Rectangular pulse splitting under the wave equation
= Rectangular pulse splitting under the wave equation
{title2=$u=[u_0(x+ct)+u_0(x-ct)]/2$}
A rectangular initial displacement with zero initial velocity splits into two equal half-height copies moving in opposite directions. The <D'Alembert formula> reads $u(x,t)=[u_0(x+ct)+u_0(x-ct)]/2$. Overlap adds the heights; separated copies each retain the original width. For discontinuous fronts the formula defines a <weak solution> of the <wave equation>.