# The Wave Equation's General Solution and D’Alembert’s

I'm going through my notes and I'm stuck with quite a basic part of the wave equation.

I have that $$u_{tt}=c^2u_{xx}$$ for $$x ∈ (0,1), t>1$$

subject to the following boundary conditions:

$$u(0, t) = u(1, t) = 0 , t > 0$$

I am also told that the initial displacement is:

$$u(x, 0) = f(x), x ∈ (0, 1)$$

and also that it is initially at rest which I believe means that

$$u_{t}(x, 0) = g(x) = 0$$

All I needed to do from here was find an expression for $$h_1(z)$$ and $$h_2(z)$$ for $$0 ≤ z ≤ 1$$

When deriving D'Alembert's formula I noticed that:

$$h_1(z)+h_2(z)=f(z)$$ and $$-h_1(z)+h_2(z)=\frac{1}{c}\int_{0}^{z}g(s)ds-\frac{1}{2}A$$

And from this I deduced that $$h_1(z)=h_2(z)=\frac{1}{2}f(z)$$

After this I need to substitute in the boundary condition at x = 0 into the general solution derive an expression for $$h_1(z)$$ in terms of $$h_2$$ for z < 0, and then also substitute in the boundary condition at x = 1 into the general solution derive an expression for $$h_2(z)$$ in terms of $$h_1$$ for z > 1

But this is where I get stuck, I know the answer to the first bit is $$h_1(z)=−h_2(−z)$$ but I'm just not sure how to get there.

Any help would be greatly appreciated.