{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "import numpy as np                    # matrices & etc\n",
    "from matplotlib import pyplot as plt  # general plotting \n",
    "from matplotlib import patches        # plotting shapes\n",
    "from datetime import date             # date and time arithmetic\n",
    "from math import exp, pi, sin, cos, tan, sqrt, floor, log10, nan   # math functions\n",
    "import scipy.linalg as la             # linear algebra functions\n",
    "import os                             # operating system\n",
    "import matplotlib.cm as cm\n",
    "from matplotlib.colors import ListedColormap\n",
    "import scipy.signal as sg\n",
    "import scipy.stats as st\n",
    "import scipy.sparse.linalg as las\n",
    "import matplotlib\n",
    "\n",
    "# function to make a numpy N-by-1 column vector\n",
    "# c=eda_cvec(v1, v2, ...) from a list of several\n",
    "# array-like entities v1, v2, including a number\n",
    "# a list of numbers, a tuple of numbers, an N-by-0 np array\n",
    "# and a N-by-1 np array. The function\n",
    "# also insures that, if v is an np array, that\n",
    "# c is a copy, as contrasted to a view, of it\n",
    "# It promotes integers to floats, and integers\n",
    "# and floats to complex, by context.\n",
    "# This version concatenates many argments,\n",
    "# whereas c=eda_cvec1(v1) takes just one argiment.\n",
    "# I recommend always using eda_cvec(v1, v2, ...)\n",
    "def gda_cvec(*argv):\n",
    "    t = int;\n",
    "    Nt = 0;\n",
    "    for a in argv:\n",
    "        v = gda_cvec1(a);\n",
    "        N,M = np.shape(v);\n",
    "        Nt = Nt + N;\n",
    "        if( N==0 ):\n",
    "            continue; # skip vector of zero length\n",
    "        if (t==int) and isinstance(v[0,0],float):\n",
    "            t=float;\n",
    "        elif isinstance(v[0,0],complex):\n",
    "            t=complex;\n",
    "    w = np.zeros((Nt,1),dtype=t);\n",
    "    Nt = 0;\n",
    "    for a in argv:\n",
    "        v = gda_cvec1(a);\n",
    "        N,M = np.shape(v);\n",
    "        w[Nt:Nt+N,0:1] = v;  # patch 20230418 was #w[Nt:Nt+N,0] = v[0:N,0];\n",
    "        Nt = Nt + N;\n",
    "    return w;\n",
    "\n",
    "# function to make a numpy N-by-1 column vector\n",
    "# c=gda_cvec1(v) from entity v that is array-like,\n",
    "# including a number, a list of numbers, a tuple\n",
    "# of numbers, an N-by-0 np array and a N-by1 np array.\n",
    "# It promotes integers to floats, and integers\n",
    "# and floats to complex, by context. The function\n",
    "# also insures that, if v is an np array, that\n",
    "# c is a copy, as contrasted to a view, of it.\n",
    "# This version takes just one input argment.\n",
    "# whereas c=gda_cvec(v1,v2,...) concatenates\n",
    "# many argiments.\n",
    "def gda_cvec1(v):\n",
    "    if isinstance(v, int) or isinstance(v, np.int32):\n",
    "        w = np.zeros((1,1),dtype=int);\n",
    "        w[0,0] = v;\n",
    "        return w;\n",
    "    elif isinstance(v, float):\n",
    "        w = np.zeros((1,1),dtype=float);\n",
    "        w[0,0] = v;\n",
    "        return w;\n",
    "    elif isinstance(v, complex):\n",
    "        w = np.zeros((1,1),dtype=complex);\n",
    "        w[0,0] = v;\n",
    "        return w;\n",
    "    elif isinstance(v, np.ndarray):\n",
    "        s = np.shape(v);\n",
    "        if len(s) == 1:\n",
    "            return np.copy(np.reshape(v,(s[0],1)));\n",
    "        else:\n",
    "            [r,c]=s;\n",
    "            if( c==1 ):\n",
    "                return(np.copy(v));\n",
    "            elif(r==1):\n",
    "                return(np.copy(v.T));\n",
    "            else:\n",
    "                raise TypeError(\"gda_cvec: %d by %d ndarray not allowed\" % (r, c));\n",
    "    elif isinstance(v, list):\n",
    "        r = len(v);\n",
    "        t = int;\n",
    "        for vi in v:\n",
