y(x, t) = A sin (\frac{πnx}{L}) cos(ω_{n}t)

y_{1}(x, t) = \frac{A}{2} sin (\frac{πnx}{L} − ω_{n}t)

y_{2}(x, t) = \frac{A}{2} sin (\frac{πnx}{L} + ω_{n}t)

y_{i}(x, t) = A sin (kx − ωt)

y_{1}(x, t) = B sin (kx − ωt)

y_{2}(x, t) = C sin (kx + ωt)

y_{3}(x, t) = D cos (kx − ωt)

y_{4}(x, t) = E cos (kx + ωt)

y_{i}(0, t) + y_{2}(0, t) + y_{4}(0, t) = 0

A sin (0 − ωt) + C sin (0 + ωt) + E cos (0 + ωt) = 0

\frac{dy_{i}(0, t)}{dt} + \frac{dy_{2}(0, t)}{dt} + \frac{dy_{4}(0, t)}{dt} = 0

y_{r}(x, t) = −A sin (kx + ωt)

y_{r}(x, t) = A sin (kx + ωt)

y(x, t) = 0