Complex waves
2.2 Sound
Complex Sound Waves
Beats
Doppler Effect.
Complex waves consist of different frequency
components , i.e. harmonics.
• In general sound waves are a combination
of different frequencies.
• The different frequencies can be
determined by mathematical procedure
called a Fourier Transform.
Frequency dependence of Hearing
displacement
Time
relative
amplitude
Frequency
Beats
Superposition of two waves with different frequencies produce
oscillation in amplitude.
Due to constructive and destructive interference.
Beat Frequency
Tb
fb =
Tuning musical instruments
The beat frequency for two musical instruments is zero
when the two are in tune. (have the same frequency)
1
= f2 − f1
Tb
1
Observer moving toward a Stationary source
(Relative Velocity Increases)
Doppler Effect
Doppler effect- the shift in frequency of a
wave where the source and observer are
moving relative to one another.
Two different cases:
Observer moving – source stationary
Source moving- observer stationary.
V = speed of sound
Vo =speed of observer
•Relative velocity of wave (vo + v ) increases.
•Frequency increases
fo =
v + vo v + vo
v
=
fs = (1 + o )fs
λs
v
v
fo − fs =
Observer moving away from a stationary
source (Relative velocity decreases)
vo
fs
v
Source moving toward a stationary observer
(wavelength in the medium decreases)
Vo
v
Relative velocity of wave (v-vo) decreases.
Frequency decreases.
fo =
v − vo v − vo
v
=
fs = (1 − o )fs
λs
v
v
fo − fs = −
•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.
vo
fs
v
Source Moving Toward observer A
Source Moving Away from observer B
λ
Moving Source
•Wavelength decreases
•Frequency increases
fo =
v
fs
v − vs
Observer
Distance traveled by
source in one period
λ = λ s − v sT
v
v
fo =
=
λ s − v s Ts vTs − v s Ts
Moving Source
Observer
•Wavelength increases
•Frequency decreases
fo =
λ = λ s + v sT
fo =
v
v
=
λ s + v s Ts vTs + v s Ts
v
fs
v + vs
2
Observer and source moving
Observer and source moving
v
source
v
source
vs
vo
vs
observer
vo
⎛ v + vo ⎞
fo = fs ⎜
⎟
⎝ v − vs ⎠
• The frequency increases when the source and observer are moving toward
each other.
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
⎛ v − vo ⎞
fo = fs ⎜
⎟
⎝ v + vs ⎠
• The frequency decreases when the source and observer are moving away
each other.
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
Approaching source
vo
vs positive
⎛ v + vo ⎞
⎛ v ⎞
⎛ 340 ⎞
fo = fs ⎜
⎟ = fs ⎜
⎟ = 700 ⎜
⎟ = 700 (1.09 ) = 763Hz
⎝ 340 − 30 ⎠
⎝ v − vs ⎠
⎝ v − vs ⎠
Departing source
fo =
⎛ 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.
v
v
1
f
fo =
fs =
f =
v s
v s
v − vs
v(1 − s )
1− s
v
v
At low speed vs <<v
vs
)fs
v
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
Using the relation
1
≈ 1+ x
1− x
v + vo
340 + 34
fs =
fs = 1.22fs
v + vs
340 − 34
fo = 1.22 x10 3 Hz
vs is negative
fo ≈ (1 +
observer
When x<<1
• 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.
3
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 velocities of source and observe.
c = speed of light
Positive sign when approaching
Negative sign when moving away.
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.
negligible
f1 = fs (1+u/c)
f2 = f1(1+u/c) = fs
(1+u/c)2 =
fs (1+2u/c +
(u/c)2)
beat frequency = f2 – fs = 2 u fs
c
4