# Wave analysis of linear PDE with boundary conditions

I'm playing with the shallow-water equations \begin{align*} h_t &= -u_x + -v_y \\ u_t &= -h_x - r u + \nu u_{xx} + \nu u_{yy} \\ v_t &= -h_y - r v + \nu v_{xx} + \nu v_{yy} \end{align*} where $(h,u,v)$ are functions of $(x,y,t)$ and $r$ and $\nu$ are parameters related to bottom drag and viscosity. The system has solid walls at $x=(0,1)$ and $y=(0,1)$. One set of boundary conditions is the no-flow condition, so that $u=0$ at $x=(0,1)$ and $v=0$ at $y=(0,1)$. These would be all the boundary conditions needed when $\nu = 0$. When $\nu > 0$, I expect to be able to add the no-slip conditions, $u=0$ at $y=(0,1)$ and $v=0$ at $x=(0,1)$. However, I can't find a spatially wave-like solution that satisfies these boundary conditions.

Are these the right boundary conditions to look at? Are there too many? Am I missing something in the transition from general wave-like solutions to specific, real-valued solutions?

Doing a wave analysis of this system, where $$(h,u,v) \propto \exp(i (kx + ly - \omega t) + \sigma t)$$ tells us that the decay rate is $\sigma = -(r + \nu K^2)/2$ where $K^2 = k^2 + l^2$ and the phase speed is slightly modified to $c^2 = (\omega/K)^2 = 1 - \sigma^2$.

For $\nu=0$ (no viscosity) the equations are first-order in $(x,y)$ and these should be all the needed boundary conditions. However, $u_t$ would be nonzero at $x=0$ unless $h_x=0$ there as well. Thus the normal gradients of $h$ should also be zero at the boundaries, i.e. $h_x=0$ at $x=(0,1)$ and $h_y=0$ at $y=(0,1)$. This works fine if $h \propto \cos(kx) \cos(ly)$, $u \propto \sin(kx) \cos(ly)$, and $v \propto \cos(kx) \sin(ly)$, with $(k,l)=2\pi(m,n)$ for $(m,n)$ nonzero integers. These solutions also give the expected result that with no viscosity, there should be free-slip conditions at the boundary. There should be no normal shear in the tangential velocity, so $u_y=0$ at $y=(0,1)$ and $v_x=0$ at $x=(0,1)$.

With nonzero viscosity, the equations become second-order in each spatial dimension and I would expect to be able to add four more boundary conditions, the no-slip conditions. However, if $u=0$ all across $y=0$, then the equation for $u$ applied here gives $$0 = -h_x + \nu u_{yy}$$ since $u, u_t, u_{xx}$ must all be zero. But for any spatially wave-like solution, $u_{yy} \propto u = 0$ and we would still get $h_x = 0$. This works fine if we let $u \propto \sin(kx) \sin(ly)$ and $h \propto \cos(kx) \sin(ly)$. But the same analysis applies to the equation for $v$, which would now have a term $h_y \propto \cos(kx) \cos(ly)$ which is maximized on the boundaries! This boundary equation seems fine in general but does not work on spatially wave-like functions.

I thought the solution might be combinations of the solutions with wavenumbers $(k,l), (-k,l), (k,-l), (-k,-l)$ but that hasn't worked out for me yet. It could also include half-integer wavenumbers but it's not clear to me how I would start working that out. If this doesn't work for any given wavenumber, it seems like it also shouldn't work if I sum over many wavenumbers either.

Are these too many boundary conditions for these equations? Am I missing something in the transition from general wave-like solutions to specific, real-valued solutions? What other boundary conditions would be better suited or more physically realistic for a shallow-water model? Do spatially wave-like solutions just not apply here, and if so, what else can I look at?