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I am trying to formulate a problem for 1D wave propagation. The governing PDE is as follows:

$$u_{tt} = c^2 u_{xx}$$

Where subscripts indicate partial differentiation with respect to the variable subscripted. In this problem I have a one dimensional bar which is fixed at one end but free at the other. I would like to prescribe a velocity at the end. The velocity will not be a function of position within the bar. It may however be a function of time.

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For this problem this is what I came up with. The x axis begins at the left end of the bar and projects rightward:

BOUNDARY CONDITIONS: $$u_t(0,t) = v_0(t)$$ $$u(L,t) = 0$$

INITIAL CONDITIONS $$u(x,0) = 0$$ $$u_t(x,0) = 0$$

I know this is not sufficient to solve the problem. Notice in the two initial conditions are really just one, since the second is a consequence of the first. This is well documented in a previous question I posted on the math stackexchange.

Therefore I would like to determine another condition to work with. One that comes to mind might be $u_x(0,t) = 0$ since that is a free end. However there are two problems with this. Firstly it is a third boundary condition, so I would be solving with 3 BCs and 1 IC which I dont know if it will work. Secondly I am not sure if this is a free end since there is a velocity prescribed there.

I am using the separation of variables method to solve this and it will be a tedious one to solve. How can I be sure of the boundary/initial conditions I am using?

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    $\begingroup$ The second initial condition is not a consequence of the first. In the previous question that you reference, you have boundary conditions that are functions of time. Here, you have initial conditions at a single instance in time. For example, $u(x,t)=\sin (\omega t)$ satisfies $u(x,0)=0$, but not $u_t(x,0)=0$. I think you should be fine the way you have it. $\endgroup$ – LedHead Jul 31 '17 at 14:29
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    $\begingroup$ But, if the bar is fixed at one end, say at $x=0$ and free at the other, isn't this just a 1/2-plane problem, where $0 \leq x < \infty$, for which it is some version of D'Alembert' solution? $\endgroup$ – Dr. Ikjyot Singh Kohli Jul 31 '17 at 16:53

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