PhysicsLAB: The Doppler Effect

The Doppler Effect is the apparent change in a wave's frequency resulting from the relative velocity between the source of the waves and the observer. Although the Doppler Effect is generally associated with sound waves, it is applicable to any type of wave. As a rule of thumb, if the distance between the source and the observer decreases, the apparent frequency (called "f prime" or f') is higher than the actual, real frequency of the source.

Case 1: Observer moving.

If the observer is moving but the source is stationary, the apparent frequency change is evidenced directly and can be calculated with the formula:

Δf / f = v / v_w

where

Δf is the apparent change in frequency,
f is the original frequency,
v is the velocity of the observer with respect to the stationary source,
v_w is the speed of the wave.

To calculate the final frequency the observer records you would use the relationships:

when the observer approaches a wave source: f ' = f + Δf --- a higher frequency
when the observer recedes from a wave source: f ' = f - Δf --- a lower frequency

Remember that the actual frequency of the source remains unchanged and that this apparent change in frequency is due to the relative velocity between the source and the observer. Also remember that it is the frequency that is manipulated mathematically when the observer is moving either towards or away from the source.

An analogy. Suppose that you are a parent watching your child play at the beach. If the child stands still in the shallow water, you note that one wave reaches your child's position each second. However, suppose that the child decides to "rush out to meet" the waves. The child will encounter the waves more frequently as he rushes out towards the deeper water. Instead of one wave reaching him each second, he might meet two or three each second. Conversely, if the child "runs away from the waves" back into the shore, instead of one wave reaching him every second, a wave might only reach him once every 1.5 to 2 seconds. The child can change the "apparent frequency" of the oncoming waves through his motions. How much the frequency changes depends on the child's relative speed.

Refer to the following information for the next three questions.

Suppose there is a stationary air raid siren that is emitting a frequency of 880 hz. You may use 340 m/sec as the speed of sound.

	If an emergency official (listener/observer) is racing at 34 m/sec to reach his command post then what frequency would he hear as he approaches the siren's source?

	What frequency would he hear if he overshot his turn off and continued at his same rate of speed past his command post?

	Describe how a graph of apparent frequency vs position would look for this scenario.

Case 2: Source moving.

When the source is moving, the wavelength is the quantity that is directly affected by the relative motion, not the frequency. Our formula becomes:

Δλ / λ = v / v_w

The following diagram shows a source moving towards the right side of the screen at a constant velocity.

As this source moves towards the right, observers located behind the source would receive fewer waves per second so they would perceive an apparent frequency that is lower than the source's true frequency.

Notice that the wavelengths are being drawn farther apart. Longer wavelengths "produce" a lower frequency.

As this source moves towards the right, observers located in front of the source would receive more waves per second so they would perceive an apparent frequency that is higher than the source's true frequency.

Notice that the wavelengths are being crowded closer together. Shorter wavelengths "produce" a higher frequency.

Before you examine the variables in this equation and work an example, spend some time with this physlet animation of the Doppler Effect by Wolfgang Christian at Davidson College. Notice what happens to the separation in the wavelengths as you adjust the speed of the source. Also notice the shape of the bow wave if you choose a relative speed greater than "1" -- that is, a speed greater than the wave speed.

In this formula,

Δλ is the apparent change in wavelength,
λ is the original wavelength when the source is stationary,
v is the velocity of the source with respect to the stationary observer, and
v_w is the speed of the wave.

In this case, four steps are needed to calculate the apparent frequency.

Determine the original wavelength using

λ = v_w/ f

Determine the apparent change in wavelength using

Δλ = λ * (v / v_w)

Determine the new apparent wavelength using

λ' = λ ± Δλ

receding source:
wavelengths drawn apart
λ' = λ + Δλ

approaching source:
wavelengths crowded together
λ' = λ - Δλ

Determine the new apparent frequency using

f ' = v_w / λ'

ALERT! The Doppler Effect for the relative motion of a source does NOT yield the same results as for an observer moving at the same rate. You MUST choose the correct formula when solving your problem!

Refer to the following information for the next three questions.

Suppose there is an ambulance emitting a frequency of 880 hz as it rushes to the hospital at 34 m/sec. You may use 340 m/sec as the speed of sound.

	What frequency would a bystander hear as the ambulance approaches his location?

	What frequency would he hear once the ambulance has passed his location?

	Describe how the graph of apparent frequency vs position in this scenario of the source moving at 34 m/sec would differ from the graph in the previous scenario involving the listener moving at 34 m/sec.