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



standing waves: superposition of waves traveling in opposite direction (not a traveling wave) is one type of interference



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



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...)



In general, neither exactly out of nor in phase

Application to thin films



reflection from boundary at which index of refraction increases (speed of light decreases) phase shift of π



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)





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...



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

• •

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



Note heard is favg. = ωavg. /(2π) ≈ f1, 2, but loud-soft-loud: modulation (periodic variation of variation of amplitude)



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