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