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# Doppler Effect with math

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Class notes to accompany introduction to Doppler effect with equations derived.

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### Doppler Effect with math

1. 1. An Introduction to the Doppler Effect <ul><li>E. Alexander Burt </li></ul><ul><li>Prepared for Potomac School 2/18/09 </li></ul>
2. 2. What is the Doppler Effect? <ul><li>Definition: a shift in the observed frequency of a wave due to relative motion of either the source or the observer. </li></ul><ul><li>That's nice. What does it mean? Examples: </li></ul><ul><ul><li>Video of British fire truck going past </li></ul></ul><ul><ul><li>Tuning fork </li></ul></ul><ul><ul><li>Ripple tank </li></ul></ul>
3. 3. How does the doppler effect work? <ul><li>There are three cases: </li></ul><ul><ul><li>Moving source, stationary observer </li></ul></ul><ul><ul><li>Moving observer, stationary source </li></ul></ul><ul><ul><li>Source and observer both moving </li></ul></ul><ul><li>Notes: </li></ul><ul><ul><li>By ”moving” we mean moving with respect to the wave medium. </li></ul></ul><ul><ul><li>The Doppler effect also works with light and other electromagnetic waves, but we will not cover that today. </li></ul></ul>
4. 4. Moving Source, Stationary Observer <ul><li>As the source moves, it ”follows” the wave in front of it and moves away from the wave behind it </li></ul>
5. 5. Moving Source, Stationary Observer (part deux) <ul><li>The wavelength in front of the moving source is shorter because the source moves in between each wave. </li></ul><ul><li>As long as the speed of the source is slower than the wave speed, the new wavelength is: </li></ul><ul><li>l ' = l – v source T </li></ul><ul><li>v source is positive for source moving toward observer and negative for source moving away. T is the wave period. </li></ul>
6. 6. Moving Observer, Stationary Source (part trois) <ul><li>Taking our equation from before: </li></ul><ul><li>l ' = l – v source T </li></ul><ul><li>And combining it with v wave = f l we can get an equation for the shifted frequency f'. (lots of algebra skipped here...) </li></ul>f'= f 1-(v source / v wave )
7. 7. Moving Observer, Stationary Source <ul><li>Now the source remains stationary and the observer moves towards or away from the source. </li></ul><ul><li>To the observer, the wave speed is changed and therefore the frequency changes also. </li></ul>S O
8. 8. Moving Observer, Stationary Source (part deux) <ul><li>To the observer, the wave speed appears to be v wave +v obs . The wavelength remains the same. </li></ul><ul><li>Since f = v/ l , the shifted frequency f' will now be: </li></ul><ul><li>f' = (v wave +v obs ) / l </li></ul><ul><li>As before, positive v obs means motion toward the source, negative means away. </li></ul>
9. 9. Moving Observer, Stationary Source (part trois) <ul><li>Again, taking our equation from the last page: f' = (v wave +v obs ) / l </li></ul><ul><li>And again, combining it with: v wave = f l </li></ul><ul><li>And again, skipping a lot of algebra, we get: </li></ul>f'= f (1+ v obs /v wave )
10. 10. What if both the source and the observer are moving? <ul><li>In a nutshell, it means twice as much work. </li></ul><ul><ul><li>Figure out the doppler shifted frequency that the observer would see if the observer were not moving. </li></ul></ul><ul><ul><li>Apply the moving observer doppler shift to your answer. </li></ul></ul>
11. 11. How does it work for light waves? <ul><li>The equations are slightly different because light is not a mechanical wave – therefore we cannot measure speeds with respect to a medium. </li></ul><ul><li>The effects are similar: source moving towards observer produces higher observed frequency, source moving away produces lower frequency </li></ul><ul><li>We'll save the math for another day. </li></ul>