The Doppler Effect

If you’ve ever been standing on the curb as a car or truck races past with its horn blaring, you’ve noticed that the tone of the horn appears to change as the vehicle goes by. The driver inside, leaning on the horn, doesn’t notice any change in frequency at all as she whizzes past at 40 mph – but for you, standing on the curb at a velocity of zero, the frequency change is very noticeable. You’ve just witnessed a well-known phenomenon in physics called “the Doppler effect”. The Doppler effect is taken advantage of by humans in everything from radar, to radio direction finding devices, to lifesaving medical equipment.

To understand how it works, imagine that you’re a tiny person sitting inside a toy boat in the middle of a still lake. Someone throws a rock into the lake. If your boat isn’t moving, the ripples made by the rock will strike your boat at regular intervals. However, if your toy boat is heading toward the point of entry of the rock, then as you approach the point of entry you will strike ripples more frequently. As you leave, you will strike the ripples less frequently. Therefore, the frequency as you head toward the ripples is higher than the frequency as you head away. This is the Doppler effect.

How do we explain all this using math? The water waves travel from the source at a velocity v toward your boat, and the observer in the boat moves toward the waves at velocity v_o. The waves and the boat are approaching each other at a velocity v + v_o.

The wavelength l_s (meters/cycle) of a wave is equal to the velocity v of the wave (meters/second) times its period T_s (seconds/cycle), or l_s = vT_s . To the observer in the moving boat, the period T_o until the next wave is shorter, meaning the wavelength can also be written as l_s = (v + v_o) T_o . We can set these two equations equal to one another to come up with:

vT_s = (v+ v_o) T_o

The periods T_s and T_o are the reciprocals of frequency f_s and f_o.  Therefore,

v / f_s = (v+ v_o) / f_o

and

f_o = (v + v_o) f_s / v           (1)

So now we’ve seen that if the boat is approaching the wave, the observed frequency is higher than original source frequency. It’s now easy to convince yourself that if the boat is receding from the wave, the observed frequency is lower than that of the source and given by

f_o = (v – v_o) f_s / v

To understand how the Doppler effect is used in real life, consider a plane flying towards a stationary airport radar. Since radar uses an EM wave, the velocity v is now the speed of light c. The frequency of the source radar is f_s , and f_o is the frequency of the echoed (observed) radar. We can solve equation (1) for the velocity of the plane v_o  to get

v_o = c (f_{o –} f_s )/ f_s

Such Doppler radar allows us to find the velocity of the plane relative to the stationary air traffic control tower. We say that the frequency has red-shifted (shifted toward the infrared in the spectrum) from its original transmitted frequency to a larger echoed frequency. Had the plane been going away from the tower, we’d say that the frequency has blue-shifted to a smaller echoed frequency (toward the ultraviolet).

Suppose now both the source and the observer are moving. Can the source’s radar determine the observer’s velocity? Yes indeed. In fact, the generalized Doppler shift (where the first sign denotes approaching and the second sign denotes departing) can be written as:

f_o = f_s (c +/- v_o) / (c -/+ v_s)

So if you are the source, you know f_s, your velocity v_s, and you observe the echoed frequency f_o. Since c is just the speed of light, you can now solve this equation for v_o, the velocity of the observer. If you want to read more about the Doppler effect, the following website is a good place to start:

https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Map%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/17%3A_Sound/17.08%3A_The_Doppler_Effect

The Doppler effect is typically explained using velocity, a vector with magnitude (speed) and direction. We made some simplifying assumptions so we could ignore direction and focus on the relative speed between two objects. We will come back and talk about direction, so have no fear! But before doing that, we will take a brief excursion into the sonic boom, a first cousin of the Doppler effect caused by aircraft flying faster than the speed of sound.