# 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:

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

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

Then we simplified the equations. They would be

\$latex 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\$
\$latex 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)\$
\$latex 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)\$