1DWave_Simulation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# One Dimensional Wave Simulation\n",
    "\n",
    "#####Sajil C K \n",
    "This demo is for explaining one dimensional wave propagation with respect to spatial points\n",
    "#####June 16, 2015"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "####Lets construct a one dimensional signal of sinusoidal shape. This signal changes its value spatialy as well as with respect to time. So the signal y is a function of space and time."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "\\begin{align}\n",
    "y(x,t) & = f(x-ct)\\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "####Import Modules"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import math\n",
    "import sympy\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "####Parameter Specifications"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "delta = 1000        # Set step size here\n",
    "A     = 1;          # Set wave amplitude here   \n",
    "f     = 1000;       # Ser frequency of basic wave here\n",
    "c     = 343.0;      # Speed of sound in air\n",
    "t     = 0.0;        # Select the time instant\n",
    "lamda = c/f;        # Find the wavelength"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "####Signal Generation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = np.arange(0, lamda, (1.0/delta) )\n",
    "y = A*np.sin((2*np.pi/lamda)*x-(c*t));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "####Pad it with zeros for better visual appeal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "L     = len(y)\n",
    "#pad  = np.zeros((L,))\n",
    "#y    = np.concatenate((pad,y,pad), axis=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "####Plot the resulting signal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fcb490>]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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       "47ZxAAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2ca6d30>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "y(x,t) & = f(x-ct)\\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "y(x,t) & = f(x-f \\lambda t)\\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "y(x,t) & = Acos(\\frac{2\\pi}{\\lambda}(x-\\frac{\\lambda t}{T})) \\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "y(x,t) & = Acos(\\frac{2\\pi x}{\\lambda}-\\frac{2\\pi t}{T})) \\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "####Now its a sinusoid of Amplitude A, Frequency f, Velocity c...etc. Its a function of space and time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def OneDplot(t):\n",
    "    \n",
    "    delta = 100      # Set step size here\n",
    "    A     = 1;        # Set wave amplitude here   \n",
    "    f     = 1000;     # Ser frequency of basic wave here\n",
    "    c     = 343.0;    # Speed of sound in air\n",
    "    \n",
    "    lamda = c/f;      # Find the wavelength \n",
    "    x = np.arange(0, lamda, (1.0/delta) )\n",
    "    y = A*np.sin((2*np.pi/lamda)*x-(c*t));\n",
    "    \n",
    "    \n",
    "    L    = len(y)\n",
    "    #pad = np.zeros((L,))\n",
    "    #y = np.concatenate((pad,y,pad), axis=1)\n",
    "    plt.plot(y)\n",
    "    plt.title('1D Wave')\n",
    "    plt.xlabel('Points in Space')\n",
    "    plt.ylabel('Amplitue')\n",
    "    plt.grid(1)\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      ":0: FutureWarning: IPython widgets are experimental and may change in the future.\n"
     ]
    }
   ],
   "source": [
    "from IPython.html.widgets import interact"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "/8QZRgghhIbEGUYIIYSGRMEIIYTQkCgYIYQQGhIFI4QQQkOiYIQQQmhIFIwQQggN+f8Qk3ddHi6w\n",
       "9AAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x91efbb0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interact(OneDplot, t= (0.0,0.01,(1.0/delta)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Clearly its a right going wave. As time progress the wave propagates to the positive x or to right side "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Similarly we can have a left going wave. As time progress the wave propagates to the negative x or to the left side "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def OneDplotR(t):\n",
    "    \n",
    "    delta = 100      # Set step size here\n",
    "    A     = 1;        # Set wave amplitude here   \n",
    "    f     = 1000;     # Ser frequency of basic wave here\n",
    "    c     = 343.0;    # Speed of sound in air\n",
    "    \n",
    "    lamda = c/f;      # Find the wavelength \n",
    "    x = np.arange(0, lamda, (1.0/delta) )\n",
    "    y = A*np.sin((2*np.pi/lamda)*x+(c*t));\n",
    "    \n",
    "    \n",
    "    L    = len(y)\n",
    "    #pad = np.zeros((L,))\n",
    "    #y = np.concatenate((pad,y,pad), axis=1)\n",
    "    plt.plot(y)\n",
    "    plt.title('1D Wave')\n",
    "    plt.xlabel('Points in Space')\n",
    "    plt.ylabel('Amplitue')\n",
    "    plt.grid(1)    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from IPython.html.widgets import interact"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "9AAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9167d30>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interact(OneDplotR, t= (0.0,0.01,(1.0/delta)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "### Lets try standing waves"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def standingwave(t):\n",
    "    \n",
    "    delta = 10000      # Set step size here\n",
    "    A     = 1;        # Set wave amplitude here   \n",
    "    f     = 1000;     # Ser frequency of basic wave here\n",
    "    c     = 343.0;    # Speed of sound in air\n",
    "    \n",
    "    lamda = c/f;      # Find the wavelength \n",
    "    x = np.arange(0, lamda, (1.0/delta) )\n",
    "    y1 = A*np.sin((2*np.pi/lamda)*x-(c*t));\n",
    "    y2 = A*np.sin((2*np.pi/lamda)*x+(c*t));\n",
    "    y=y1+y2\n",
    "    \n",
    "    L    = len(y)\n",
    "    #pad = np.zeros((L,))\n",
    "    #y = np.concatenate((pad,y,pad), axis=1)\n",
    "    plt.plot(y)\n",
    "    plt.ylim([-2.0,2.0])\n",
    "    plt.title('1D Wave')\n",
    "    plt.xlabel('Points in Space')\n",
    "    plt.ylabel('Amplitue')\n",
    "    plt.grid(1)    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "CCGElkRBEkIIoSVRkIQQQmjJ/wfWnmJhaYujEgAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x81fb590>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interact(standingwave, t= (0.0,0.01,(1.0/delta)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}