Beats
2.2 Doppler Effect
Superposition of two waves with different frequencies produce
oscillation in amplitude.
Beats
Doppler Effect.
Shock Waves
Tb
Beat Frequency
Tuning musical instruments
The beat frequency for two musical instruments is zero
when the two are in tune. (have the same frequency)
Doppler effect
fb =
1
= f2 − f1
Tb
Doppler Effect
Doppler effect- the shift in frequency of a
wave where the source and observer are
moving relative to one another.
Observer moving toward a Stationary source
(Relative Velocity Increases)
Positive direction toward source
Two different cases:
Observer moving – Relative velocity changes
Source moving- Wavelength changes
v
f=
λ
•Relative velocity of wave (vo + v ) increases.
•Frequency increases
fo =
v + vo
λs
=
v + vo
v
fs = (1 + o )fs
v
v
1
Observer moving away from a stationary
source (Relative velocity decreases)
Vo
Source moving toward a stationary observer
(wavelength in the medium decreases)
Negative direction away from source
v
Relative velocity of wave (v+vo) decreases.
Frequency decreases.
fo =
•When the source is moving the wavelength of the wave in the media is changed
• source approaches observer A
•Wavelength decreases and frequency heard by observer A increases
• Source moves away from observer B.
•Wavelength increases and frequency heard by observer B decreases.
v + vo
v
fs = (1 + o )fs
v
v
vo is negative
Source Moving Toward observer A
Source Moving Away from observer B
vs negative direction for observer B
vs Positive direction toward observer
λ
Moving Source
•Wavelength decreases
•Frequency increases
fo =
Observer
Distance traveled by
source in one period
λ = λ s − v sT
fo =
v
v
=
λ s − v s Ts vTs − v s Ts
v
fs
v − vs
Moving Source
Observer
•Wavelength increases
•Frequency decreases
fo =
v
fs
v − vs
vs is negative
Observer and source moving
Observer and source moving
v
source
λ = λ s − v sT
v
source
vs
vo
⎛ v + vo ⎞
fo = fs ⎜
⎟
⎝ v −vs ⎠
observer
vs positive
vo positive
• The frequency increases when the source and observer are moving toward
each other.
vs
vo
⎛ v + vo ⎞
fo = fs ⎜
⎟
⎝ v − vs ⎠
observer
vo negative
vs negative
• The frequency decreases when the source and observer are moving away
each other.
2
Example
A fire truck is approaching an observer with a speed of 30
m/s. The siren has a frequency of 700 Hz. What
frequency does the observer hear as the truck
approaches? What frequency is heard after the truck
passes. speed of sound 340 m/s
Two trains are approaching each other each moving
at 34 m/s. One train sounds a whistle at a frequency
of 1000 Hz. Find the frequency of sound heard by
an observer on the other train.
v
vs
vs positive
Approaching source
⎛ v + vo ⎞
⎛ v ⎞
⎛ 340 ⎞
fo = fs ⎜
⎟ = fs ⎜
⎟ = 700 ⎜
⎟ = 700 (1.09 ) = 763Hz
⎝ 340 − 30 ⎠
⎝ v − vs ⎠
⎝ v − vs ⎠
Departing source
fo =
v + vo
340 + 34
fs =
fs = 1.22fs
v + vs
340 − 34
fo = 1.22x10 3 Hz
vs is negative
⎛ v + vo ⎞
⎛ v ⎞
fo = fs ⎜
⎟ = fs ⎜
⎟
⎝ v − vs ⎠
⎝ v − vs ⎠
⎛ 340 ⎞
= 700 ⎜
⎟ = 700 ( 0.92 ) = 643Hz
⎝ 340 + 30 ⎠
Note each motion increases the frequency by about vtrain/v =10%.
The net increase is about 20% This is true when vsource << v.
Approximate solution for two
trains approaching
Approximate solution at low speeds.
Source moving toward observer.
fo =
vo
vo =0
v
v
1
fs =
f =
f
vs s
v s
v − vs
v(1 − )
1− s
v
v
Using the relation
1
≈ 1+ x
1− x
When x<<1
At low speed vs <<v
fo ≈ (1 +
vs
)fs
v
Doppler shift of Electromagnetic
waves
• Electromagnetic waves are also shifted by the Doppler
effect.
• Since EM waves travel in a vacuum the equations
governing the shift are different.
• The same shift is observed for moving source or moving
observer.
• For motion with speeds less than the speed of light the
relation is the same as for the approximate shift for sound
waves when u<<v.
u
f = fs (1 ± )
c
u = relative velocity of source and observer.
c = speed of light =3.00x108 m/s
Positive sign when approaching
Negative sign when moving away.
fo =
v + vo
⎛ v + v o ⎞⎛ v ⎞
⎛ v ⎞⎛ v ⎞
⎟⎟fs ≅ ⎜1 + o ⎟⎜1 + s ⎟fs
fs = ⎜
⎟⎜⎜
v − vs
v ⎠⎝
v ⎠
⎝ v ⎠⎝ v − v s ⎠
⎝
negligible
v
v v ⎞
v ⎞
⎛ v
⎛ v
fo = ⎜1 + s + o + s 2 o ⎟fs ≅ ⎜1 + s + o ⎟fs
v
v
v ⎠
v
v ⎠
⎝
⎝
v ⎞
⎛v
fo − fs ≅ ⎜ s + o ⎟fs
v ⎠
⎝ v
• The shift in frequency is approximately proportional to the ratio of the train
velocities to speedof sound as we found in the previous example.
• This is a good approximation when the train velocities are slow compared
to the speed of sound.
• This is a good approximation for the Doppler shift of electromagnetic waves.
Doppler Radar
Doppler radar is used to determine the speed of a car.
f1
f
u
f2
The beat frequency between the Doppler shifted frequency and the initial
frequency is measured to determine the speed of the car.
f1 = fs (1+u/c)
negligible
f2 = f1(1+u/c) = fs (1+u/c)2 = fs (1+2u/c + (u/c)2)
u
beat frequency = f2 – fs = 2 fs
c
3
Applications Doppler ultrasound
Shock Waves
When vs > speed of sound, shock waves are formed.
θ
v
Sinθ =
vs
Ultrasonic waver 2-18 MHz.
vt
vst
Doppler shift used to measure blood flow speed.
Sonic Boom
When a sonic boom is heard the observer sees the plane at an
angle of θ = 450 What is the speed of the plane relative to the
speed of sound?
v
= sin 45 = 0.707
vs
v
v
vs =
=
= 1.41v
sin θ 0.707
sin θ =
The plane travels at
1.41 time the speed of
sound.
4