    "            if isinstance(vi,int) or isinstance(vi, np.int32): #patch v->vi 20230418\n",
    "                pass;\n",
    "            elif isinstance(vi,float):\n",
    "                t=float;\n",
    "            elif isinstance(vi,complex):\n",
    "                t=complex;\n",
    "                break;\n",
    "            else:\n",
    "                raise TypeError(\"gda_cvec: list contains unsupported type %s\" % type(vi));\n",
    "        w = np.zeros((r,1),dtype=t);\n",
    "        w[:,0] = np.array(v); # patch v -> np.array(v)\n",
    "        return w;\n",
    "    elif isinstance(v, tuple):\n",
    "        r = len(v);\n",
    "        t = int;\n",
    "        for vi in v:\n",
    "            if isinstance(vi,int) or isinstance(vi, np.int32): #patch v->vi 20230418\n",
    "                pass;\n",
    "            elif isinstance(vi,float):\n",
    "                t=float;\n",
    "            elif isinstance(vi,complex):\n",
    "                t=complex;\n",
    "                break;\n",
    "            else:\n",
    "                raise TypeError(\"gda_cvec: tuple contains unsupported type %s\" % type(vi));\n",
    "        w = np.zeros((r,1),dtype=t);\n",
    "        w[:,0] = np.array(list(v)); # patch v -> np.array(list(v));\n",
    "        return w;\n",
    "    else:\n",
    "        raise TypeError(\"gda_cvec: %s not supported\" % type(v));\n",
    "\n",
    "        \n",
    "# gda_draw function makes a \"pictorial matrix equation\"\n",
    "# arguments are vectors, matrices and strings\n",
    "# which are plotted in the order that the appear\n",
    "# except that strings starting with 'title ' are plotted\n",
    "# under the subseqeunt matrix or vector\n",
    "# always returns a status of 1\n",
    "def gda_draw(*argv):\n",
    "    DOCOLOR=True;\n",
    "    if( DOCOLOR ):\n",
    "        bwcmap = matplotlib.colormaps['jet'];\n",
    "    else:\n",
    "        bw = np.zeros((256,4));\n",
    "        v = 0.9*(256 - np.linspace( 0, 255, 256 ))/255;\n",
    "        bw[:,0] = v;\n",
    "        bw[:,1] = v;\n",
    "        bw[:,2] = v;\n",
    "        bw[:,3] = np.ones(256);\n",
    "        bwcmap = ListedColormap(bw);\n",
    "    # size of plot\n",
    "    W = 16;\n",
    "    H = 4;\n",
    "    fig1 = plt.figure(1);\n",
    "    # figsize width and height in inches\n",
    "    fig1.set_size_inches(W,H);\n",
    "    ax1 = plt.subplot(1,1,1);\n",
    "    plt.axis([0, W, -H/2, H/2]);\n",
    "    plt.axis('off');\n",
    "    LM = W/6;    # matrix width and heoght\n",
    "    LV = W/40;   # vector width\n",
    "    FS = 0.12;    # character width\n",
    "    TO = 0.4;    # title vertical offset\n",
    "    SP = 0.2;    # space between objects\n",
    "    LS = 0.2;    # leading space\n",
    "    p = LS; # starting x-position\n",
    "    istitle=0; # flags presence of a title\n",
    "    for a in argv:\n",
    "        if isinstance(a,np.ndarray):\n",
    "            sh = np.shape(a);\n",
    "            if len(sh) == 1:  # conversion to nx1 array\n",
    "                n = sh[0];\n",
    "                m = 1;\n",
    "                ap = a;\n",
    "                a = np.zeros((n,1));\n",
    "                a[:,0] = ap;\n",
    "            else:\n",
    "                n = sh[0];\n",
    "                m = sh[1];\n",
    "            if m==1:\n",
    "                pold=p;\n",
    "                left=p;\n",
    "                right=p+LV;\n",
    "                bottom=-LM/2;\n",
    "                top=LM/2;\n",
    "                plt.imshow( a, cmap=bwcmap, vmin=np.min(a), vmax=np.max(a), extent=(left,right,bottom,top) );\n",
    "                p = p+LV;\n",
    "                pm = (p+pold)/2;\n",
    "                if istitle:\n",
    "                    plt.text(pm,-(LM/2)-TO,titlestr,horizontalalignment='center');\n",
    "                    istitle=0;\n",
    "                p = p+SP;\n",
    "            else:\n",
