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# Doppler effect for a moving observer

What happens when just the observer is moving? Explore the Doppler effect in sound waves as we examine how the frequency changes when an observer moves towards or away from a sound source. Understand the formulas used to calculate observed frequency, and how speed influences the pitch of the sound. Created by David SantoPietro.

## Want to join the conversation?

• Why is this not same as saying that the source is moving towards the observer? • The best way to visualize what is happening is by doing the following:
The source emits a frequency of X or a period of 1/X. Now if you are standing still the F(0) = F(S) (O = observed frequency; S = source Frequency). Now lets think in terms of time per wave (sec/Wave) I.E. the period; Lets say you hear a constant tone for which the time between each wave is 5 seconds (5 seconds/Wave), now as you are moving toward the source you are effectively reducing the time with which you perceive each wave even though the source is still emitting a wave every 5 seconds. This will give you the equation | T-observed = T-source[V-wave/(V-wave + V-observer) |. Notice what this equation is saying: It is saying that you have allowed the sound (each wave) to reach you faster than what the speed (velocity) of sound would normally allow [the faster you move toward the source--larger denominator--you experience an increasingly smaller period (observed) of the original period (Source, 5). Now think of the opposite for the equation | T-observed = T-source[(V-wave - V-source/V-wave) | where the source is moving toward you and you are standing still. When the source moves toward you it is still emitting a wave every 5 seconds but because the speed of sound is relatively constant and the moving source can not increase the speed of sound it causes an increase in frequency when moving toward you by "chasing" the previous wave emitted, effectively reducing the distance between each wave (wavelength) in front of it. So Basically, The first equation can be interpreted as how much faster the sound emitted from the source reaches the observer compared to what the speed of sound allows; whereas the second equation can be interpreted as how much the source has caught up with the speed of sound. Notice that both equations decrease the period or increase the frequency when he distance between the source and observer becomes less, however, the first does it by making the speed of sound "faster" and the other does it by making the speed of sound "slower".
• I'm sure I've probably missed something, but doesn't the formula expressed in this video seem different to the one Sal expresses in the video: https://www.khanacademy.org/science/physics/mechanical-waves-and-sound/doppler-effect/v/doppler-effect-formula-when-source-is-moving-away • How do you get the wave velocity if they only give you the wave frequency and source velocity • in the video named doppler effect formula when the source is moving away the formula for frequency seems to be different.which one do i use in my problems or numericals • What would the formula be if the two objects are moving
1) towards each other?
2) away from each other?
3) both in the same direction?

I'm using the formula
Fo = Fs ( (V+VL) / (V-VS) )

Fo = freq of observer
Fs = freq of source
V = speed of sound
Vo = speed of observer
Vo = speed of source

Operating under the idea that the values of Vo and Vs are + (moving toward) or - (moving away) from each other.

Is this correct?
If so,
What is the formula if they move in the same direction?
(1 vote) • I think you made a typo in defining your velocities. If you mean to say VL = Speed of observer and VS = speed of source than I believe that formula is correct.

I also agree with your assignment of + and - to Vo and Vs provided the + and - are such as in your formula.

If they move in the same direction you can still use this formula? FOr instance if the source was moving at 5ms^-1 in the positive x direction and an observer at 10ms^-1 also in the positive x direction, Vo= +10 and Vs = +5 (Important - this is they velocity relative to the air that is at rest - do NOT e.g. use Vo=+5 and Vs=0 because the observer is moving 5ms^-1 faster than the source!).

Hope that helped :).
• How would you solve for the velocity of the observer if the observer first goes towards the source then away from the source? For example a motor bike approaching a fire engine then passing it?
(1 vote) • What if the observer and the source emitting the sound are both moving at different/same velocities?How would we then tackle the situation?
(1 vote) • Good question.
The only important think its the relative speed, think about it , when you say you are "still" and an ambulance passes at 100km/h you are really talking about relative speed because you are not still you are moving across the solar sistem , and the galaxy etc...
So if you both move at the same speed you wouldn notice the doppler effect for example
• What will happen if both the source and the observer moves
a ) Towards each other
b) Away from each other
(1 vote) • how does the speaker strike your location?
(1 vote) • At about , why is time taken by the wave to reach the observer TOBS?
(1 vote) • Maybe this is not what you are asking, but T_obs is just the name of a variable he chooses for the time taken by the next wave crest to reach the observer. "T" for "time" and "obs" is short for "observer". It will be the PERIOD that the OBSERVER measures for the waves. Because they are moving, they will meet the next crest SOONER than if they were at rest, so T_obs is different from T_source (the period of the wave as measured by the source of the sound, which would also be measured by an observer at rest).
(1 vote)

