diff --git a/notebooks/lab7/01-lab7.ipynb b/notebooks/lab7/01-lab7.ipynb index 020b6b5..309f2ba 100644 --- a/notebooks/lab7/01-lab7.ipynb +++ b/notebooks/lab7/01-lab7.ipynb @@ -634,20 +634,25 @@ "$$\\frac {h_{i}(t+dt)-h_{i}(t-dt)}{2 dt} + H \\frac {u_{i}(t) - u_{i - 1}(t)}{dx} = 0$$\n", "\n", "\n", - "The initial conditions are: At $t = 0$ and $t = dt$, all points have\n", + "The initial conditions are: At $t = 0$, all points have\n", "zero elevation except at $h_{3}$, where \n", "$$h_{3}(0) = h_{0}$$\n", + "At $t = dt$, \n", + "$$h_{2}(dt) = h_{0} Hg \\frac{dt^2}{2 dx^2}$$ \n", "$$h_{3}(dt) = h_{3}(0) - h_{0} Hg \\frac{dt^2}{dx^2}$$ \n", + "$$h_{4}(dt) = h_{0} Hg \\frac{dt^2}{2 dx^2}$$ \n", + "\n", "At $t = 0$ and\n", "$t = dt$, all points have zero velocity except at $u_{2}$ and $u_{3}$,\n", "where \n", "$$u_{2}(dt) = - h_{0} g \\frac{dt}{dx}$$\n", "$$u_{3}(dt) = - u_{2}(dt)$$ \n", "This time we assume there is a wall at\n", - "$u_{1}$ and $u_{4}$, so we will ignore the value of $h_{1}$. The\n", - "boundary conditions are: \n", + "$u_{1}$ and $u_{4}$. The\n", + "boundary conditions in $u$ are: \n", "$$u_{1}(t) = 0$$ \n", "$$u_{4}(t) = 0$$\n", + "We can use $\\partial h/\\partial x=0$ at the wall to put boundary conditions on $h$. An exercise for the reader is to decide which two $h$ points straddle the walls.\n", "\n", "### Problem One\n", "[lab7:prob:staggered]:(#Problem-One)\n",