A simple wave for the linear shallow water equations

Question

I'm looking for a simple, right-traveling wave for the linear shallow water equations (1D). My question: what are the initial conditions (velocity $U_0(x)$ and/or average water height, average velocity) for a simple (single!) right-traveling wave?

I want to create a simple wave solution for the 1D shallow water equations by choosing appropriate initial conditions, just like in this other post. I'm comparing a staggered grid with a non-staggered grid. On the staggered grid, if I have a simple wave traveling to the right, there should not be a left-traveling wave, whereas on the non-staggered grid, there can still be a component with negative group velocity. Just like in the other question, I'm also working with a Gaussian initial wave profile. The problem is that when I use their equation for $m_0(x)$, I still get a (small) left-traveling component on both grids. I don't have access to the book by Leveque, so I don't understand where the equation for $m_0(x)$ comes from. Could someone please tell me what the reasoning behind $m_0(x)$ is or share with me the relevant page(s) from Leveque?

I'm working with linearized shallow water equations, which, I think, is different from their example. It's pretty much the simplest case of shallow water equations: linear equations and no bottom topography. On the staggered grid, the dispersion relation with my discretization scheme (centered-space, leapfrog-time) is

$$\sin(\omega\Delta t) = \frac{(U \pm 2c)\Delta t}{\Delta x}\sin\Big(\frac{k\Delta x}{2}\Big)$$

Where $c = \sqrt{gH}$ with $H$ the average water height and $g$ the gravitational constant. I thought I could just set my initial velocity $U_0(x)$ equal to $2\sqrt{g H}$ or to $2\sqrt{g H_0(x)}$ to eliminate the "-" solution, and have only the right-traveling wave left, but when I do there is a left-traveling trough and I don't know what to make of it. The pictures below are for $U_0(x) = 2\sqrt{g H_0(x)}$. The other thing I tried gives a forward and a backward wave.

What are the initial conditions for a simple (single!) right-traveling wave?

Your insights will be very much appreciated.

Answer 1

The other question you have referred to is about the (nonlinear) shallow water equations. Here you are just asking about the linear wave equation, which is quite different.

To get a purely right-going solution of the 1D wave equation, your initial condition $(\eta, u)^T$ at each value of $x$ should be a multiple of a certain vector. For the linearized shallow water equations with gravitational constant $g$, that vector is

\begin{pmatrix} 1 \\ \sqrt{g/H(x)} \end{pmatrix}

Thus if $\eta(x)$ and $u(x)$ are your initial surface height and velocity, you should have $u(x) = \eta(x)\sqrt{g/H(x)}$. For this initial condition, the exact solution is purely right-going. Numerically, if you are using a multistep method (it sounds like you are) then you may see a very small part going to the left. The magnitude of that part will decrease as you refine your grid.