Boundary Conditions for Wave Solution

Classical Mechanics

Given an arbitrary harmonic solution to the wave equation

$y(x,t) = A \sin (x-vt) + B \sin (x+vt),$

find the general solution, i.e. find the coefficients $A$ and $B$ given the following boundary conditions:

$y(0,t) = 0, \qquad y(L,0) = 1.$

y(x,t) =\frac{\sin x}{\sin L}

y(x,t) =\sin x \cos L \cos vt

y(x,t) =\sin x \sin (L-vt)

y(x,t) = \frac{\sin x \cos vt}{\sin L}