How can I prove that d'Alembert's general solution satisfies the boundary conditions:

for the 1d Vibrating String equation:

Question

How can I prove that d'Alembert's general solution satisfies the boundary conditions:

for the 1d Vibrating String equation:

anonymous · Answer

Boundary Conditions:
$$y(0,t) = y(L,t) = 0$$

Vibrating String Equation:
$$y_{tt} = c ^{2}*y _{ss}$$

anonymous · Answer

D'Alembert solution has (by defintion) the boundary condition described in equation (19) [ http://mathworld.wolfram.com/dAlembertsSolution.html ]. It is then easy from that to show that y(0,t) = 0, and y(L,t) = 0.

anonymous · Answer

i'm not following... plugging in y=0 into the equation i get:

$$y(0,t) = 0 = (1/2)y _{0}(ct) + (1/2)y _{0}(-ct) + 1/2c * (v _{0}\prime(ct) - v _{0}\prime(-ct))$$
which doesn't cancel out to 0.