waves (many Fourier modes)

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Supplement: Non-periodic Waves• Discrete Fourier Transform

– Often our signal consists of a set of discrete points (e.g. from a digital detector)

– The Fourier integral is approximated by a summation• The resulting transform can then be filtered to remove unwanted frequencies

(e.g. interference) and then transformed back.– This techniques can be further generalized to muti-dimensional data, such

as a 2-d images• See spatial filtering examples on the web.

– Two excellent references:• The Fourier Transform and Its Application (Bracewell)• A Student’s Guide to Fourier Transforms: with Applications in Physics and

Engineering (James)

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Summary and Key Concepts

• Waves Add Together via Superposition• Phasors Provide a Means for Adding

Amplitudes and Phases• Simple Cases for Just Two Waves• Fourier Analysis for Adding Many Waves– We Can Model Any Wave as a Series of Sines and

Cosines (Basis Set)• Finite Wave Packets Require Frequency Range

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Reading this Week

By Friday:Finish Ch. 15: Addition of Waves of the Same

Frequency, Addition of Waves of Different Frequency

Supplement: Anharmonic Periodic Waves and Fourier Techniques

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Homework this Week

SZ Chapter 15: #22, 32, 35, 39, 40, 42, 46, 47Due Monday Sept. 18