Lecture 8: Interference
• superposition of waves in same direction graphical and mathematical phase and path-length difference application to thin films
• in 2/3 D • Beats: interference of slightly different frequencies
Summary of traveling vs. standing wave Wave: particles undergoing collective/correlated SHM
• Traveling: (i) same A for all x and
(ii) at given t, different x at different point of cycle
a sin (kx − ωt)
(in general function of (x − vt))
• Standing: (i) A varies with x and
(ii) at given t, all x at same point in cycle (e.g. maximum at cos ωt = 1 ) A(x)
(2a sin kx) cos ωt
(in general, function of x . function of t)
Interference: basic set-up
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standing waves: superposition of waves traveling in opposite direction (not a traveling wave) is one type of interference
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2 waves traveling in same direction:
distance of observer from source 1
D1 (x1 , t) D2 (x2 , t)
•
= a sin (kx1 − ωt + φ10 ) = a sin (kx2 − ωt + φ20 )
= a sin φ1 = a sin φ2
Phase constants tell us what the source is doing at t = 0
Interference: graphical
aligned crest-to-crest and trough-to trough: in-phase D1 (x) = D2 (x) ⇒ φ1 = φ2 ± 2πm Dnet = 2D1 (or D2 ): A = 2a
crest of wave 1 aligned with trough of wave 2: out of phase D1 (x) = −D2 (x) ⇒ Dnet (x) = 0
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Phase and path-length difference (t drops out)
net phase difference determines interference: 2 contributions (i) path-length difference:
∆x = x2 − x1 (distance between sources: extra distance traveled by wave 2; independent of observer)
(ii) inherent phase difference: ∆φ0 = φ10 − φ20
Example
• Two in-phase loudspeakers separated by distance d
emit 170 Hz sound waves along the x-axis. As you walk along the axis, away from the speakers, you don’t hear anything even thought both speakers are on. What are three possible values for d? Assume a sound speed of 340 m/s. ( ∆φ independent of observer in 1D cf. 2/3D)
Identical sources: ∆φ0 = 0 maximum constructive interference: ∆x = mλ perfect destructive interference: ∆x = (m + 1/2) λ
Interference: mathematical D = D1 + D2 = A sin (kxavg. − ωt + φ0 avg. ) xavg. = (x1 + x2 ) /2;φ0 avg. = (φ10 + φ20 ) /2 A = 2a cos ∆φ 2 (independent of x of observer) ⇒ traveling wave (xavg. = x of observer +...) A varies from 0 for ∆φ = (2m + 1)π (perfect destructive interference) to 2a for ∆φ = 2mπ (maximum constructive interference)
(confirms graphical...)
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In general, neither exactly out of nor in phase
Application to thin films
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reflection from boundary at which index of refraction increases (speed of light decreases) phase shift of π
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2 reflected waves interfere: from air-film and film-glass boundaries ( nair = 1 < nf ilm < nglass) ∆x ∆φ = 2π − ∆φ0 λf 2dn = 2π wave 2 travels λ twice thru’ film in air Use ∆φ0 = 0 (common origin) λ λf = n ⇒ constructive for λC = 2nd m , m = 1, 2, ... 2nd destructive for λD = m−1/2 , m = 1, 2, ... (anti-reflection coating for lens)
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Interference in 2/3 D
Similar to 1D with x
r: D(r, t) = a sin (kr − ωt + φ0 )
constant a approx.
after T/2, crests troughs...points of constructive/destructive interference same
∆r (⇒ ∆φ and A) depends on observer (cf. in 1 D)
measure r by counting rings...
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Picturing interference in 2/3 D
Antinodal and nodal lines: line of points with same ∆r , A traveling wave along it
Strategy for interference problems
• •
assume circular waves of same amplitude
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note ∆φ0
draw picture with sources and observer at distances r1, 2
( ∆r → ∆x for 1D)
Example
• Two out-of-phase radio antennas at x = +/- 300 m on the x-axis are emitting 3.0 MHz radio waves. Is the point (x, y) = ( 300 m, 800 m ) a point of maximum constructive interference, perfect destructive interference, or something in between?
Beats
• superposition of waves of slightly different f
(so far, same f): e.g. 2 tones with ∆f ∼ 1 Hz single tone with intensity modulated: loud-soft-loud
• Simplify: a
1
D1 = a1 sin (k1 x − ω1 t + φ10 ) D2 = a2 sin (k2 x − ω2 t + φ20 ) = a2 = a; x = 0; φ10 = φ20 = π !
"
D = D1 + D2 = 2a cos ωmod. t sin ωavg. t 1 ωavg. = 2 (ω1 + ω2 ): average frequency 1 ωmod. = 2 (ω1 − ω2 ): modulation frequency
Conditions for beats ω1 ≈ ω2 ⇒ ωavg. ≈ ω1 or 2 ωmod. # ωavg. 2a cos ωmod. t is slowly changing amplitude for rapid oscillations sin ωavg. t
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Note heard is favg. = ωavg. /(2π) ≈ f1, 2, but loud-soft-loud: modulation (periodic variation of variation of amplitude)
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waves alternately in and out of phase: initially crests overlap A = 2 a, but after a while out of step (A = 0) due to unequal f’s
Beats frequency I ∝ A2 ∝ cos2 ωmod. t =
Each loud-soft-loud is 1 beat
1 2
(1 + cos 2ωmod. t) fbeat = 2fmod. = f1 − f2