Wave equation (travelling)

Intermediate

A sinusoidal wave function of space and time.

Key relationy = A sin(kx − ωt)

y(x, t) = A sin(kx − ωt) describes a wave moving in +x at v = ω/k. Using (kx + ωt) gives motion toward −x.

Worked example

A = 0.1 m, k = 2π, ω = 4π: v = 2 m/s.

Common mistake

Sign of ωt gives the direction.

In more detail

A travelling wave satisfies y(x,t) = A sin(kx − ωt), which describes a disturbance moving with speed ω/k. The wave equation governs all such waves.

Engineers use the travelling wave equation to model pipelines, cables and seismic data.
Audio processing describes sound as travelling waves.
A physicist writes the displacement of a water wave as a function of position and time to predict where a float will be.

What does the minus sign mean?

It means the wave moves in the positive x direction.

Is the wave speed fixed?

It is set by the medium, so it is constant for given conditions.

What do the terms in kx and ωt mean?

kx gives the phase change with position and ωt the phase change with time, so the minus sign shows the wave moves in the positive direction.