Wave equation for a driven string and standing waves I'm having trouble trying to figure out what conditions should I apply to solve the wave equation for a driven string.
The string should follow the wave equation:
$$\dfrac{\partial^2 u(x,t)}{\partial x^2}=\dfrac{1}{v^2}\dfrac{\partial^2 u(x,t)}{\partial t^2}$$
where the velocity $v$ depends on the tension and linear density of the string.
This string has length L and is fixed on both ends. One of the ends is driven, forcing the string to oscillate. This end should barely move for some given frequencies of the driven force where we have standing waves on the string. There is nice experiment showing the harmonics of a driven string.
What I want to do is to solve the wave equation for a string in order to find a solution that gives me a function for the behavior of the string for different frequencies of the driven force, including the standing wave behavior.
I know how to solve the wave equation for common boundary conditions, such as the fixed ends conditions for a plucked string.
For this problem what initial conditions and what boundary conditions should I use to model the system? My first attempts were to apply a BC for the first time derivative on the moving end with the driven frequency, but the final solution wasn't correct. Am I missing information, should the equation be non-homogeneous, or maybe there is no simple solution at all?
 A: I'd apply Dirichlet boundary conditions on both ends, the driven end with the prescribed displacement, the other one with homogeneous b.c., namely:

*

*$u(0,t) = u_0(t)$ at the driven end in $x=0$

*$u(L,t) = 0$ at the fixed end in $x=L$.

I'm assuming that the string has pre-stress since string without it can't hold compression forces, and that assumptions of small strain and displacement holds.
Meaning of the boundary condition
In this model,

*

*Dirichlet boundary conditions model prescribed displacements
$u_D(t) = u(x_D,t)$


*Neumann boundary conditions model prescribed axial force $N_N(t) = N(x_N,t)$, being $N(x,t) = EA(x,t) \frac{\partial u}{\partial x}(x,t)$,
$EA(x_N,t) \frac{\partial u}{\partial x}(x,t) = N_N(t)$
At a free end (even if it's quite hard to me to think at a free end in a string with the pre-stress needed by the string, not so hard if we're dealing with beam or rod structural elements that hold compression stress as well as tension stress), you can prescribe
$EA(x_N,t) \frac{\partial u}{\partial x}(x,t) = 0$
but it should not be the case of your set-up.
