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Wave Interference Fundamentals: Concepts and Equations to Understand Waves' Behavior

Explore interference in waves, phasors, wave phenomena, and coherence. Understand how waves interfere at specific locations and the mathematical methods to analyze wave interference. Dive into wave equations and graphical representations to study wave addition and phase differences. Improve your understanding of wave behaviors using trigonometry and interference concepts.

kaitlin-callery
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Wave Interference Fundamentals: Concepts and Equations to Understand Waves' Behavior

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  1. Agenda • This Week • Interference in waves • Today • Phasors • Tuesday • Lab, Quiz on Lenses/Mirrors/Geo Optics • Wed&Fri • Finish Chapter 35

  2. Interference • Wave Phenomenon • Speakers • Fundamentals • Multiple “sources” • Correlated – “Coherence” • Waves interfere at some location • Usual: Spatial without time dependence

  3. Waves from 2 Point Sources • Waves Interfere with each other • Notice Lines – Time independent

  4. Waves from 2 Point Sources • Path dependence for each source • Set up mathematical method - waves

  5. Set Up Interference • Single Light Source • Laser – Coherent (phase set) Intersection of Light Bottom Path travels L Top Path travels L2 = L+ D

  6. At Intersection Wave Upper Path Lower Path E is electric field amplitude Before we continue…. What is “k”? What is w?

  7. Wave Parameters Wave Equation: Generic • E is amplitude: Same for both paths • & k depend on frequency & wavelength: Same for both paths f is phase offset: Also depends on light source Same light source, same f.

  8. Wave dependencies Wave Equation: Generic w & k depend on frequency & wavelength: Same for both paths Time is just a measurement of time: As in it’s 2:15 “x” is a measure of how far wave has traveled in space Time is same for both paths, x is different

  9. Wave dependencies Wave Equation: Generic w & k depend on frequency & wavelength: Same for both paths When time t = T (period) has passed, then one cycle has occurred. Wave equation looks same for any period T: or…. This is true as when time T passes, 2p radians proceed for wave

  10. Wave AppearanceRepeats on 2pTime or distance 2p E

  11. Wave dependencies Wave Equation: Generic Equivalence over period

  12. Wave dependencies (k) Wave Equation: Generic w & k depend on frequency & wavelength: Same for both paths When time x = l (wavelength) has passed, then one cycle has occurred. Wave equation looks same for any distance l: or…. This is true as when distance l is traveled, 2p radians proceed for wave k = 2p/l (like w = 2p/T) k utilized often in wave mechanics (i.e. quantum)

  13. Interference View • Single Light Source • Laser – Coherent (phase set) Intersection of Light Bottom Path travels L Top Path travels L2 = L+ D

  14. At Intersection Wave Upper Path Lower Path For electromagnetic waves: Amplitude  E (B as well) Both waves, time is same. t refers to a “time” not how long wave has traveled Coherent means phase is set by f. Single light source, so f same for each Laser so frequency and w same for each. Trig anyone? Can add using trig identities, math handbooks, math software…. Or old school (1st time?)

  15. Examine Wave Addition Stuff inside is angle Amplitude same for both, but doesn’t have to be. Show generic method for addition Works for waves, complex #’s, AC circuits, and quantum mechanics

  16. Graphical Representation Lower Path E q = (wt + f + kL) YLX = Ecos(wt + f + kL) Hypotenuse is just EL=E

  17. Graphical Representation Lower Path Upper Path E E qL = (wt + f + kL) qU = (wt + f + kL2) YLX = Ecos(wt + f + kL) YUX = Ecos(wt + f + kL2) Hypotenuse is just EL=E Hypotenuse is just EU=E

  18. To Add: add x & y parts… Lower Path Upper Path E E qL = (wt + f + kL) qU = (wt + f + kL2) YLX = Ecos(wt + f + kL) YUX = Ecos(wt + f + kL2)

  19. To Add: add x & y parts…Good? Horrible? Combined Dq E ET Let’s find Dq… Angle (phase) difference E qL = (wt + f + kL) YUX = Ecos(wt + f + kL2) YLX = Ecos(wt + f + kL)

  20. Interference View • Single Light Source • Laser – Coherent (phase set) Intersection of Light Bottom Path travels L Top Path travels L2 = L+ D

  21. Phase Difference • Lower Path • Angle qL = wt + kL + f • Upper Path • Angle qU = wt + kL2 + f • L2 = L + D • Angle qU = wt + kL + f + kD • Dq = qU – qL = kD

  22. To Add: add x & y parts…Good? Horrible? Combined Dq=kD E ET Let’s find Dq… Angle (phase) difference E qL = (wt + f + kL) YUX = Ecos(wt + f + kL2) YLX = Ecos(wt + f + kL)

  23. To Add: add x & y parts…Law of Cosines? Combined Dq=kD E ET Let’s find Dq… Angle (phase) difference E qL = (wt + f + kL) YUX = Ecos(wt + f + kL2) YLX = Ecos(wt + f + kL)

  24. Law of Cosines?Interior angle is p – kD. Let’s find Dq… Angle (phase) difference Combined Dq=kD E ET p-kD E qL = (wt + f + kL) ET2 = EU2 + EL2 – 2EUELcos(p-kD)

  25. Law of Cosines?Interior angle is p – kD. Let’s find Dq… Angle (phase) difference Combined Dq=kD E ET ET2 = EU2 + EL2 – 2EUELcos(p-kD) ET2 = 2E2 – 2E2cos(p-kD) ET2 = 2E2 + 2E2cos(kD) [Trig fun part 2] ET2 = 2E2 (1+cos(kD)) p-kD E qL = (wt + f + kL) Notice Final wave only depends on phase difference!

  26. Law of Cosines?Interior angle is p – kD. Let’s find Dq… Angle (phase) difference Combined Dq=kD E ET ET2 = EU2 + EL2 – 2EUELcos(p-kD) ET2 = 2E2 – 2E2cos(p-kD) ET2 = 2E2 + 2E2cos(kD) [Trig fun part 2] ET2 = 2E2 (1+cos(kD)) 1 + cosq = 2cos2(q/2) [Trig fun part 3] ET2 = 2E2 (2cos2(kD/2)) ET2 = 4E2 (cos2(kD/2)) ET = 2Ecos(kD/2) Theorists find that… enjoyable…. Experimentalists find that … in a book p-kD E qL = (wt + f + kL)

  27. Interference Implications • Single Coherent Light Source • Split paths Final Amplitude ET = 2E cos(kD/2) D is path length difference k = 2p/l. Max when cos(kD/2) = +/-1 Min when cos(kD/2) = 0

  28. Interference ImplicationsMax  constructive Final Amplitude ET = 2E cos(kD/2) D is path length difference k = 2p/l. Max when cos(kD/2) = +/-1 kD/2 = np, n =0,1,2…. Maxima Check at home: Intensity: I = 0.5e0cET2 Minima when D=(n+1/2)l

  29. What did we learn? • Light is electromagnetic wave • Electric Field part most important • All you need for intensity • Varies in time & space • Interference • Defined by path LENGTH difference • time independent • Path length referenced to wavelength • Coherent, linked sources

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