Let $u(x, t)$ be the displacement of a string, at position $x$ and time $t$, which is stretched between two fixed points at $x = 0$ and $x = 1$. Its motion satisfies the wave equation, $$u_{tt} = c^2u_{xx}\qquad x ∈ (0, 1),\: t > 0$$
where $c$ is a real positive constant, and is subject to boundary conditions $$u(0, t) = u(1, t) = 0 \qquad t > 0$$
The string has an initial displacement $u(x, 0) = f(x), x ∈ (0, 1)$ and is initially at rest.
Starting with the general solution to the wave equation $$u(x, t) = h_1(x − ct) + h_2(x + ct)$$
for arbitrary functions $h_1(z)$ and $h_2(z)$,
Derive expressions for $h_1(z)$ and $h_2(z)$ for $0 ≤ z ≤ 1.$
I understand d'Alemberts formula but Im getting confused when inputting my initial conditions.
Any help will be appreciated.
You have $$ u(x,t) = h_1(x-ct) + h_2(x+ct), \\ u_t(x,t) = c( -h_1(x-ct)+h_2(x+ct) ). $$ The initial conditions tell you that for $0<x<1$, $$ u(x,0) = f(x), \\ u_t(x,0) = 0. $$ Substituting the d'Alembert solution in, $$ h_1(x)+h_2(x)=f(x) \\ -h_1(x)+h_2(x)=0, $$ so we find $$ h_1(x) = h_2(x) = \frac{1}{2}f(x). $$
The boundary conditions are in a sense a red herring for this: what they tell you is what happens when $x \pm ct$ crosses the boundary, namely $$ h_1(-ct)+h_2(ct) = 0, \\ h_1(1-ct)+h_2(1+ct) = 0, $$ and with these, we can extend the definitions of $h_1$ and $h_2$ to other $z$.