solving wave equation by fourier transform

in solving wave equation by Fourier transform after taking fourier transform of wave equation $$\frac{\partial^2y}{\partial^2x}=\frac{1}{v^2} \frac{\partial^2y}{\partial t^2}$$ we get $$(-ia)^2 Y(a,t)=\frac{1}{v^2}\frac{d^2Y(a,t)}{dx^2}$$ this is ok. now how we can get to this: $Y(a,t)=F(a)e^{(+-)ivat}$

tanks.

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HINT: Rearrange your expression to yield $(va)^{2}Y(a,t)+\frac{d^{2}Y(a,t)}{dt^{2}}=0$.
so now the descriptor equation is $r^2+(va)^2=0$ and this yield to $Y(a,y)=F(a)(e^{ivat}+e^{-ivat})$ and this equals to $Y(a,t)=F(a)(e^{(+-)ivat})$. i am right? – r.zarei Jan 12 '12 at 8:33