I am trying to solve a wave equation on unit interval $x \in [0,1]$ with a specific boundary condition which I can't see how to tackle. The problem is:

Find $(u, P)$ subject to \begin{align} u_t + P_x &= F(x,t), \\ P_t + u_x &= 0. \end{align}

Here, $F(x,t) = x \cdot U(t)$ is a known force term. For simplicity let's say for now the initial conditions are zero. The boundary conditions are:

\begin{align} u(x=1, t) &= 0, \\ \frac{P(x=0, t)}{m} &= V_{tt} + \omega^2 V, \\ u(x=0, t) &= V_t, \quad \text{and} \\ V(t=0) &= V_0. \end{align}

The boundary condition at $x=0$ is a way to model a spring connected to the main domain. $V$ is the spring displacement and is only a function of time $t$. $\omega$ and $m$ are just some constants.

My main problem is that I get a second order boundary condition in time (something like $u_{tt} + \omega^2 u = u_x/m$, and I don't whether this problem is solvable in general.

Ideally, I would like to find an analytical solution to this.

Thank you!

Edit $1$.

I think I've completed the first step, and it ends up with the following equation:

$$s^2 \left( P - \frac{1-x}{\omega^2} \left(\frac{P'}{m} + P'' \right) \right) - P'' = f $$

Here I write the system in terms of pressure mode amplitude $P$. Basically, I've moved to the frequency domain $(\partial/\partial t \to s)$. The new forcing term $f$ is considered to be known.

I can assume the 'spring' is very slow, so $\omega^2 \gg 1$. This can give me a fourth order wave equation. Maybe there's anything else I could do with it instead?


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