I understand what the general solution is to a wave equation, but am unsure of the general solution for a damped wave equation. If someone knows what that is, or the steps to find it, that would be greatly appreciated. I can give the rest of the problem if needed but I think the part that matters for the general solution is:
$$U_t+U_{tt} = U_{xx}, \;\; 0 < x < 4, 0 < t < \infty$$
This PDE is separable as written. Assume solutions of the form $U(x,t)=X(x)\cdot T(t)$ $$U_{tt}+U_t=U_{xx}$$ $$XT''+XT'=X''T$$ $$\frac{T''}{T}+\frac{T'}{T}=\frac{X''}{X}=-\lambda$$ The equation separates into $$T''+T'+\lambda T=0 \mbox{ and } X''+\lambda X = 0$$ The solution to the first equation is the span of $$\left\{e^{-t/2}\cos\left(\frac{\sqrt{4\lambda -1}}{2} t\right) \ ,\ e^{-t/2}\sin\left(\frac{\sqrt{4\lambda -1}}{2} t\right)\right\}$$ and the solution to the second equation is the span of $$\{\cos\left(\sqrt{\lambda}\cdot x\right)\ ,\sin\left(\sqrt{\lambda}\cdot x\right)\}$$ Thus, the general solution consists of linear combinations of products of these sets, with appropriate values for $\lambda$. If we choose to impose 0-Dirichlet boundary condition $X(0)=X(4)=0$, then the spatial solution reduces to the sine function, and the appropriate values for $\lambda$ are $\sqrt{\lambda_n}=\frac{n\pi}{4}$ for positive integers $n$. This yields a solution of the form: $$\displaystyle\sum_{n=1}^{\infty}a_n e^{-t/2}\cos\left(\frac{\sqrt{n^2\pi^2/4-1}}{2} t\right)\sin\left(\frac{n\pi x}{4}\right)+b_n e^{-t/2}\sin\left(\frac{\sqrt{n^2\pi^2/4-1}}{2} t\right)\sin\left(\frac{n\pi x}{4}\right)$$ for indexed constants $a_n, b_n$. The constants $a_n$ are the Fourier sine coefficients for an imposed initial condition $U(x,0)=f(x)$. The constants $b_n$ are rescaled to produce the Fourier sine coefficients of an imposed initial velocity $U_t(x,0)=g(x)$.
For example, a fundamental mode $u(x,0)=\sin(\frac{\pi x}{4})$ with $u_t(x,0)=0$