Periodic reflection for a fixed-end string ($y=f(ct-x)-f(ct+x),f(s+2L)=f(s)$)

Zero values at both endpoints of a length-$L$ string allow the D'Alembert formula to be written with opposite travelling waves and a $2L$-periodic profile. This represents reflection with sign reversal at each fixed endpoint. For positive-time boundary conditions, periodicity concerns the arguments visited by the string solution; unused profile values can be defined by periodic extension.