    "                pold=p;\n",
    "                left=p;\n",
    "                right=p+LM;\n",
    "                bottom=-LM/2;\n",
    "                top=LM/2;\n",
    "                plt.imshow( a, cmap=bwcmap, vmin=np.min(a), vmax=np.max(a), extent=(left,right,bottom,top) );\n",
    "                p = p+LM;\n",
    "                pm = (p+pold)/2;\n",
    "                if istitle:\n",
    "                    plt.text(pm,-(LM/2)-TO,titlestr,horizontalalignment='center');\n",
    "                    istitle=0;\n",
    "                p = p+SP;\n",
    "        elif isinstance(a,str):\n",
    "            ns = len(a);\n",
    "            istitle=0;\n",
    "            if( ns>=6 ):\n",
    "                if 'title ' in a[0:6]:\n",
    "                    istitle=1;\n",
    "                    titlestr=a[6:];\n",
    "            if( istitle != 1):\n",
    "                plt.text(p,0,a);\n",
    "                p = p + ns*FS + SP;\n",
    "    plt.show();\n",
    "    return 1;\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
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      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Check normalization of eigenfunctions. Normalization should be close to unity.\n",
      "min norm 0.9999999999999997  max norm  1.0000000000000007\n",
      "Excitations:\n"
     ]
    },
    {
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      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
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     "metadata": {},
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    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "min  -0.0023443908757935492  max  0.0023487017026553115\n"
     ]
    },
    {
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       "<Figure size 1200x500 with 2 Axes>"
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   ],
   "source": [
    "# Excitation of an 2D rectangular membrane by an impulsive force\n",
    "\n",
    "# force direction\n",
    "q = 0.0; # force orientation, angle in degrees from x-axis\n",
    "cosq = cos( (pi/180.0)*q);\n",
    "sinq = sin( (pi/180.0)*q);\n",
    "\n",
    "# material constants\n",
    "rho = 1.0;\n",
    "lam = 1.0;\n",
    "c2 = lam/rho;\n",
    "c = sqrt(c2);\n",
    "\n",
    "# standard set-up of position x, y\n",
    "Nx=512;\n",
    "Dx=0.5;\n",
    "x=gda_cvec(Dx*np.linspace(0,Nx-1,Nx));\n",
    "xmin=0.0;\n",
    "xmax = Dx*(Nx-1);\n",
    "Ny=Nx; # requiring equal allows one to use a single D matrix\n",
    "Dy=0.5;\n",
    "y=gda_cvec(Dy*np.linspace(0,Ny-1,Ny));\n",
    "ymin=0.0;\n",
    "ymax = Dy*(Ny-1);\n",
    "\n",
    "# eigenfrequencies and eigenfunction of organ pipe\n",
    "Nw = 50;\n",
    "n = gda_cvec( np.linspace(1,Nw,Nw) );\n",
    "wnm = np.zeros((Nw,Nw));\n",
    "for iw in range(Nw):\n",
    "    for jw in range(Nw):\n",
    "        wnm[iw,jw] = pi*c*sqrt((n[iw,0]/xmax)**2 + (n[jw,0]/ymax)**2);\n",
    "\n",
    "gda_draw('title wnm',wnm);\n",
    "pn = sqrt(2*lam/xmax) * np.sin( (pi/xmax)*np.matmul(x, n.T) );  # eigenfunctions p(ix,n)\n",
    "pm = sqrt(2*lam/ymax) * np.sin( (pi/ymax)*np.matmul(y, n.T) );  # eigenfunctions p(iy,n)\n",
    "\n",
    "# 1st derivative operator\n",
    "myr = gda_cvec( 1, -1, np.zeros((Nx-2)) );\n",
    "myc = gda_cvec( 1, np.zeros((Nx-1)) );\n",
    "D1 = la.toeplitz( myr.ravel(), myc.ravel() );\n",
    "# first dervative at right hand end of interval [..., 0, -1, 1]\n",
    "D1[Nx-1,Nx-2]=(-1.0);\n",
    "D1[Nx-1,Nx-1]=1.0;\n",
    "\n",
    "# 2nd derivative operator\n",
    "myr = gda_cvec( -2, 1, np.zeros((Nx-2)) );\n",
    "myc = gda_cvec( -2, 1, np.zeros((Nx-2)) );\n",
    "D2 = la.toeplitz( myr.ravel(), myc.ravel() );\n",
    "# second derivatice zero at ends of interal\n",
    "D2[0,0]=0.0;\n",
    "D2[0,1]=0.0;\n",
    "D2[Nx-1,Nx-1]=0.0;\n",
    "D2[Nx-1,Nx-2]=0.0;\n",
    "\n",
    "# check noralization of eigenfunctions\n",
