Doppler Effect

2.3 Sound Doppler effect- the shift in frequency of a wave where the source and observer are Doppler Effect. moving relative to one another. Shock Waves Two different cases for sound: Observer moving – source stationary Source moving- observer stationary.

Observer moving away from a stationary Observer moving toward a Stationary source source (Relative velocity decreases) (Relative Velocity Increases) Vo

V = speed of sound

Vo =speed of observer

v f − f = o f o s v s

Relative velocity of wave (v-vo) decreases. •Relative velocity of wave (vo + v ) increases. Frequency decreases. •Frequency increases vv− vv− v vv+ vv+ v f = o oo f = o oo o ==−f(1)fss o ==+f(1)fss λs vv λs vv

Source moving toward a stationary observer Source Moving Toward observer A (wavelength in the medium decreases)

v

Moving Source Observer Distance traveled by source in one period

•When the source is moving the wavelength of the wave in the media is changed •Wavelength decreases λ=λss −vT • source approaches observer A vv •Wavelength decreases and frequency heard by observer A increases •Frequency increases f == • Source moves away from observer B. o λ−vT vT − vT •Wavelength increases and frequency heard by observer B decreases. ssssss v ffos= vv− s

1 Source Moving Away from observer B Observer and source moving v

source vs λ vo observer Moving Source ⎛⎞vv Observer + o ffos= ⎜⎟ ⎝⎠vv− s •Wavelength increases λ=λss +vT vv •Frequency decreases fo == • The frequency increases when the source and observer are moving toward λ+ssssssvT vT + vT each other. v ffos= vv+ s

Question Observer and source moving A fire truck is going down the street toward a stationary v observer sounds an alarm with a frequency fs. Which of these is true of the frequency heard by the observer.

A. The frequency is higher because the wavelength of the source vs vo observer sound in air is lengthened. B. The frequency is higher because the wavelength of the ⎛⎞ vv− o sound in air is shortened. ffos= ⎜⎟ C.The frequency is higher but the wavelength of the sound ⎝⎠vv+ s in air is the same as for a stationary truck. D. The frequency is higher because the speed of sound in • The frequency decreases when the source and observer are moving away air is faster. each other.

Two trains are approaching each other each moving at 34 m/s. One Example train sounds a whistle at a frequency of 1000 Hz. Find the frequency A fire truck is approaching an observer with a speed of 30 shift of sound heard by an observer on the other train. m/s. The siren has a frequency of 700 Hz. What frequency does the observer hear as the truck approaches? speed of sound 340 m/s v

vs vo

2 Approximate solution for two Approximate solution at low speeds. trains approaching

Source moving toward observer. v + v ⎛ v + v ⎞⎛ v ⎞ ⎛ v ⎞⎛ v ⎞ f = o f = ⎜ o ⎟⎜ ⎟f ≅ ⎜1+ o ⎟⎜1+ s ⎟f o v − v s v ⎜ v − v ⎟ s v v s v v 1 s ⎝ ⎠⎝ s ⎠ ⎝ ⎠⎝ ⎠ f = f = f = f o v − v s v s v s s v(1− s ) 1− s negligible v v ⎛ v v v v ⎞ ⎛ v v ⎞ f = ⎜1+ s + o + s o ⎟f ≅ ⎜1+ s + o ⎟f At low speed v <

vs fo ≈ (1+ )fs v ⎛ vs vo ⎞ fo − fs ≅ ⎜ + ⎟fs Using the relation ⎝ v v ⎠

1 • The shift in frequency is approximately proportional to the ratio of the train ≈ 1+ x When x<<1 1− x velocities to speed of 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 shift of Electromagnetic Doppler Radar waves • Electromagnetic waves are also shifted by the Doppler Doppler radar is used to determine the speed of a car. effect. • Since EM waves travel in a vacuum the equations governing the shift are different. f1 f • The same shift is observed for moving source or moving observer. u f2 • 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<

u = relative velocities of source and observe. 2 2 f2 = f1(1+u/c) = fs (1+u/c) = fs (1+2u/c + (u/c) ) c = speed of light u Positive sign when approaching beat frequency = f –f = 2 f 2 s c s Negative sign when moving away.

Question Shock wave A Doppler shifted radar beam is reflected off a car going At super-sonic speed the pressure amplitude is large 30m/s coming directly toward a stationary police car. If the 10 Shock wave front frequency of the radar is 1.00x10 Hz. The beat frequency of Sonic boom the reflected beam with the stationary source is. (speed of light is 3.00x108m/s) θ vt A. 1x102 Hz ut B. 2x102 Hz M < 1 M=1 M>1 C. 1x103 Hz u Mach number M = D. 2x103 Hz v Mach angle θ v sinθ = u

3 Question

An observer feels a shock wave and sees that the supersonic jet is at an angle of 20O above the horizon flying in the horizontal direction. What is the speed of the jet (take the speed of sound to be 340 m/s)

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