Lecture 6
• 2D/3D waves
• Sound and light
• Power and Intensity
• Doppler effect for
(i) mechanical waves e.g. sound
(ii) EM waves
Example
• A 2.31 kg rope is stretched between supports 10.4 m
apart. If one end of the rope is tweaked, how long will
it take for the resulting disturbance to reach the
other end? Assume that the tension in the rope is
42.8 N.
2D/3D waves
•
2D circular waves: wavefronts (lines
locating crests), small section
appear as straight lines far away
•
3D spherical waves...appear as
planes far away, described by D(x, t)
(same at every point in yz plane)
2D/3D waves
D(r, t) = A(r) sin (kr − ωt + φ0 )
with A(r) decreasing with r
Phase and phase difference
phase, φ = kx − ωt + φ0
D(x, t) = A sin φ
• wavefronts are surfaces of same
displacement constant phase
∆x
2π λ
phase difference, ∆φ =
∆φ = 2π between adjacent wavefronts
(separated by λ)
Sound waves
vsound in air at 20 = 343 m/s
(larger in liquid/solid)
◦
• human ears: 20 Hz to
20 k Hz
• ultrasound: > 20 k Hz
•
•
Electromagnetic (EM) waves
oscillations of EM field, can travel in vacuum
e.g. light from stars
vlight = c = 3 × 108 m/s in vacuum
(" vsound )
visible spectrum: 400 nm (violet/blue
to 700 nm (orange/red)
! λsound ⇒ flight # fsound
•
EM spectrum: visible + higher frequencies
(UV/X rays) + lower frequencies
(IR/micro/radio waves)
•
index of refraction (light slowed down):
speed of light in vacuum
n=
=
speed of light in material
frequency does not change (e.g.,
sound wave hitting water):
!
c
v
c
fvac. = λvac.
⇒ λmat. < λvac.
"
!
= fmat. =
vmat.
λmat.
"
•
•
Power and Intensity
Power is rate of transfer of energy by wave
Brightness/loudness depends also on area
receiving power:
intensity, I = Pa = power-to-area ratio
(SI units: W/m2 )
•
Uniform spherical wave
I=
(from energy conservation:
total energy crossing wavefront is same)
r22
I1
I2 = r 2
Psource
4πr 2
1
2
I∝A
(energy of oscillations E = 12 kA2 )
Decibels (for wide range of human hearing)
β = 10 dB log
! "
I
I0
(dimensionless)
10−12 W / m2 (threshold)
• β = 0 at threshold
• increasing by 10 dB
factor of 10
I increases by a
Example
• The ``planet’’ Pluto’s average distance from the sun
is 5.9 × 10.12 m . Calculate the sun’s intensity at the
distance of Pluto, assuming the sun radiates a total power
of 4.0 × 1026 W.
•
Doppler effect
relative motion between observer and wave
source modifies frequency e.g. pitch of ambulance
siren drops as it goes past
• moving source: Pablo detects (λ−, f−) ,
Nancy detects (λ+ , f+ ) vs. (λ0 , f0 ) if source at
rest
•
Doppler effect:derivation
motion of wave crest (once leaves source)
governed by medium (not affected by source
moving)
wave crests bunched up in front/
stretched out behind: λ+ < λ0 < λ− + speed v ⇒ f+ > f0 > f−
In time t = 3T ,
source moves 3vs T ,
wave (crest 0) moves 3vT
⇒ 3 wave crests in
3vT ∓ 3vs T ...= 3λ±
f±
=
=
v
λ±
v
v ∓ vs
Doppler effect for moving source
Doppler effect: moving
observer
• not same as source moving: motion relative
to medium (not just source vs. observer)
matters...
Doppler effect for EM waves
• no medium: use Einstein’s theory of
relativity
!
1+vs /c
1−vs /c
λred =
(receding
source:
longer
wavelength,
red
shift)
!
1−vs /c
1+vs /c
λblue =
(approaching source: shorter wavelength, blue shift)
where vs is the speed of the source relative to the observer