Doppler Effect — Approaching Ambulance Siren (Worked Example)

Medium

Problem

An ambulance drives toward you at 30 m/s sounding its 700 Hz siren. The speed of sound in air is 343 m/s. What frequency do you hear as it approaches?

Given

f = 700 Hz (source)
v_s = 30 m/s toward observer
v = 343 m/s
observer stationary

Find

Observed frequency f′.

Principle

A source moving toward the observer squeezes wavefronts together, shortening the observed wavelength and raising the heard frequency.

Formula

f' = f\frac{v}{v - v_s}

f′ = f · v/(v − v_s) for approaching source

Solution

Write the Doppler formula for an approaching source: $f' = 700 \times 343/(343 - 30)$
Denominator: $343 - 30 = 313\ \mathrm{m/s}$
Ratio: $343/313 = 1.0958$
Multiply: $f' = 700 \times 1.0958 = 767\ \mathrm{Hz}$

≈ 767 Hz

Unit check

(m/s)/(m/s) is unitless, so Hz × unitless = Hz. Consistent.

What this means physically

You hear 767 Hz instead of 700 — about a semitone and a half higher in musical terms. After it passes, the sign flips and the pitch drops below 700 Hz.

Common mistake

Using (v + v_s) for an approaching source. Approaching compresses waves: denominator gets smaller (v − v_s), frequency goes up.