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Unformatted text preview: _a‘_ Superposition  CHAPTER 21 217 KEV 9. The ﬁgure shows a snapshot graph at t = 0 s of loudspeakers emitting triangular—shaped sound
waves. Speaker 2 can be moved forward or backward along the axis. Both speakers vibrate in
phase at the same frequency. The second speaker is drawn below the ﬁrst, so that the ﬁgure is
clear, but you want to think of the two waves as overlapped as they travel along the xaxis. 21.5 Interference in One Dimension 21.6 The Mathematics of Interference a. On the left set of axes, draw the t: 0 s snapshot graph of the second wave if speaker 2 is
placed at each of the positions shown. The ﬁrst graph, with xspeaker = 2 m, is already drawn. 4 m
#1 “ —>
I I I I I I I I x (m)
—4 4 8
I: 0 Net wave to the right of bark speakers l 51 It
#2 p
I I I I I I I r I X l I
—4 0 4 —4 O 4 8
—>
#2 #1
I I I  I I I I I I I I' I I )5 I I I T I I
4 4 8 / 4/1
I’ng U
#2 i #1 w KW ’
E W
I I I i I I T . I I I I I I X h x (m)
~4 0 4 8 —4 4
#2 I i‘x(m) 8
‘ #1 \\
I l I I I I l I I I I I l I I I I I 'I X0“)
0 4 8 —4 4 b. On the right set of axes, draw the superposition Dnet = D] + D2 of the waves from the
two speakers. Dnet exists only to the right of both speakers. It is the net wave traveling to the right. , ,
C. What separauons between the speakers glve constructive interference? :9 r7 f d. What are the Ax/A ratios at the points of constructive interference? _~_.J___ e. What separations between the speakers give destructive interference? "M  t l
f. What are the Ax/l ratios at the points of destructive interference? w ....m..umu.:.:an.mmm... .. , Mama. Superposition  CHAPTER 21 219 ll. Two loudspeakers are shown at t = 0 8. Speaker 2 is #1
. B D
4mbeh1ndspeakerl. ‘_.4l l A C4‘ ‘8. x(m)
a. Is the interference constructive or destructive? ,2 0 S
a 'H #2 B . D 1 x lOA C4l T8I x(m)
O b. What is the phase constant $10 for wave 1? 7:( What is the phase constant ¢20 for wave 2?
c. At points A, B, C, and D on the xaxis, what are:
0 The distances x1 and x2 to the two speakers?
 The path length difference Ax = x2 — x] ?
 The phases ab} and (p2 of the two waves at the point (not the phase constant)?
0 The phase difference Ad) 2 $2 — (121?
Point A is already ﬁlled in to illustrate. x1 X2 AX $1 (152 A4)
Point A l m 5 In 4 In 0.5% rad 2.521: rad 27rrad
PointB Z W‘ (a w L! m 77’ TU“ TIT
PointC 3 m 1L"? HM {'W 3'52”, Hr
PointD H ’V‘ (‘W‘ L/W‘ Ur “T Z N
d. Now speaker 2 is placed only 2 m behind speaker l. #1
Is the interference constructive or destructive? x . T . .  B o D i 1 l x (m)
_4 A C 4 g
I: O s
l ‘#2 l . B ' D T l l x e. Repeat step c for the points A, B, C, and D. ‘4 0 A C 4 8
x1 X2 AX (P1 (P2 M)
PointA M 3m 2w 05% WW «V
Point B 2 M L‘ “4 2 "M 7” “V TT
POintD H M Q “A 2 M bf 3 7], /( f. When the interference is constructive, what is AXI/l? What is A¢? M g. When the interference is destructive, what is Ax/l? What is Ag)? MT
7,
m in f7 ru‘ E a.
.
3
3 Superposition  CHAPTER 21 2111
4 Q}, /
21.7 Interference in Two and Three Dimensions Key—m, gwymyﬂ. 14. This is a snapshot graph of two plane waves passing through a \x region of space. Each has a 2 mm amplitude. At each lettered point,
what are the displacements of each wave and the net displacement? a.PointA:D1= 21”“ D2: ZW‘I‘“ Dnet: "lwm OWM b.PointB:D1= ZMM D2: ’1’” Dnet= 0. Point C: D1 = ’lw‘“ D2: ‘1“ Due; *‘W‘ 15. Speakers 1 and 2 are 12 m apart. Both emit
identical triangular sound waves with A = 4 In and
(1)0 = 7r/2 rad. Point A is r1 = 16 m from speaker I. a. What is distance r2 from speaker 2 to A? v~,~———~‘—" t: a. M 12m
/.1:"l'<m‘>‘+ WM} — 20"” b. Draw the wave from speaker 2 along the dashed
line to just past point A. crest
(weft d. What is the path length difference Ar = r2 — r1? _”l_WL What is the ratio Ar/ﬁt? ._l__ c. At A, is wave 1 a crest, trough, or zero? At A, is wave 2 a crest, trough, or zero? e. Is the interference at point A constructive, destructive, or in between? an Sf‘Y (A ($14 16. Speakers 1 and 2 are 18 m apart. Both emit #1 4 m g m
identical triangular sound waves with A = 4 m and
(1)0 = 7r/2 rad. Point B is r1 = 24 m from speaker 1. 12m 16m 20m B a. What is distance r2 from speaker 2 to B? wmw a 7.. ~ "01:" {(15va ‘+ (2%)“ 30m M b. Draw the wave from speaker 2 along the dashed
line to just past point A. l
. (N '
c. At B, is wave 1 a crest, trough, or zero? g . 1
At B, IS wave 2 a crest, trough, or zero? £1;— d. What is the path length difference Ar = r2 — r1? _(_0‘M_ What is the ratio Arl/l? ‘w'nt : . . . I I f
e. Is the interference at point B constructlve, destructlve, or 1n between? 6’65 W‘W ‘ ' i’ﬁ‘r'l.(_é, Superposition A CHAPTER 21 2113 18. The ﬁgure shows the wavefront pattern emitted by two loudspeakers. a. Draw a dot c at points where there is constructive interference. These will be points where
two crests overlap or two troughs overlap. b. Draw an open circle 0 at points where there is destructive interference. These will be points
where a crest overlaps a trough. *2“ g ,_~ E \\ \ \a /
c ’1 ‘; a 1 v VD] Position along dotted line
60\lA b
c. Use a hhlﬂeline to draw each “ray” of constructive interference. Use ared line to draw
each “ray” of destructive interference. (1. Draw a graph on the axes above of the sound intensity you would hear if you walked along
the horizontal dotted line. Use the same horizontal scale as the ﬁgure so that your graph
lines up with the ﬁgure above it. e. Suppose the phase constant of speaker 2 is increased by 7: rad. Describe what will happen to
the interference pattern. {PailWl/‘t WIN/owl {butill +9 #14 MM mﬂx Max/A44
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 Summer '08
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