Intensity
• Intensity =
(average power)/
Example 16.6
(unit area)
I = <P>/A = <F"y>/A = <p(x,t)"y(x,t)>
• But p(x,t)"y(x,t) = BkAcos(%t–kx) %Acos(%t–kx)
= B%kA2cos2(%t – kx)
• So
or
!
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I=
I=
1
2
B"kA 2 =
1
2
Example 16.7
#B" 2 A 2
2
"pmax
p2
p2
= max = max
2B
2 #" 2 #B
Note: <x> means
average of x.
<cos2(%t – kx)> = 1/2
Oscillations & Waves
§16.3
1
Find I from Ex.16.1 if the air has ! = 1.20kgm-3
( pmax = 3x10-2Pa from Ex. 16.1)
• Solution:
Use I = pmax2/2!"
Find A and pmax for a 20 Hz wave with the same
intensity as the 1000Hz wave from Ex. 16.1
• Solution: For constant I with ƒ, %A must be
constant. i.e. 2# x 20 x A20 = 2# x 1000 x A1000
= (3.0 x 10-2Pa)2/(2 x 1.20kgm-3 x 344 ms-1)
So A20 = (1000/20)A1000 = 6.0 x 10-7m
= 0.60µm (c.f. 0.012µm @ 1000Hz)
pmax must be the same as in 16.1 (3 x 10-2Pa)
since I is the same.
= 1.1 x 10-6Wm-2
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Oscillations & Waves
§16.3
2
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Oscillations & Waves
§16.3
3
!
Example 16.9
Intensity (2)
We want I = 1Wm-2 at 20m from speakers in all
directions. What speaker power is needed?
• Solution: “all directions” means hemisphere
above ground, with area 1/2 x 4#(20m)2.
• For a point sound source, I = P/4!r2
“Inverse
§16.3
4
Interference in Travelling Sound
Waves
• If two sources of produce identical sound signals in-phase
(coherent) then depending on difference in path lengths
from sources to listener, constructive (loud sound) or
destructive (soft sound) interference may occur.
• If path difference d is whole
number of wavelengths d = n$,
then constructive. If path
difference is half integer
d = (n +1/2)$ then destructive.
Note: Also works for 2 light sources
e.g. Young’s double slit
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Oscillations & Waves
§16.6
I0 is roughly the softest audible sound @ 1000 Hz
• B = 0 dB when I = I0 , and B = 120 dB
when I = 1Wm–2.
• Doubling intensity " 3dB increase
actually need more electrical power than this since
speakers are inefficient.
Oscillations & Waves
square law” similar to light.
• Sound intensity level often measured in
decibels: B = 10dB log10 (I/Io)
where I0 = 10–12Wm–2 is a reference intensity.
So acoustic power = 1 Wm-2 x 2513m2 = 2.5kW;
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Example 16.11
7
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Oscillations & Waves
§16.3
5
By how many dB does the intensity drop
when you move twice as far away from a
singing bird?
• Solution: The difference is given by B2–B1
= 10dB [log(I2/I0)–log(I1/I0)]
= 10dB log(I2/I1) = 10dB log(r12/r22)
= 10dB log[r12/(2r1)2]
= 10dBlog(1/4) = –6.0dB
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Interference in Travelling Sound
Waves
• Here we DON’T have standing waves. Both
waves travel in roughly the same direction
• Interference of travelling waves from two
identical sources, but:
Oscillations & Waves
§16.6
§16.3
6
Example 16.15
Speakers A & B emit in-phase
sinusoidal waves. v = 350 ms-1.
Find what frequencies yield
a) constructive & b) destructive interference at
point P
• Solution: 1st find difference between path
lengths AP & BP.
AP = !(22+42) " 4.47m, BP = !(12+42) " 4.12m.
Path diff d = 4.47 - 4.12 = 0.35m
– Energy flow is channeled
– Pressure & displacement have the same nodes
– Nodes where paths differ by d = (n +1/2)$
(or d = m$/2 for odd m) (destructive interference)
– Antinodes where paths differ by d = n$
(constructive interference)
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Oscillations & Waves
8
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Oscillations & Waves
§16.6
9
Example 16.15 (2)
Beats
• Constructive (loud sound) when d = n$ = nv/f;
or f = nv/d
f = n x 350/0.35 = 1000, 2000, 3000 Hz, ….
• Destructive (soft sound) when d = (n+1/2)$
= (n+1/2)v/f; or f = (n+1/2) v/d
f = (n+1/2) x 350/0.35 = 500, 1500, 2500 Hz, ….
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Oscillations & Waves
10
• “Beats” occur when two sounds have almost the same
frequency. “Beats” means periodically varying amplitude
• Mathematically:
Asin%1t – Asin%2t = 2A sin[(%1–%2)t/2] cos[(%1+%2)t/2]
Beat Envelope High frequency part
• Result is an
amplitude
that varies
at the beat
frequency
ƒbeat = ƒ1–ƒ2
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Doppler effect (2)
Oscillations & Waves
§16.8
Oscillations & Waves
§16.7
11
Doppler effect (3)
• In front of S: distance travelled in 1 cycle of
sound emission is
$ = $S – dS = "/ƒS – "S/ƒS = ("–"S)/ƒS
• Behind S: distance travelled in 1 cycle
$ = $S + dS = "/ƒS + "S/ƒS = ("+"S)/ƒS
• Crests arrive at listener at speed "+"L, so
listener hears a frequency ƒL = ("+"L)/$
thus…
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Doppler effect
• The Doppler equation for a moving Source
and moving Listener:
fL =
" + "L
f
" + "S S
• Remember! " positive from L to S,
negative in opposite direction
!
13
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Oscillations & Waves
§16.8
14
• Examples: moving sources (trains, etc.)
• Define all
" as
+ve
when
moving
from L
towards S.
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Oscillations & Waves
§16.8
12