Topic 11: Wave phenomena 11.2 Doppler effect

1. 11.2.1 Describe what is meant by the Doppler effect. 11.2.2 Explain the Doppler effect by reference to wavefront diagrams for moving-detector and moving-source situations. 11.2.3 Apply the Doppler effect equations for sound. 11.2.4 Solve problems on the Doppler effect for sound. 11.2.5 Solve problems on the Doppler effect for electromagnetic waves using the approximation f = vf/c. 11.2.6 Outline an example in which the Doppler effect is used to measure speed. Topic 11: Wave phenomena11.2 Doppler effect

2. The Doppler Effect When a car passes you on the street, the frequency of its engine note appears to change. This is due to the ‘Doppler effect’: It can occur due to any relative motion; i.e. Either due to the source or the observer moving. The Doppler effect is the change in observed frequency of a wave due to relative motion between the wave source and an observer.

3. The Doppler effect Johann Christian Doppler1803-1853 The diagram below represents waves emitted by a source of sound, S, which is stationary relative to the air. The velocity of the sound waves relative to the air is v.

4. Describe what is meant by the Doppler effect. The Doppler effect has to do with moving sound sources (or moving observers). For example if a sound source of frequency f approaches you at a speed us its wave fronts will bunch together and you will hear a frequency f’ which is higher than f. ( f’ > f). On the other hand if a sound source recedes from you at a speed us its wave fronts will stretch out and you will hear a frequency f’ which is lower than f. ( f’ < f). Topic 11: Wave phenomena11.2 Doppler effect us

5. The next diagram represents waves emitted by a source which is moving(with velocity vs) relative to the air. λ λ Speed same, frequency and wavelength change Motion of the source produces a change in the wavelength, longer wavelength behind, shorter wavelength in front of source.

6. Doppler effect for sound Hyperlink

9. Doppler effect for sound with moving observer

10. Explain the Doppler effect by reference to wavefront diagrams for moving-detector and moving-source situations. Consider the ice-cream truck that parks in the hood playing that interminable music. Since the ice cream truck is not yet moving, the wave fronts are spherically symmetric. Topic 11: Wave phenomena11.2 Doppler effect The cross-section is symmetric. Planar cross-section us = 0 us = 0

11. Explain the Doppler effect by reference to wavefront diagrams for moving-detector and moving-source situations. Whether Dobson is in front of the truck, or behind it, he hears the same frequency. Topic 11: Wave phenomena11.2 Doppler effect Us = 0 FYI Even though the sound waves are drawn as transverse, they are in actuality longitudinal.

12. Explain the Doppler effect by reference to wavefront diagrams for moving-detector and moving-source situations. Suppose the ice-cream truck is now moving, but ringing the bell at the same rate as before. Note how the wave fronts bunch up in the front, and separate in the back. The reason this happens is that the truck moves forward a little bit during each successive spherical wave emission. Topic 11: Wave phenomena11.2 Doppler effect us FYI Don’t forget, the actual speed of the wavefronts through the stationary medium of the air is the speed of sound v (bunched or otherwise).

13. Explain the Doppler effect by reference to wavefront diagrams for moving-detector and moving-source situations. Dobson will hear a different frequency now, depending on his position relative to the truck. If Dobson is in front of the moving truck, f’ > f. If he is behind, he will hear f’ < f. Topic 11: Wave phenomena11.2 Doppler effect f’ < f f’ > f us

14. ’  = vT d = usT uS Explain the Doppler effect by reference to wavefront diagrams for moving-detector and moving-source situations. Now we want to look at TWO successive pulses emitted by a source that is MOVING at speed us:  is the “real” wavelength of the sound. Since the speed of sound in air is v,  = vT where T is the period of the sound source. Note that Pulse 1 has traveled  = vT in the time t = T. Topic 11: Wave phenomena11.2 Doppler effect Pulse 1 t = T Pulse 1 t = 0 Pulse 2 t = T

15. Which gives you the most chocolates?

16. ’  = vT d = usT Pulse 1 t = T Pulse 1 t = 0 Pulse 2 t = T uS Explain the Doppler effect by reference to wavefront diagrams for moving-detector and moving-source situations. ’ is the wavelength detected by the observer. The distance d between the emission of the two pulses is simply the velocity of the source us times the time between emissions T. Thus d = usT. Since  = d + ’ then ’ =  – d = vT – usT. So ’ = (v – us)T Topic 11: Wave phenomena11.2 Doppler effect

17. ’  = vT d = usT Pulse 1 t = T Pulse 1 t = 0 Pulse 2 t = T uS Explain the Doppler effect by reference to wavefront diagrams for moving-detector and moving-source situations. From v = ’f’ and ’ = (v – us)T we see that f’ = v/’ = v/[ ( v – us )T ] f’ = ( 1/T )v/( v – us ) f’ = f[ v / ( v – us ) ]. Topic 11: Wave phenomena11.2 Doppler effect [ f = 1/T ] FYI If the source is receding, replace us with –us.

