---
title: "The Wave Equation and Its Travelling-Wave Solutions"
description: "Any function of x minus vt keeps its shape while moving, and the second-order equation relating the curvature of a wave to its acceleration is what such functions satisfy. One equation covers strings,"
canonical: https://lightmysky.com/learn/science/the-wave-equation-and-its-travelling-wave-solutions-mt_Ue-v1cSxNx
source: https://lightmysky.com/learn/science/the-wave-equation-and-its-travelling-wave-solutions-mt_Ue-v1cSxNx.md
retrieved: 2026-09-12
---

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# The Wave Equation and Its Travelling-Wave Solutions

Any function of x minus vt keeps its shape while moving, and the second-order equation relating the curvature of a wave to its acceleration is what such functions satisfy. One equation covers strings, sound and light.

Subject: Science · Area: Waves, Light & Sound · Ages 18 to 19
Page: https://lightmysky.com/learn/science/the-wave-equation-and-its-travelling-wave-solutions-mt_Ue-v1cSxNx

## Ready when they can

- Checks by differentiation that a given function satisfies the wave equation
- Reads the speed of the wave off the equation
- Explains why any shape, not only a sine, can travel without distortion

## Lesson: One equation for every travelling wave

A wave that keeps its shape while moving at speed v has a simple secret. It depends on position and time only through the pair x minus vt, written y equals f of x minus vt.

Such functions pass one test, the wave equation: the second time derivative equals v squared times the second space derivative. To check a candidate, differentiate it twice each way and compare.

**Example.** A sine wave y equals A sin of kx minus omega t travels at v equals omega over k, which also equals f times lambda. The medium fixes v, the source fixes f, and lambda follows. A string shakes across its travel, called transverse, while sound shakes along it, called longitudinal. On the line below, the hop from 0 to 4 spans one full wavelength.

**Tip.** Do not reserve travel for sines. Any twice bendable shape in x minus vt moves without distortion, because the equation stays linear in that pair.

**Recap.** Shape preserving travel means x minus vt, speed reads as omega over k, and any smooth shape qualifies.

## Practice

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

## Needs first

- [Progressive Waves: Phase and Path Difference](https://lightmysky.com/learn/science/progressive-waves-phase-and-path-difference-mt_RotpycCNF8)
- [The Derivative as a Rate of Change](https://lightmysky.com/learn/mathematics/the-derivative-as-a-rate-of-change-mt_RqyLE3jrAW)

## Opens up

- [Plane Electromagnetic Waves from Maxwell's Equations](https://lightmysky.com/learn/science/plane-electromagnetic-waves-from-maxwells-equations-mt_27tbaYcxCM)
- [Wave Speed on a String from Tension and Linear Density](https://lightmysky.com/learn/science/wave-speed-on-a-string-from-tension-and-linear-density-mt_83x7vwb7UU)
- [From Coordinates to Fields: the Lagrangian Density](https://lightmysky.com/learn/science/from-coordinates-to-fields-the-lagrangian-density-mt_fRZN3ZByjr)