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    "<center><h2> Helmholtz Equation for Vibrating Membrane</h2></center>\n",
    "<p>\n",
    "A visualization is made of the solution of the Helmholtz equation for a  vibrating\n",
    " membrane:\n",
    " \n",
    " $u(x,y,t) = cos(c sqrt(5)*t)*sin(2x)* cos(y)$.\n",
    " \n",
    " Based on example in: \n",
    "http://matplotlib.org/examples/mplot3d/wire3d_animation_demo.html"
   ]
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   "source": [
    "# MembraneHelmholz.ipynb\n",
    "\n",
    "%matplotlib notebook\n",
    "from mpl_toolkits.mplot3d import axes3d \n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import matplotlib.animation as animation\n",
    "import matplotlib\n",
    "\n",
    "t = 0                                    # initial time\n",
    "den = 390.0                              #density kg/m2\n",
    "ten = 180.0                              # tension N/m2\n",
    "c = np.sqrt(ten/den)                     # propagation velocity\n",
    "s5 = np.sqrt(5)"
   ]
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    "def membrane(t,X,Y): # analytic solution vibrating membrane\n",
    "    return np.cos(c * s5* t) * np.sin( 2* X)*np.sin(Y)\n",
    "\n",
    "fig = plt.figure()\n",
    "ax  =  fig.add_subplot(111, projection = '3d')\n",
    "xs = np.linspace(0, np.pi,32)       # 0 to pi increm: pi/32\n",
    "ys = np.linspace(0, np.pi,32)        \n",
    "ax.set_zlim(-1, 1)                  # z limits\n",
    "X, Y  =  np.meshgrid(xs,ys)          \n",
    "Z  =  membrane(0, X, Y)             # initial condition \n",
    "wframe  =  None\n",
    "ax.set_xlabel('X')                 \n",
    "ax.set_ylabel('Y')\n",
    "ax.set_title('Vibrating membrane')\n",
    "\n",
    "def update(idx):                  # called by animation\n",
    "    t = tt[idx]\n",
    "    global wframe\n",
    "    if wframe:\n",
    "        ax.collections.remove(wframe) # remove last frame\n",
    "    Z = membrane(t,X,Y)               # membrane at t != 0\n",
    "    wframe = ax.plot_wireframe(X,Y,Z)  \n",
    "    \n",
    "tt = range(200)    # interval = delay between iterations\n",
    "ani  =  animation.FuncAnimation(fig, update,frames = 100,interval = 300)"
   ]
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