18. f’ = f[ v / ( vus ) ] Doppler effect moving source v is the speed of sound. Approach (-us), recede (+us). Apply the Doppler effect equations for sound. Topic 11: Wave phenomena11.2 Doppler effect EXAMPLE: A car horn has a frequency of 520 Hz. The car is traveling to the right at 25 ms-1 and the speed of sound is 340 ms-1. (a) What frequency is heard by the driver? SOLUTION: (a) The driver has a relative speed of us = 0. Thus f’ = 520 Hz.

19. f’ = f[ v / ( vus ) ] Doppler effect moving source v is the speed of sound. Approach (-us), recede (+us). Apply the Doppler effect equations for sound. Topic 11: Wave phenomena11.2 Doppler effect EXAMPLE: A car horn has a frequency of 520 Hz. The car is traveling to the right at 25 ms-1 and the speed of sound is 340 ms-1. (b) What frequency is heard by Observer 1? SOLUTION: (b) For approach use -us. f’ = 520[ 340/( 340 - 25 ) ] = 560 Hz. Be sure f’ is higher since car is approaching.

20. f’ = f[ v / ( vus ) ] Doppler effect moving source v is the speed of sound. Approach (-us), recede (+us). Apply the Doppler effect equations for sound. Topic 11: Wave phenomena11.2 Doppler effect EXAMPLE: A car horn has a frequency of 520 Hz. The car is traveling to the right at 25 ms-1 and the speed of sound is 340 ms-1. (c) What frequency is heard by Observer 2? SOLUTION: (c) For approach use +us. f’ = 520[ 340/( 340 + 25 ) ] = 480 Hz. Be sure f’ is lower since the car is receding.

21. o Explain the Doppler effect by reference to wavefront diagrams for moving-detector and moving-source situations. Suppose the source is stationary. If the observer is not moving, he will hear the true frequency of the source. Topic 11: Wave phenomena11.2 Doppler effect EXAMPLE: A sound source with wavefronts is shown. A stationary observer (the blue circle) is immersed in the sound. Observing the clock and the wavefronts, determine the frequency f of the source (and the frequency f’ detected by the observer). SOLUTION: Both frequencies are the same. Thus f = f’ = 5 cycles / 12 s = 0.42 Hz. 5 wavefronts pass observer in 12 seconds.

22. v∆t o Explain the Doppler effect by reference to wavefront diagrams for moving-detector and moving-source situations. The diagram shows which wavefronts pass the observer in the time interval ∆t (red): Because the wave speed is v, the length of the region in question is v∆t. If we divide the length v∆t by the wavelength  we get the number of cycles (or wavefronts). Thus #cycles detected = v∆t/. Topic 11: Wave phenomena11.2 Doppler effect FYI Recalling the definition of f’ we have f’ = #cycles detected /∆t =(v∆t/)/∆t = v/. But v/ = f so that f’ = f. (EXPECTED)

23. o Explain the Doppler effect by reference to wavefront diagrams for moving-detector and moving-source situations. Suppose the source is stationary. If the observer is moving TOWARD the source at uo, he will hear a higher frequency. Topic 11: Wave phenomena11.2 Doppler effect EXAMPLE: A moving observer (the blue circle) is now immersed in the sound. Observing the clock and the wavefronts, determine the frequency f of the source (and the frequency f’ detected by the observer). SOLUTION: f = 5 cycles / 12 s = 0.42 Hz. Now 8 wavefronts pass the observer in 12 s. Thus f’ = 8 cycles / 12 s = 0.66 Hz.

