Consider a semi-infinite string stretched between $2$ fixed points. Let $u(x, t)$ be the displacement of a string, at position $x$ and time $t.$
We describe the wave equation by: $$u(x, t) = h_1(x − ct) + h_2(x + ct)$$ for arbitrary functions $h_1(z)$ and $h_2(z).$
The string is subject to boundary conditions: $$u(0, t) = u(1, t) = 0 ,\: t > 0.$$
The string has an initial displacement $u(x, 0) = f(x), x ∈ (0, 1)$ and is initially at rest.
Use separation of variables to find the displacement of the string for $\textbf{t > 0.}$
Using the identity $\sin \theta \cos \phi =\frac{1}{2}(\sin(\theta − \phi) + sin(\theta + \phi))$ and the Sturm Liouville boundary value problem
$X''(x) + λX(x) = 0; X(0) = X(1) = 0$ has eigenvalues $λ_n = n^2π^2$ with corresponding eigenfunctions $X_n(x) = sin (nπx)$.
, rewrite your answer in the same form as $$u(x, t) = \frac{1}{2}(\overset{ˆ}{f}(x − ct) + \overset{ˆ}{f}(x + ct))$$
I've done the first part of this question which provided me with most of the information here but I really have no idea how to draw all this stuff together. I'm desperate at the moment so any help would be appreciated.
So far this is what I've done:
$$u_{tt} = c^2 u_{xx}$$ in $0 < x < 1$ subject to the initial and (homogeneous) Dirichlet boundary conditions given above.
Use separation of variables and let $u = P(t)Q(x)$. The problems for $P$ and $Q$ are second order ODEs:
$$ P''/(c^2 P) = Q''/Q = \lambda$$
Where $\lambda$ is a separation constant. The only functions satisfying $Q(0) = Q(1) = 0$ are
$$ Q_n(x) = \sin(n \pi x), \quad n = 1, 2, \ldots$$
where I have made $|\lambda| = n^2 \pi^2$.
I am seeking solutions $u$ in the most general form
$$ u = \sum_{n=1}^\infty P_n(t) Q_n(x), $$
because there are infinitely many eigenfunctions $Q_n$. From the PDE, one has
$$ \sum_{n=1}^\infty (Q_n P''_n - c^2 Q''_n P_n) = \sum_{n=1}^\infty Q_n [ P''_n + (n \pi c)^2 P_n ] = 0$$
Now, leverage the orthogonality properties of $Q_n$ to find an equation for $P_n(t)$ (in this case is trivial), to be solved with initial conditions \begin{align} & u = 0 \implies P(0) = 0 \\ & u_t = f(x) \implies P'_n(0) = \frac{\int^1_0 Q_n(x) f(x) \, \mathrm{d}x}{ \int^1_0 Q^2_n(x) f(x) } \end{align}
The separation of variables solution for $u_{tt}=c^2u_{xx}$ has the form $$ u(x,t)=\sum_{n=1}^{\infty}(A_n\cos(n\pi ct)+B_n\sin(n\pi ct))\sin(n\pi x). $$ where the $A_n$ and $B_n$ are determined by the sine series for $u(x,0)$ and $u_t(x,0)$ respectively. Then it's a matter of using trig identities to write $$ \cos(n\pi ct)\sin(n\pi x)=\frac{1}{2}(\sin(n\pi x-n\pi ct)+\sin(n\pi x+n\pi ct)) \\ \sin(n\pi ct)\sin(n\pi x)=\frac{1}{2}(-\cos(n\pi ct+n\pi x)+\cos(n\pi ct-n\pi x)). $$ After you substitute these back into the expression for $u(x,t)$, you get what you want.