---
title: "The Doppler Effect and Shock Waves"
description: "For sound in still air, source motion changes wave spacing and observer motion changes the rate of meeting waves. Their formulas agree approximately at low speeds. Supersonic motion produces shock wav"
canonical: https://lightmysky.com/learn/science/the-doppler-effect-and-shock-waves-mt_UVfD_coqh6
source: https://lightmysky.com/learn/science/the-doppler-effect-and-shock-waves-mt_UVfD_coqh6.md
retrieved: 2026-11-12
---

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# The Doppler Effect and Shock Waves

For sound in still air, source motion changes wave spacing and observer motion changes the rate of meeting waves. Their formulas agree approximately at low speeds. Supersonic motion produces shock waves; an ideal Mach cone relates its half angle to source speed.

Subject: Science · Area: Waves, Light & Sound · Ages 18 to 20
Page: https://lightmysky.com/learn/science/the-doppler-effect-and-shock-waves-mt_UVfD_coqh6

## Ready when they can

- Applies the correct Doppler expression for a moving source and for a moving observer
- Explains why the two cases differ and when the difference matters
- Relates the angle of a shock cone to the speed of the source

## Lesson: Why the siren changes pitch as it passes

Recall v = fλ. A steady siren keeps its emitted frequency while moving. Its motion places sound compressions closer together ahead and farther apart behind. A stationary listener hears higher pitch ahead and lower pitch behind. A moving listener changes the rate of meeting compressions without changing their spacing in the air. The formulas below use still air and motion along the line joining source and listener.

Let f_s be emitted frequency, f_o heard frequency and v sound speed. For a source moving at air-relative speed u_s < v and a stationary listener, approach gives f_o = f_s × v ÷ (v - u_s). Recession gives f_o = f_s × v ÷ (v + u_s). The approach denominator is smaller, giving a higher frequency. These are sound formulas, not formulas for light in a vacuum.

*(drawing: This is an arithmetic walkthrough, not a drawing of wavefronts. Approach subtracts source speed in the denominator.)*

For a stationary source, let u_o be the listener's air-relative speed, with u_o < v. Approach gives f_o = f_s × (v + u_o) ÷ v. Recession gives f_o = f_s × (v - u_o) ÷ v. The wavelength in the air stays v ÷ f_s. The listener meets compressions at relative speed v + u_o or v - u_o, so listener motion changes the numerator.

**Example.** Compare equal approach speeds u and put r = u ÷ v. Source motion gives f_o = f_s ÷ (1 - r); listener motion gives f_o = f_s × (1 + r). For r much smaller than 1, the source result is also approximately f_s × (1 + r). At larger r, the results differ more. With f_s = 300 Hz, v = 320 m/s and u = 80 m/s, source motion gives 400 Hz and listener motion gives 375 Hz.

A supersonic aircraft moves faster than sound and produces shock waves, sharp changes in air pressure. A listener hears a sonic boom when a shock reaches them. Shocks can continue during supersonic flight; the boom is not limited to the instant the aircraft first exceeds sound speed. Do not extend the subsonic Doppler approach formula to u_s at or above v.

For an ideal Mach cone in a uniform medium, θ is the half angle between the cone surface and its backward axis along the source's path. For an acute angle in a right triangle, sin θ = opposite side ÷ hypotenuse. Then sin θ = v ÷ u_s = 1 ÷ M, where Mach number M = u_s ÷ v. Rearranging gives u_s = v ÷ sin θ. A faster source gives a narrower cone. This weak-disturbance geometry does not give the angle of every finite-strength shock.

**Recap.** Source motion changes sound-wave spacing; listener motion changes encounter rate. Use the matching formula. For an ideal Mach cone, sin θ = v ÷ u_s.

*Lesson, questions and flashcards written for this learning dataset.*

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [The wave equation](https://lightmysky.com/learn/science/the-wave-equation-mt_wmKhTiXt1p)

## Opens up

- [Huygens's principle and the laws of reflection and refraction](https://lightmysky.com/learn/science/huygenss-principle-and-the-laws-of-reflection-and-refraction-mt_xxdniF2__w)