How To Solve Doppler Effect Problems

If you’re having difficulty solving classical Doppler shift problems on your physics homework, you’re not alone! Many textbooks portray the Doppler shift formula in confusing ways that obscure real understanding. It is often stated as either several different formulas (one formula for a moving source and stationary observer, one formula for a stationary observer and moving source, etc) or one formula with a cruel and impossible-to-remember sign convention (for example, “source velocity is positive when the source moves in the direction of the emitted wave”).

I prefer to use what one of my past professors used to call the “Method of Thinking” to solve Doppler shift problems. (This is just one use of a more general Method of Thinking, in which you simply think about a problem to find the answer.) The basic ingredients are one easy-to-remember “incomplete formula,” two simple rules, and an instruction to think:

  1. \displaystyle f_{observed} = f_{emitted} \frac{|v|_{wave} \blacksquare |v|_{observer}}{|v|_{wave} \blacksquare |v|_{source}}, where the velocities are relative to the medium and \blacksquare stands for either plus or minus. The following two rules determine which sign to choose. (The formula is written in a general way to emphasize that the Doppler shift formula applies to many kinds of waves, but it is good to think of sound as a representative example. In that case, the wave speed is the speed of sound and the medium is air.)
  2. If the source and receiver move towards each other (and one is stationary), the observed frequency increases.
  3. If the source and receiver move away from each other (and one is stationary), the observed frequency decreases.
  4. Think!

Examples

  1. Suppose the observer is stationary with respect to the medium and the source moves towards it. Then v_{observer} = 0 and Rule 2 tells us that the frequency must increase. So the correct formula to use is \displaystyle f_{observed} = f_{emitted} \frac{|v|_{wave}}{|v|_{wave} - |v|_{source}}. We choose the minus sign because shrinking the denominator of a fraction makes it grow.
  2. Suppose the source is stationary with respect to the medium and the source moves away from it. Then v_{source} = 0 and Rule 3 tells us that the frequency must decrease. So the correct formula to use is \displaystyle f_{observed} = f_{emitted} \frac{|v|_{wave} - |v|_{observer}}{|v|_{wave}}. We choose the minus sign because shrinking the numerator of a fraction makes it decrease.
  3. Suppose both the source and observer are moving with respect to the air. This situation involves a little bit more Rule 4 than the first two examples. The trick is to imagine a third entity (call it “The Phantom”) at rest with respect to the air. The Phantom listens to the sound emitted by the source (call this “Stage 1”) and emits an exact copy to the observer (call this “Stage 2”). Thus, The Phantom is both an observer and a receiver. In Stage 1, The Phantom hears a frequency given by \displaystyle f_{phantom} = f_{emitted} \frac{|v|_{wave}}{|v|_{wave} \blacksquare |v|_{source}}, where the sign is determined as in the previous examples. In Stage 2, the observer hears a frequency given by \displaystyle f_{observed} = f_{phantom} \frac{|v|_{wave} \blacksquare |v|_{observer}}{|v|_{wave}}, where the sign is determined as before. Inserting the equation for the Phantom frequency into the equation for the observed frequency gives the correct formula. Note that the top of the Phantom frequency fraction will cancel with the bottom of the observed frequency fraction.
  4. If the source and the observer are at rest relative to the ground, there can still be a Doppler shift if the medium is moving. This is equivalent to the previous case. (This means that on a windy day, your voice could appear to be lower or higher.)

It is not useful to remember the specific equations; instead, try to learn the process that leads to them. This allows you generate formulas as you need them from a few basic concepts, instead of cluttering your head with multiple and/or confusing equations. Note: this is a good example of the power of the physicist’s mode of thinking.

Continue forward to learn where Rules 1, 2, and 3 come from!

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