24. v∆t uo∆t o Explain the Doppler effect by reference to wavefront diagrams for moving-detector and moving-source situations. The diagram shows which wavefronts pass the moving observer in the time interval ∆t: The red regionv∆t is still due to the wave speed itself (still 5 cycles in our example). The blue regionuo∆t is due to the oberver’s speed uo(3 cycles in our example). Now for the observer #cycles detected = (v∆t +uo∆t)/. Thus f’ = #cycles detected /∆t =[(v∆t + uo∆t) /  ]/∆t = v/ + uo/. Topic 11: Wave phenomena11.2 Doppler effect

25. f’ = f[ ( vuo ) / v ]. Doppler effect moving observer v is the speed of sound. Approach (+uo), recede (-uo). Explain the Doppler effect by reference to wavefront diagrams for moving-detector and moving-source situations. From f’ = v/ + uo/ and the relation v = f we can write f’= v/ + uo/ f’= f + uo/(v/f) f’= fv/v + fuo/v f’ = f[ ( v+uo ) / v ]. If the observer is moving AWAY from the source, substitute –uo for uo. Topic 11: Wave phenomena11.2 Doppler effect FYI Pick the sign that makes f’do what is expected.

26. f’ = f[ ( vus ) / v ]. Doppler effect moving observer v is the speed of sound. Approach (+us), recede (-us). Explain the Doppler effect by reference to wavefront diagrams for moving-detector and moving-source situations. Topic 11: Wave phenomena11.2 Doppler effect EXAMPLE: A car horn has a frequency of 520 Hz. The car is stationary and the speed of sound is 340 ms-1. (a) An observer approaches the car at 25 ms-1. What frequency is heard by the observer? SOLUTION: (a) For approach use +us. f’ = 520[ ( 340 + 25 ) / 340 ] = 560 Hz. Be sure f’ is higher since the observer is approaching the car.

27. f’ = f[ ( vus ) / v ]. Doppler effect moving observer v is the speed of sound. Approach (+us), recede (-us). Explain the Doppler effect by reference to wavefront diagrams for moving-detector and moving-source situations. Topic 11: Wave phenomena11.2 Doppler effect EXAMPLE: A car horn has a frequency of 520 Hz. The car is stationary and the speed of sound is 340 ms-1. (b) An observer recedes from the car at 25 ms-1. What frequency is heard by the observer? SOLUTION: (b) For receding use -us. f’ = 520[ ( 340 - 25 ) / 340 ] = 480 Hz. Be sure f’ is lower since the observer is receding from the car.

28. ∆f = (v/c)f Doppler effect for light v is the relative speed between the source and the observer. Solve problems on the Doppler effect for electromagnetic waves using the approximation f = vf/c. We have discussed the Doppler effect for sound waves, but we can also discuss it for electromagnetic waves, with this major difference: The speeds of source usand observer uo are very much less than the speed of light c. We define v to be the relative velocity between source and observer. Thus v = us – uo. Then f’, the frequency detected by the observer, is related to f, the frequency of the light source, by the formula f’ = f + (v/c)f. Topic 11: Wave phenomena11.2 Doppler effect

29. Solve problems on the Doppler effect for electromagnetic waves using the approximation f = vf/c. Topic 11: Wave phenomena11.2 Doppler effect EXAMPLE: A star in another galaxy is traveling away from us at a speed of 5.6106 ms-1. It has a known absorption spectrum line that should be located at 520 nm on an identical stationary star. Where is this line located on the moving star? SOLUTION: Use c = f to find f = c/: f = c/ = 3108 / 52010-9 = 5.81014 Hz. The star is moving away from us so the effect is to decrease the frequency. Thus v = -5.6106 ms-1. f’ = f + (v/c)f = f[ 1 + (v/c)] f’ = 5.81014 ( 1 - 5.6106/ 3108 ) = 5.71014 Hz. ’ = c/f’ = 3108 / 5.71014 = 5.310-7 = 530 nm.

30. Moving observer Wavelength fixed, speed and frequency change

32. Doppler shift for SOUND source Be careful to apply the + or – the correct way.

33. Doppler sound problems

34. Answers

35. Questions Tsokos page 248 Q’s 1-7 Hamper page 136 Q’s 17-19.

36. Solve problems on the Doppler effect for electromagnetic waves using the approximation Students should appreciate that the approximation may be used only when v << c.

37. Caught by the fuzz. Take the case of the car and the radar speed trap. The emitted frequency of the police radar gun is f, the car acts like a moving observer receiving it at f'. It reflects it back at f', acting now as a moving source and the police receive it back shifted once again as f''.The total change in frequency (f'' - f) is therefore doubled.

