{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Numerical solution to the wave equation in 1D and 2D\n", "\n", "The wave equation for variable wave speed is given by (for surface waves, EM waves obey different form):\n", "\n", "$$\\ddot{\\psi} = \\nabla^2\\left(c^2 \\psi \\right)$$\n", "\n", "\n", "In 1D, the wave equation for variable wave speed is solved using a Lax-Wendroff scheme\n", "\n", "In 2D, a Lax scheme is used instead\n", "\n", "The numerical methods used are discussed fully in the [documentation](http://pycav.readthedocs.io/en/latest/api/pde/lax_wendroff.html)\n", "\n", "Animations take ~ 1 min to create and display when ran in notebook\n", "\n", "### Importing modules and creating functions needed by PDE solver\n", "\n", "Functions which create the initial wavepacket, its gradients along the axes and its velocity as well as the wavespeed as a function of position" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "#NAME: Wave Equation\n", "#DESCRIPTION: Numerically solving the wave equation in 1D and 2D.\n", "\n", "import pycav.pde as pde\n", "import numpy as np\n", "\n", "from mpl_toolkits.mplot3d import Axes3D\n", "import matplotlib.pyplot as plt\n", "import matplotlib.animation as anim\n", "\n", "import pycav.display as display\n", "\n", "def twoD_gaussian(XX,YY,mean,std):\n", " return np.exp(-((XX-mean[0])**2+(YY-mean[1])**2)/(2*std**2))\n", "\n", "def oneD_gaussian(x,mean,std):\n", " return np.exp(-((x-mean)**2)/(2*std**2))\n", "\n", "def c(x):\n", " return np.ones_like(x)\n", "\n", "def velocity_1d(x,mean,std):\n", " def V(psi_0):\n", " return -c(x)*(x-mean)*oneD_gaussian(x,mean,std)/std**2\n", " return V\n", "\n", "def gradient_1d(x,mean,std):\n", " def D(psi_0):\n", " return -(x-mean)*oneD_gaussian(x,mean,std)/std**2\n", " return D\n", "\n", "def gradient_2d(x,y,mean,std):\n", " XX,YY = np.meshgrid(x,y, indexing='ij')\n", " def D(psi_0):\n", " dfdx = -(XX-mean[0])*twoD_gaussian(XX,YY,mean,std)/std**2\n", " dfdy = -(YY-mean[1])*twoD_gaussian(XX,YY,mean,std)/std**2\n", " return dfdx,dfdy\n", " return D\n", "\n", "def velocity_2d(x,y,mean,std):\n", " XX,YY = np.meshgrid(x,y, indexing='ij')\n", " def V(psi_0):\n", " return -c_2d(x,y)*(x-mean[0])*twoD_gaussian(XX,YY,mean,std)/std**2\n", " return V\n", "\n", "def c_2d(x,y):\n", " XX,YY = np.meshgrid(x,y, indexing='ij')\n", " return np.ones_like(XX)" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "dx = 0.01\n", "x = dx*np.arange(101)\n", "\n", "N = 150" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1D wave with reflective boundary conditions" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mu = 0.75\n", "sigma = 0.05\n", "psi_0_1d = oneD_gaussian(x,mu,sigma)\n", "\n", "psi_1d,t = pde.LW_wave_equation(psi_0_1d,x,dx,N,c, \n", " init_vel = velocity_1d(x,mu,sigma), init_grad = gradient_1d(x,mu,sigma),\n", " bound_cond = 'reflective')\n", "\n", "fig1 = plt.figure(figsize = (9,6))\n", "line = plt.plot(x,psi_1d[:,0])[0]\n", "plt.ylim([np.min(psi_1d[:,:]),np.max(psi_1d[:,:])])\n", "\n", "def nextframe(arg):\n", " line.set_data(x,psi_1d[:,arg])\n", "\n", "animate1 = anim.FuncAnimation(fig1,nextframe, frames = N, interval = 50, repeat = False)\n", "animate1 = display.create_animation(animate1, temp = True)\n", "display.display_animation(animate1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1D wave with fixed boundary conditions" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "psi_1d,t = pde.LW_wave_equation(psi_0_1d,x,dx,N,c, \n", " init_vel = velocity_1d(x,mu,sigma), init_grad = gradient_1d(x,mu,sigma),\n", " bound_cond = 'fixed')\n", "\n", "fig2 = plt.figure(figsize = (9,6))\n", "line = plt.plot(x,psi_1d[:,0])[0]\n", "plt.ylim([np.min(psi_1d[:,:]),np.max(psi_1d[:,:])])\n", "\n", "def nextframe(arg):\n", " line.set_data(x,psi_1d[:,arg])\n", "\n", "animate2 = anim.FuncAnimation(fig2,nextframe, frames = N, interval = 50, repeat = False)\n", "animate2 = display.create_animation(animate2, temp = True)\n", "display.display_animation(animate2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2D wave with reflective boundary conditions" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = 200\n", "x = dx*np.arange(101)\n", "y = dx*np.arange(201)\n", "\n", "XX,YY = np.meshgrid(x,y,indexing='ij')\n", "\n", "psi_0_2d = twoD_gaussian(XX,YY,[0.5,0.8],0.1)\n", "\n", "psi_2d,t = pde.LW_wave_equation(psi_0_2d,[x,y],dx,2*N,c_2d, a = 0.5,\n", " init_grad = gradient_2d(x,y,[0.5,0.8],0.1),\n", " bound_cond = 'reflective')\n", "\n", "fig3 = plt.figure(figsize = (9,9))\n", "ax = fig3.gca(projection='3d')\n", "image = ax.plot_surface(XX.T,YY.T,psi_2d[:,:,0].T,cmap = 'plasma')\n", "\n", "def nextframe(arg):\n", " ax.clear()\n", " ax.set_zlim3d([np.min(2*psi_2d[:,:,:]),np.max(2*psi_2d[:,:,:])])\n", " ax.plot_surface(XX.T,YY.T,psi_2d[:,:,2*arg].T,cmap = 'plasma')\n", " \n", "animate3 = anim.FuncAnimation(fig3,nextframe, frames = int(N/2) ,interval = 50, repeat = False)\n", "animate3 = display.create_animation(animate3, temp = True)\n", "display.display_animation(animate3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2D wave with reflective boundary conditions and variable wavespeed\n", "\n", "The below simulation has the same initial conditions as the above example but the wavespeed has the function form:\n", "\n", "$$c(x) = \\frac{1}{2}+\\frac{x}{2}$$" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "