= Solution
With zero initial time derivative, the integral term in the <D'Alembert formula> vanishes. The <Heaviside step function> representation gives
$$
\boxed{\zeta(x,t)=\frac12\big[H(x+ct+1)-H(x+ct-1)+H(x-ct+1)-H(x-ct-1)\big].}
$$
The <rectangular pulse splitting under the wave equation> consists of \b[two width-two pulses of height $1/2$, moving left and right at speed $c$]. At $t=0$ they coincide and add to the initial height-one pulse. Before $ct=1$ the overlapping interval has height one; after $ct=1$ the two pulses are separated. Values exactly at the fronts depend on the chosen $H(0)$ convention. Away from the fronts the equation holds classically, and across the fronts it holds as a <weak solution> of the <wave equation>.
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