40. Question It can be shown that the relative Doppler shift for electro-magnetic radiations like light, radio waves etc is given (approximately) by where c is the speed of light and v is the relative speed of source and observer. The speed of a car is being measured by a police-person using a "radar speed-measuring gun". The frequency of the transmitted signal is 5GHz. When "mixed" the transmitted and received signals beat with a frequency of 750Hz. If the speed limit for the road is 110kmh-1, should the driver be fined or not?

41. Adding colour

42. Analysing blood flow

43. Questions Hamper page 138 Q’s 20-22.

44. A B C D Outline an example in which the Doppler effect is used to measure speed. PRACTICE: The absorption spectra of stars of varying distances from Earth are shown here. The sun is the bottom spectrum (D). Which star has the highest relative velocity to Earth? Is it moving towards us, or away from us? SOLUTION: From ∆f = (v/c)f we see that the higher the relative velocity v the greater the shift ∆f. Thus A is our candidate. As a reference look at the heavy black line in D. It is shifting to the red region. Thus its frequency is less, and thus A is receding. Topic 11: Wave phenomena11.2 Doppler effect

45. This is the so-called “redshift” and it is used as evidence for an expanding universe. A B C D Outline an example in which the Doppler effect is used to measure speed. PRACTICE: The absorption spectra of stars of varying distances from Earth are shown here. The sun is the bottom spectrum (D). What is the approximate speed at which star D is receding from Earth? SOLUTION: Use c = f ( f = c/ ): From A, f= 3108/39010-9 = 7.71014 Hz. From D, f’ = 3108/49010-9 = 6.11014 Hz. Then ∆f =f’ – f = (6.1–7.7)1014 = -1.11014 Hz. Finally from ∆f = (v/c)f we get v = c∆f/f = (3108)(-1.11014)/7.71014 = -4.31107 ms-1. Topic 11: Wave phenomena11.2 Doppler effect Receding…

46. Outline an example in which the Doppler effect is used to measure speed. EXAMPLES: The following are all examples of the Doppler effect. 1. The radar gun used by police. It uses the form v = c∆f/f to find the velocity of the car. It actually measures the difference in frequency between the emitted radar beam, and the reflected-and- returned one. Topic 11: Wave phenomena11.2 Doppler effect

47. Outline an example in which the Doppler effect is used to measure speed. EXAMPLES: The following are all examples of the Doppler effect. 2. A Doppler ultrasound test uses reflected sound waves to see how blood flows through a blood vessel. It helps doctors evaluate blood flow through major arteries and veins, such as those of the arms, legs, and neck. It can show blocked or reduced blood flow through narrowing in the major arteries of the neck that could cause a stroke. It also can reveal blood clots in leg veins (deep vein thrombosis, or DVT) that could break loose and block blood flow to the lungs (pulmonary embolism). During pregnancy, Doppler ultrasound may be used to look at blood flow in an unborn baby to check the health of the fetus. Topic 11: Wave phenomena11.2 Doppler effect

48. Outline an example in which the Doppler effect is used to measure speed. EXAMPLES: The following are all examples of the Doppler effect. 3. A Doppler radar is a specialized radar that makes use of the Doppler effect to produce velocity data about objects at a distance. It does this by beaming a microwave signal towards a desired target and listening for its reflection, then analyzing how the frequency of the returned signal has been altered by the storm’s motion. This variation gives accurate measurements of the radial component of a storm’s velocity. Topic 11: Wave phenomena11.2 Doppler effect

49. Outline an example in which the Doppler effect is used to measure speed. EXAMPLES: The following are all examples of the Doppler effect. 4. If you look at a rotating luminous object like the sun, you see that one side is moving away from the observer while the other side is approaching the observer. Once again, the formula v = c∆f/f comes into play. The sun’s right side will be red shifted, whereas the left will be blue shifted. We can then find the speed of rotation of the sun (or any star, even if it is quite distant). Topic 11: Wave phenomena11.2 Doppler effect

50. Solve problems on the Doppler effect for sound. Topic 11: Wave phenomena11.2 Doppler effect Since the source is receding ’ had better increase. It will increase by how far d = VT the source has traveled. Note that the source is NOT traveling at v, the speed of sound!