An ambulance siren sounds higher as it comes toward you and lower as it goes away. The siren never changes; the waves reaching you do. The speed of sound is set by the air (What Sets a Wave's Speed), so motion changes the spacing of the crests you receive. As in Sound: Pitch and Loudness, the frequency you hear is the pitch.
Key idea
$$f' = f\,\frac{v \pm v_o}{v \mp v_s}$$ Use the upper signs when source and observer move toward each other, the lower signs when they move apart. Here $v$ is the speed of sound, $v_o$ the observer's speed and $v_s$ the source's. Approaching raises the pitch.
Why it works
A source with frequency $f$ sends a crest every $T = 1/f$. If it moves toward you at $v_s$, each crest starts $v_sT$ closer than the one before, so the crests arrive $\lambda' = (v - v_s)T$ apart. They still move at $v$, so you hear $f' = \frac{v}{\lambda'} = f\frac{v}{v - v_s}$. A moving observer meets crests more often: $f' = f\frac{v + v_o}{v}$.
Common mistake
Thinking the pitch keeps rising as an approaching source gets closer. At constant speed the higher pitch is steady; it drops suddenly to a lower steady pitch as the source passes you. Check with the formula: nothing in it depends on the distance.
Worked example: An ambulance
A $600\ \text{Hz}$ siren moves at $30\ \text{m/s}$. What do you hear as it approaches and as it moves away? ($v = 343\ \text{m/s}$.)
- Approaching (upper sign in the denominator): $$f' = 600\,\dfrac{343}{343 - 30} = 600(1.096) = 658\ \text{Hz}.$$
- Moving away: $$f' = 600\,\dfrac{343}{343 + 30} = 600(0.920) = 552\ \text{Hz}.$$
- Higher while approaching, lower after it passes.
Answer: $658\ \text{Hz}$, then $552\ \text{Hz}$
Worked example: A moving listener
You drive at $25\ \text{m/s}$ toward a factory whistle of $500\ \text{Hz}$ that stands still. What do you hear?
- Moving toward the source: $f' = f\,\dfrac{v + v_o}{v} = 500\,\dfrac{343 + 25}{343}$.
- $f' = 500(1.073) = 536\ \text{Hz}$.
Answer: $536\ \text{Hz}$
Worked example: Measuring a speed
A train whistle of $400\ \text{Hz}$ sounds like $420\ \text{Hz}$ as the train approaches. How fast is the train?
- $420 = 400\,\dfrac{343}{343 - v_s}$, so $343 - v_s = \dfrac{400(343)}{420} = 326.7$.
- $v_s = 343 - 326.7 = 16\ \text{m/s}$.
Answer: About $16\ \text{m/s}$
Light does the same. Light from a galaxy moving away arrives with longer wavelengths (redshift), which is how astronomers measure the motions of stars and galaxies.