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    "<center><h2>Lossless Transmission Line</h2></center>\n",
    "<p>\n",
    " The telegraph equation describing lossles propagation on a \n",
    " transmission line is solved and visualized with MatPlotLib:\n",
    " \n",
    " $$\n",
    "\\frac{\\partial^2 V}{\\partial t^2}-v_p^2\\frac{\\partial^2 V}{\\partial x^2}=0, \\quad v_p= \\frac{1}{\\sqrt{LC}}.\n",
    "$$ "
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    "\"\"\" From \"COMPUTATIONAL PHYSICS\" & \"COMPUTER PROBLEMS in PHYSICS\"\n",
    "    by RH Landau, MJ Paez, and CC Bordeianu (deceased)\n",
    "    Copyright R Landau, Oregon State Unv, MJ Paez, Univ Antioquia, \n",
    "    C Bordeianu, Univ Bucharest, 2021. \n",
    "    Please respect copyright & acknowledge our work.\"\"\"\n",
    "   \n",
    "# TelehMat.py:  Lossless transmission line animation, Matplotlib\n",
    "\n",
    "%matplotlib notebook\n",
    "\n",
    "from numpy import *; from matplotlib import animation\n",
    "import numpy as np, matplotlib.pyplot as plt\n",
    "\n",
    "L = 0.1; C = 2.5; c = 1/sqrt(L*C); dt = 0.025;dx = 0.05\n",
    "R = (c*dt/dx)**2          \n",
    "V  = np.zeros( (101, 3), float)                # V(Nx, Nt)\n",
    "xx = 0\n",
    "fig = plt.figure()                            \n",
    "ax = fig.add_subplot(111,autoscale_on=False,xlim=(0,100), ylim=(-40,40))\n",
    "ax.grid()                                                       \n",
    "plt.title(\"Wave Transmission Via Telegrapher's Equations\")\n",
    "plt.xlabel(\"x\");  plt.ylabel (\"V(x,t)\")\n",
    "line, = ax.plot([],[], lw=2)     # x axis, y values, linewidth=2\n",
    "                  \n",
    "def init():                               \n",
    "    line.set_data([],[])\n",
    "    return line,\n",
    "\n",
    "for i in arange (0,100):\n",
    "    V[i,0] =  10*exp(-(xx**2)/0.1)   \n",
    "    xx = xx + dx\n",
    "for i in range(1, 100): V[i,2] = V[i,0] + R*(V[i+1,1]+V[i-1,1]-2*V[i,1])\n",
    "V[:,0] = V[:,1];  V[:,1] = V[:,2]           # Recycle array\n",
    "     \n",
    "def animate(dum): \n",
    "        i =   arange(1, 100)   \n",
    "        V[i, 2] = 2.*V[i,1]-V[i,0]+R*(V[i+1,1]+V[i-1,1]-2*V[i,1])\n",
    "        line.set_data(i,V[i,2])\n",
    "        V[:,0] = V[:,1]; V[:,1] = V[:,2]     # Recycle array\n",
    "        return line,\n",
    "ani = animation.FuncAnimation(fig, animate,init_func=init,frames=10000,\n",
    "\tinterval=20,blit=True)  \n",
    "plt.show()"
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