Linear Frictional Shallow Water Equations

First of all, this passage requires a basic understanding of the classical shallow water theory. The depth-averaged governing equations in x-z plane with friction are

$$ \frac{\partial u}{\partial t} + g\frac{\partial \eta}{\partial x} + Ru=0, \\ \frac{\partial \eta}{\partial t} + H\frac{\partial u}{\partial x} = 0, $$

where $\eta$ is surface elevation, R is bottom stress, H is water depth.

Providing the frictionless condition, i.e. $R=0$, one will obtain a wave equation

$$ \frac{\partial^2 \eta}{\partial t^2}-gH\frac{\partial^2 \eta}{\partial x^2}=0, $$

with the corresponding wave speed

$$ c=\frac{\omega}{k} = \sqrt{gH} $$

for a wave of the form $\eta = a\cos(kx-\omega t)$. In terms of a, c, H, and $\eta$, we have

$$ u=\frac{ac}{H}\eta+u_0, $$

where $u_0$ is the constant background flow, and we assume $u_0=0$ hereafter. In general with friction, the solution to the linear system forced at frequency w will only respond at that frequency. A reasonable guess of the solutions:

$$ u=Re\{U\exp(-i\omega t)\},\\ \eta=Re\{E\exp(-i\omega t)\}, $$

where U, E are unknown complex function of x only. We proceed by working out the full complex solution and only evaluate the real part when satisfying the real boundary condition at the end.

Substitute the solutions into the governing equation, we get

$$ -i\omega U + g \frac{\partial E}{\partial x} + Ru =0,\\ -i\omega E + H \frac{\partial U}{\partial x} = 0. $$

With simple manipulations, we can obtain

$$ \frac{\partial u}{\partial x}=\frac{i\omega}{H}E,\\ -i\omega\frac{\partial U}{\partial x} + g\frac{\partial^2 E}{\partial x^2} + R\frac{\partial U}{\partial x} = 0. $$

By eliminating U to get a single equation for E,

$$ \frac{\partial^2 E}{\partial x^2}+(\frac{\omega^2}{gH}+\frac{i\omega R}{gH})E=0. $$

$$ \frac{\partial^2E}{\partial x^2}+k^2E=0, $$

where we have defined the possibly complex wave number $k=\frac{\omega}{c}\sqrt{1+i\frac{R}{\omega}}$, where $c=\sqrt{gH}$. The equation above has the solution of the form

$$ E = \alpha^+\exp{i(kx)} + \alpha^-\exp{i(kx)}, $$

and $\alpha^{+/-}$ are complex constants we can choose to satisfy the boundary conditions.

Recall that the boundary condtion at the mouth of the channel: $\eta=a\cos\omega t ,x=0$. If we assume the channel is infinite, so there is no reflected wave, then we have $\alpha^-=0$, and we choose $\alpha^+$ to satisfy the boundary condition $\alpha^+=a$, where $a$ is real. Therefore, the surface function

$$ \eta=Re\{a\exp{i(kx-\omega t)}\}=a\cos(kx-\omega t), $$

providing that $R=0$ so $k$ is real.

Linear Frictional Shallow Water Equations

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