Linearise shallow water wave equation to get decoupled equations

Question

I am trying to construct a set of decoupled system of equations from the following linear shallow water wave model which aims to capture the activity of a tsunami:

\begin{align*} h_t +(Hu)_x &= 0\\ u_t + (gh)_x &=0 \end{align*} Where $H$ is the height of the ocean; assumed to be constant, and $H+h$ is the total height after the tsunami. We also take $g = 10m/s^2$

This is my working so far:

1. I wrote the system as a matrix equation:

\begin{align*} \begin{bmatrix} h\\u \end{bmatrix}_t + \begin{bmatrix} 0 & H\\ g& 0 \end{bmatrix}\begin{bmatrix} h\\u \end{bmatrix}_x = 0 \end{align*}

2. I diagonalized $$A = \begin{bmatrix} 0 & H\\ g& 0 \end{bmatrix}$$ by $$ P = \begin{bmatrix} \sqrt{gH}/g & -\sqrt{gH}/g\\ 1& 1 \end{bmatrix}, \hspace{0.1cm} P^{-1} =\begin{bmatrix} \frac{g}{2\sqrt{gH}} & \frac{1}{2}\\ -\frac{g}{2\sqrt{gH}}& \frac{1}{2} \end{bmatrix}, \hspace{0.1cm} D =\begin{bmatrix} \sqrt{gH} & 0\\ 0& -\sqrt{gH} \end{bmatrix} $$

Now I am not sure how to proceed, I have tried utilising this diagonalization to get some kind of decoupled system by substituting this diagonalization into the matrix equation, but cant get anywhere. Thanks

Answer 1

You have $A=PDP^{-1}$. Define $s:=P^{-1}\begin{bmatrix}h\\u\end{bmatrix}$. Therefore, we get $$\frac{\partial s}{\partial t}+D\,\frac{\partial s}{\partial x}=0\,.$$ Let $s=:\begin{bmatrix}s_+\\s_-\end{bmatrix}$. Then, we have $$\frac{\partial s_+}{\partial t}+\sqrt{gH}\,\frac{\partial s_+}{\partial x}=0$$ and $$\frac{\partial s_-}{\partial t}-\sqrt{gH}\,\frac{\partial s_-}{\partial x}=0\,.$$ I hope that you know how to solve the two partial differential equations above.

We have $s_+(x,t)=f_+(x-\sqrt{gH}\,t)$ and $s_-(x,t)=f_-(x+\sqrt{gH}\,t)$ for some univariate differentiable functions $f_+$ and $f_-$. Since $\begin{bmatrix}h\\u\end{bmatrix}=Ps=\begin{bmatrix}+\sqrt{\frac{H}{g}}&-\sqrt{\frac{H}{g}}\\1&1\end{bmatrix}\,\begin{bmatrix}s_+\\s_-\end{bmatrix}$, we conclude that $$h(x,t)=\sqrt{\frac{H}{g}}\,\big(f_+(x-\sqrt{gH}\,t)-f_-(x+\sqrt{gH}\,t)\big)$$ and $$u(x,t)=f_+(x-\sqrt{gH}t)+f_-(x+\sqrt{gH}t)\,.$$