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

• Traveling: (i) same A for all x and (ii) at given t, different x at different point of cycle

• Standing: (i) A varies with x and (ii) at given t, all x at same point in cycle (e.g. maximum at )

Wave: particles undergoing collective/correlated SHM

a sin (kx! !t)

(2a sin kx) cos !tA(x)

(x! vt)

cos !t = 1

(in general function of )

(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:

• Phase constants tell us what the source is doing at t = 0

D1 (x1, t) = a sin (kx1 ! !t + "10) = a sin "1

D2 (x2, t) = a sin (kx2 ! !t + "20) = a sin "2

distance of observer from source 1

Interference: graphical

aligned crest-to-crest and trough-to trough: in-phase

crest of wave 1 aligned with trough of wave 2: out of phase

D1(x) = D2(x) ! !1 = !2 ± 2"mDnet = 2D1 (or D2): A = 2a

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: !!0 = !10 ! !20

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.

( independent of observer in 1D cf. 2/3D)!!

Interference: mathematical

• In general, neither exactly out of nor in phase

Identical sources: !!0 = 0

maximum constructive interference: !x = m!perfect destructive interference: !x = (m + 1/2) !

(xavg. = x of observer +...)

(confirms graphical...)

D = D1 + D2 = A sin (kxavg. ! !t + "0 avg.)xavg. = (x1 + x2) /2;"0 avg. = ("10 + "20) /2A = 2a cos !!

2 (independent of x of observer)" traveling waveA varies from0 for !" = (2m + 1)# (perfect destructive interference) to2a for !" = 2m# (maximum constructive interference)

Application to thin films• reflection from boundary at which index of refraction

increases (speed of light decreases) phase shift of

• 2 reflected waves interfere: from air-film and film-glass boundaries ( )

!! = 2"!x

#f!!!0

= 2"2dn

#

Use !!0 = 0 (common origin)"f = !

n

! constructive for "C = 2ndm , m = 1, 2, ...

destructive for "D = 2ndm!1/2 , m = 1, 2, ...

(anti-reflection coating for lens)

nair = 1 < nfilm < nglass

!

wave 2 travels twice thru’ film in air

Interference in 2/3 D• Similar to 1D with x r:

• after T/2, crests troughs...points of constructive/destructive interference same

• measure r by counting rings...

!r (! !! and A) dependson observer (cf. in 1 D)

D(r, t) = a sin (kr ! !t + "0)

constant a approx.

Picturing interference in 2/3 D• Antinodal and nodal lines: line of points

with same , A traveling wave along it

• assume circular waves of same amplitude

• draw picture with sources and observer at distances

• note

•

Strategy for interference problems

!r

!!0

( for 1D)!r ! !x

r1, 2

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 single tone with intensity modulated: loud-soft-loud

• Simplify:

D1 = a1 sin (k1x! !1t + "10)D2 = a2 sin (k2x! !2t + "20)

a1 = a2 = a; x = 0; !10 = !20 = "

D = D1 + D2 =!2a cos !mod.t

"sin!avg.t

!avg. = 12 (!1 + !2): average frequency

!mod. = 12 (!1 ! !2): modulation frequency

!f ! 1 Hz

Conditions for beats

• Note heard is , 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

Each loud-soft-loud is 1 beat

Beats frequency

!1 ! !2 " !avg. ! !1 or 2

!mod. # !avg.

2a cos !mod.t is slowly changing amplitudefor rapid oscillations sin!avg.t

fbeat = 2fmod. = f1 ! f2

I ! A2 ! cos2 !mod.t = 12 (1 + cos 2!mod.t)

favg. = !avg./(2") ! f1, 2