## Video transcript

- [Voiceover] The frequency that you'll observe when standing next to a speaker is determined by the rate at which wave crests strike your location. If the speaker moves toward you, you'll hear a higher frequency. If the speaker moves away from you, you'll hear a lower frequency. But what will happen if you run toward the speaker? You'll hear a higher frequency because more wave crests will strike you per second. If you run away from the speaker, you'll hear a lower frequency because less wave crests will strike you per second. How do we figure out exactly what frequency you'll hear? To find out, let's zoom in on what's going on. Say a wave crest has just made it to your location. The time it takes until another wave crest hits you will be the period that you'll observe since that will be the time you observed between wave crests. If you're at rest, you'll just have to wait until another wave crest gets to your location. The period you'd observe would be the actual period of the wave emitted by the speaker. If you're running towards the speaker, or wave source, you don't have to wait as long since you'll meet the next wave crest somewhere in between. If you can figure out how long it takes for the next crest to hit you, that would be the period that you'd observe and experience. Let's say you're moving at a constant speed that we'll call VOBS, for the speed of the observer. The distance you'll travel in order to reach the next crest will be your speed times the time required for you to get there. This time is just going to be the period you observe since it'll be the time you experience between wave crests. We'll write the time as TOBS for period of the observer. Similarly, the distance the next wave crest will travel in meeting you, will be the speed of the wave VW times that same amount of time, which is the period you are observing. Now what do we do? We know that the distance between crests is the actual wavelength of the wave, not the observed wavelength but the actual source wavelength emitted by the speaker at rest. If we add up the distance that we ran plus the distance that the next wave crest traveled to meet us, they have to equal one wave length in this case. We can now pull out a common factor of TOBS. If we solve this for the period of the observer, we find that it will be equal to the wavelength of the source, divided by the speed of the wave, plus the speed of the observer. This is a perfectly fine equation for the period experienced by a moving observer but one side's in terms of period and the other side's in terms of wavelength. If we want to compare apples to apples we can put this wavelength in terms of period by using this formula. The velocity of the wave must equal the wavelength of the source divided by the period of the source. Since this wavelength was the actual wavelength emitted by the source or the speaker, we have to also use the actual period emitted by the source not the observed period. If we solve for the wavelength, we'd get that the speed of the wave times the period of the source has to be equal to the wavelength of the source. We can plug in this expression for wavelength and we get a new equation that says that the observed period will be equal to the speed of the wave times the period of the source divided by the speed of the wave plus the speed of the observer. This is a perfectly fine equation to find the observed period, but physicists and other people actually prefer talking about frequency more than period. We can turn this statement that relates periods into a statement that related frequencies by just inverting both sides, or taking one over both sides. We'll get one over the observed period equals the speed of the wave plus the speed of the observer divided by the speed of the wave times the period of the source. But look, one over the observed period is just the frequency experienced by the observer. On the right hand side I'm going to pull out a factor of one over the period of the source, which leaves the velocity of the wave plus the velocity of the observer divided by the velocity of the wave. For the final step, we can put this entirely in terms of frequencies by noting that one over the period of the source is just the frequency of the source. Phew, there it is. This is the formula to find the frequency experienced by an observer moving toward a source of sound. Note that the faster the observer moves, the higher the note or pitch. This formula only works for the case of an observer moving toward a source. What do we do if the observer is moving away from the source? Let's start all over from the very beginning. Just kidding. Since you're running away from the speaker instead of toward it, you can just stick in a negative sign in front of the speed of the observer. So here we have it, a single equation that describes the Doppler shift experienced for an observer moving toward or away from a stationary source of sound. Use the plus sign if you're moving toward the source of the sound and use the negative sound if you're moving away from the source of the sound.