(Wave Pulses and Fourier Transforms) Lecture...
Transcript of (Wave Pulses and Fourier Transforms) Lecture...
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Lecture 8(Wave Pulses and Fourier Transforms)
Physics 2310-01 Spring 2020
Douglas Fields
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Wave Pulses
• So, if our solution to the wave equation is:
• How do we get a wave pulse???
• Using sinA + sinB = 2cos[(A-B)/2]sin[(A+B)/2]
Since dk << k and dω << ω
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Wave Pulses (Beats)
•
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Wave Pulses
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Bounded Problem
• For a bounded problem, there are discrete (quantized) normal modes. Let’s say we have some mix of those normal modes:
• How do we find the values a1, a
2, …?
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Bounded Problem
• Let’s multiply both sides by one of the normal mode wave functions:
• Now, let’s integrate over all space (or time) on both sides of the equation:
• But the integrals on the right are either 0 (if n not equal to the index), or 1 (if it is). So,
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Wave Pulses
k and x are conjugate variables
In the case of an unbounded problem, with continuous values of wave numbers:
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Wave Pulses
ω and t are conjugate variables
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Phasors
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Phasors
• One can represent sin(ωt) or cos(ωt) as a x- or y-projection of a phasor which rotates at an angular velocity ω.
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Phasors
• Let’s just focus on cos(ωt).• The phasor is rotating with
angular velocity ω, so the angle w.r.t. the x-axis is just ωt.
• Then, the x-component of the phasor is just cos(ωt).
• Or, the y-component of the phasor is just sin(ωt).
• Why is this useful?
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Phasors
• Let’s add two waves, in this case, two waves with the same frequency, but with different relative phases.
• Note that the each wave can be represented by a phasor, and that the sum of the waves are represented by the vector sum of the phasors!
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Phasors
• But this still doesn’t seem too useful…
• What about if the phasors rotate at different angular velocities?
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Square Wave
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Saw-tooth Wave
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• https://youtu.be/QVuU2YCwHjw
• https://www.youtube.com/watch?v=qS4H6PEcCCA
• And continue down the rabbit hole.