# Solutions

```Superposition and In terference
Problem 1
Two loudspeakersarevibrating in phase.They aresetup asin the figure below, andpoint
C is locatedas shownthere.The speedof soundis 343 m/s. The speakersplay the same
tone.What is the smallestfrequencythat will producedestructiveinterferenceat point C?
^ft6
==
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,tt\, ^I^;*,,/Me +r-'-
p&quot;-+
C h ;iPa-'l.-1
ft t'&quot; nay&quot;^-l Eo :
A C = 4.oor&quot;,
::---
USL=
n=Orl)1,'--
n\
n=or\r4,&quot;&quot;
AL= Cn+'A)\
WI
---) \=u/5
\f=v
AL= (n&quot;7r) \
AL= (n&quot;'t*)v/5
C)-
(nira)v
:-----:---.-
AL
A L= Ac-
BC = 4.oocq- ?&quot;9or.&quot;)
A L = |&quot;Goro
'YW
(_
I-
( n+r/).) /
:
AL
'4 =
f&quot;'n=
f&quot;,r = lo7 Hz
11*L3u3n/s)
1.60 m
Problem2
(a) A cello string hasa fundamentalfrequencyof 65.40Hz. What beatfrequencyis heard
whenthii cello string is bowedat the sametime asa violin string with frequencyof 196.0IIz?
(Hint: Thebeatsoccurbetweenthe third harmonicof the cello string andthe fundamentalof
the violin.)
,1-l-4t- :
Fr&quot; ns -0^'o',.^&quot;.-o^t&lt;
+\'T)-t'rJv\X )r
€
,&quot;
^
lp*
n bruz
/'
tn- ntr
.-.-
9^&quot;a^ = )9b'o Hz
f&quot;&quot;rr.
W
-- 3 (G5.'.|oil&quot;) :
l9{o-}Hz
= | ( '- { rl
;r&quot; -lx-* yvq!',''.n v 2,r'&quot;'ua&quot; 9*&quot;x
f etux)=f lla.o*.*le6,aHzl--,
I@
(b) A tube 1.20m long is closedat oneend.A stretchedwire is placednearthe openend.
Thewire is 0.330m longandhasa massof 9.60g. It is fixed at both endsandoscillates
in its firndamentalmode. By resonance,it setsthe air column in the tube into oscillation
at the tubesfundamentalflequency.Find that frequencyandthe tensionin the wire.
W
^u/4y n=1,3,5,&quot;'
-a^&quot;tr
tn=
..i.o,-|&quot;.b-'o
&quot;7k
ful&quot;
u
&amp;^
-affi s*JA f&quot;=19^, 6=o21,),
k ^ iln LpUffi, &lt;) #,/t) &quot;t
h,,r*i s,= Q!:].li') --F--
lA€l
4(r'a&quot;-')
4W.
ge.odrJa4 t\$ rc+orro&quot;'^rr t{r4 -'h^^'€-LU}n-r
pto
},{,o -^'&quot;d r9\'
T_
9&quot;= nu/^,
.t^)A^e,.
---) S,=uAL
v-l5rr,
= [F,, --.. F= 4 +,L(^L) = 4 t&quot;, Lm
V = l S,L
'
(inlt)
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a
r=--.::
1
=
xt
Cq.z
tt
tcl)
(Dr.su, fio&quot;lrD.)C9&quot;t,ovto3
/
F
l/-
```