This is a rudimentary question and has the potential to be better suited in physics's stackexchange (feel free to redirect me). Suppose that I have solved a homogeneous wave equation initial value boundary problem. Isn't it then true that the solution $u(x,t)$ in terms of the Fourier-coefficients, describes the motion of the string across all time, such that no energy is lost during the motion. That is, if the system of equations is for example
$\begin{cases}u_{tt} - c^2u_{xx} = 0&: x \in (0, \pi), t > 0\\\ u_t(x, 0) = f(x), &: x \in [0, \pi]\\\ u(0, t) = u(\pi, t) = 0, &: t \in [0, \infty)\\\ u(x, 0) = g(x),&: x \in [0, \pi]\end{cases}$
then friction, heat etc. play no role here in the sense that with suitable boundary conditions, $u(x,t) \not\to 0, t\to\infty$?
My PDE reference does a great job at motivating mathematics of the different solution approaches to the problem, but fails to give any qualitative description of the system(s) as the parameters are varied across their domain.
the wave equation can be derived from a Lagrangian density:
$\int{(c^2 u_x^2-u_t^2)dxdt}$
since the Lagrangian does not explicitly depend on time, energy is conserved.