# shallow water wave equations

cuma mindahin catatan thesis

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Weiyan, in Shallow Water Hydrodynamics: Mathematical Theory and Numerical Solution for Two-Dimensional System of Shallow Water Equations, described Saint Venant System as:

$h_{t}+left(huright)_{x}+ left(hvright)_{y} = 0$
$left(uhright)_{t}+left(u^{2}h+frac{1}{2}gh^{2}right)_{x}+left(uhvright)_{y} = -ghleft(So_{x}-Sf_{x}right)$
$left(vhright)_{t}+left(uhvright)_{y}+left(v^{2}h+frac{1}{2}gh^{2}right)_{y} = -ghleft(So_{y}-Sf_{y}right)$

where, $h$ is water depth, $u$ is water velocity in the$x$ co-ordinate direction, $v$ is water velocity in the $y$ co-ordinate direction, $g$ is the acceleration due to gravity, $So$ is bed slope and $Sf$ is friction slope.

Then we simplified the equations. They would be

$dfrac{partial h}{partial t}+hdfrac{partial u}{partial x}+udfrac{partial h}{partial x}+hdfrac{partial v}{partial y}+vdfrac{partial h}{partial y}= 0$
$dfrac{partial u}{partial t}+ udfrac{partial u}{partial x}+vdfrac{partial u}{partial y}+gdfrac{partial h}{partial x}= -gleft(So_{x}-Sf_{x}right)$
$dfrac{partial v}{partial t}+ udfrac{partial v}{partial x}+vdfrac{partial v}{partial y}+gdfrac{partial h}{partial y}= -gleft(So_{y}-Sf_{y}right)$