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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
Shape preserving travel means x minus vt, speed reads as omega over k, and any smooth shape qualifies.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.