Lecture 8

• standing waves on string and tubes

• Interference in 1D: superposition of waves in same direction graphical and mathematical phase and path-length difference application to thin films

• Interference in 2/3D

Transverse standing waves• generate standing waves on string fixed at

both ends: traveling wave encounters a boundary...

(phase change of on reflection)

!

Standing waves on string• Reflection does not change f,

• boundary condition (constraint obeyed at edge): D = 0 at ends (nodes) at all times

• allowed standing waves (traveling for other...):

• normal modes: fundamental (smallest) frequency:

only envelope shown

f1 = v2L , ! = 2L (only half ! and no node in-between)

harmonics: fm = mf1 (m = 2, 3, ...) (m! 1 nodes and m antinodes in-between)

!

master formulae

A(x) = 2a sin kx = 0 at x = 0 and at x = Lif sin kL = 0 ! kL = m!

Example• A 121-cm-long, 4.0 g string oscillates in its m=3 mode

with a frequency of 180 Hz and a maximum amplitude of 5.0 mm. What are (a) the wavelength and (b) the tension in the string?

Standing EM waves

• light wave has node at each mirror...similar to string....

• microwave oven: turntable to avoid part of food being always a node

! ! cm vs. size of oven ! 10’s cm "

Standing Sound Waves (I)

• closed end: node (air can’t oscillate)

• open end (sound wave partly reflected back into tube): antinode not literally a snapshot, size of

tube unrelated to A

(...like string)

Musical Instruments

• fundamental mode: 1/4 in length L (cf.1/2 for closed-closed or open-open) frequency is half of open-open/closed-closed...

• stringed: fundamental frequency: change density or tension

• wind: change effective length by covering holes/opening valves

Standing Sound Waves (II)!

speed along string

f1 = v2L = 1

2L

!Tsµ

sound speed in air

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

same !, a

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

D1(x, t) = !D2(x, t) " 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, depends on distance between sources in 1D cf. 2/3D)

!x ! !!

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) !

(confirms graphical...)

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

2 (" independent of x of observer: see e.g.)

traveling

A varies from0 for !! = (2m + 1)" (perfect destructive interference) to2a for !! = 2m" (maximum constructive interference)

! wave (due to xavg. " vt) withxavg. = x of observer +x of midpoint of sources

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

+ two ! phase shifts

Mnemonic for phase shifts at boundaries

• “Low to high, phase shift of ’’

• “High to low, phase shift no’’

• Low and high above refers to index of refraction high and low speeds of light

• works for strings as well...

!

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...

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

constant a approx.

!r (! !!) dependson observer

as in 1D...but...

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?

Interference in 1 vs. 2/3 D

In 2/3D, Dnet = D1 + D2 = A sin (kravg ! !t) withA = 2a cos (!"/2)...as in 1D......but A (due to !") depends on observer in 2/3D...

!r (! !!) dependson observer in 2/3 (vs. not in 1D)

In general, not a traveling wave (cf. 1D)

based on figure, demo., example

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