Travelling waves
A travelling wave is a spatial profile that moves with a constant shape and speed.
Wave form
A solution is sought in the form
\[u(x,t)=U(z),\qquad z=x-ct,\]where \(c\) is the wave speed.
Substituting this form into a reaction–diffusion equation converts the PDE into an ODE for \(U(z)\).
Biological examples
Travelling waves can represent invasion fronts, epidemic spread, movement of genetic variants or propagation of biochemical signals.
Key idea. Travelling-wave analysis reduces a moving spatial pattern to a fixed profile viewed in a moving coordinate system.