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I have a 2D wave equation $u_{tt} =c^2u_{xx}$ subject to the boundary conditions $$a_0u(0, t)-b_0u_x (0, t) = 0$$ $$a_1u(1, t)+b_1u_x (1, t) = 0$$ for constants $a_0, b_0, a_1, b_1$ with $b_0\neq0$ and $b_1\neq 0$.

  1. Calculate the derivative $E'(t)$ of the energy.
  2. How should $E(t)$ be modified in order to remain constant in time with these new boundary conditions?
  3. What constraints should be imposed on $a_0, b_0, a_1, b_1$ to ensure the modified energy is a nonnegative function of $u$?

By differentiating with respect to $t$, we find, $$\frac{dE}{dt}=2\int_0^1((u_tu_{tt}+c^2u_xu_{xt})dx.$$ Next integrating the second term on the right side by parts gives \begin{align*} \frac{dE}{dt} &=2\int_0^1 u_t(u_{tt}-c^2u_{xx})dx+c^2u_xu_t\Big|_0^1 \\ \end{align*} Given our PDE and boundary conditions, \begin{align*} a_0u(0,t)-b_0u_x(0,t)=0 \implies u_x(0,t)=\frac{a_0}{b_0}u(0,t) \\ a_1u(1,t)+b_1u_x(1,t)=0 \implies u_x(1,t)=-\frac{a_1}{b_1}u(1,t) \end{align*} we find, \begin{align*} \frac{dE}{dt} &= 2\int_0^1 u_t(0)dx-c^2\Big[\frac{a_1}{b_1}+\frac{a_0}{b_0}\Big]uu_t\Big|_0^1 \\ &= -c^2\Big[\frac{a_1}{b_1}+\frac{a_0}{b_0}\Big]uu_t\Big|_0^1. \end{align*}

If we define $c^2=\frac{a_1}{b_1}-\frac{a_0}{b_0}$, we ensure that $E'(t)=0$ and $E(t)$ remain constant in time.

And letting $\frac{a_1}{b_1}\ge\frac{a_0}{b_0}$, we ensure $E$ is non-negative: $$E(t)=\int_0^1 (u_t)^2+\big(\frac{a_1}{b_1}-\frac{a_0}{b_0}\big)(u_x)^2dx.$$

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