    "norm = np.zeros((Nw,Nw));     \n",
    "for iw in range(Nw):\n",
    "    px = pn[0:Nx,iw:iw+1];\n",
    "    for jw in range(Nw):\n",
    "        py = pm[0:Ny,jw:jw+1];\n",
    "        z = (Dx*Dy) * np.sum( np.power(np.matmul(px,py.T),2) / lam );\n",
    "        norm[iw,jw] = z;\n",
    "print('Check normalization of eigenfunctions. Normalization should be close to unity.');\n",
    "print( 'min norm', np.min(norm), ' max norm ', np.max(norm) );\n",
    "\n",
    "# assign excitation function for force, f\n",
    "# the source in the differential equation is -c2 divergence(force)\n",
    "sfx = 2.0;\n",
    "sfx2 = sfx**2;\n",
    "sfy = 2.0;\n",
    "sfy2 = sfy**2;\n",
    "x0 = xmax/2.0;\n",
    "y0 = ymax/2.0;\n",
    "Gx = (1.0/(sqrt(2*pi)*sfx)) * np.exp( -0.5*np.power(x-x0,2)/sfx2 ); # Gaussian in x\n",
    "Gxx = -c2 * cosq * np.matmul(D1/Dx,Gx);\n",
    "Gy = (1.0/(sqrt(2*pi)*sfy)) * np.exp( -0.5*np.power(y-y0,2)/sfy2 ); # Gaussian in y\n",
    "Gyy = -c2 * sinq * np.matmul(D1/Dy,Gy);\n",
    "Sxy = np.matmul(Gxx,Gy.T)+np.matmul(Gx,Gyy.T);\n",
    "excit = np.zeros((Nw,Nw));\n",
    "for iw in range(Nw):\n",
    "    px = pn[0:Nx,iw:iw+1];\n",
    "    for jw in range(Nw):\n",
    "        py = pm[0:Ny,jw:jw+1];\n",
    "        pxy = np.matmul(px,py.T);\n",
    "        # excit = (1/lam) (integral) pn(x) f(s)\n",
    "        excit[iw,jw] = (1.0/lam) * (Dx*Dy) * np.sum( np.multiply(pxy,Sxy) );\n",
    "print('Excitations:');\n",
    "gda_draw('title excit',excit);\n",
    "print( 'min ', np.min(excit), ' max ', np.max(excit) )\n",
    "\n",
    "Nt = 100; # time frames to compute\n",
    "Nplot = 10; # plot every Nplot frame\n",
    "PP = np.zeros((Nx,Ny,Nt));\n",
    "# compute pressure at time t0 by performing mode sum\n",
    "for itime in range(1,100):\n",
    "    t0 = 3.0*itime;\n",
    "    P = np.zeros((Nx,Ny));\n",
    "    for iw in range(Nw):\n",
    "        px = pn[0:Nx,iw:iw+1];\n",
    "        for jw in range(Nw):\n",
    "            py = pm[0:Ny,jw:jw+1];\n",
    "            pxy = np.matmul(px,py.T);\n",
    "            tfun = np.sin( wnm[iw,jw] * t0 );\n",
    "            P = P + (1.0/wnm[iw,jw]) * excit[iw,jw] * pxy * tfun;\n",
    "    PP[:,:,itime] = P;\n",
    "    if( (itime==1) or ((itime%Nplot)==0) or (itime==(Nt-1)) ):\n",
    "        fig1 = plt.figure(figsize=(12,5));   # smallish figure\n",
    "        ax1 = plt.subplot(1, 1, 1);\n",
    "        mycm = plt.get_cmap('jet');\n",
    "        plt.axis( [ymin, ymax, xmin, xmax ] );\n",
    "        plt.imshow(P,cmap=mycm,aspect='auto',extent=(ymin,ymax,xmax,xmin));\n",
    "        plt.colorbar();\n",
    "        plt.xlabel('y');\n",
    "        plt.ylabel('x');\n",
    "        plt.title(\"time %.2f and source at %.2f %.2f \" % (t0,x0,y0) );\n",
    "        plt.savefig(\"frame_%02d.jpg\" % (itime));\n",
    "        plt.show();\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "[encoded data, 14940 characters]",
      "text/plain": [
       "<Figure size 1200x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "[encoded data, 14976 characters]",
      "text/plain": [
       "<Figure size 1200x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "[encoded data, 16064 characters]",
      "text/plain": [
       "<Figure size 1200x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Ladder plots for lecture\n",
    "\n",
    "fig1 = plt.figure(figsize=(12,5));\n",
    "ax1 = plt.subplot(1, 1, 1);\n",
    "plt.axis( [0, 20, 0, 1 ] );\n",
    "plt.xlabel('frequency');\n",
    "n=1;\n",
    "for m in range(1,20):\n",
    "    w = sqrt( n**2 + m**2 )\n",
    "    plt.plot(gda_cvec(w,w),gda_cvec(0,1),'r-');\n",
    "plt.show();\n",
    "\n",
    "fig1 = plt.figure(figsize=(12,5));\n",
    "ax1 = plt.subplot(1, 1, 1);\n",
    "plt.axis( [0, 20, 0, 1 ] );\n",
    "plt.xlabel('frequency');\n",
    "n=5;\n",
    "for m in range(1,20):\n",
    "    w = sqrt( n**2 + m**2 )\n",
    "    plt.plot(gda_cvec(w,w),gda_cvec(0,1),'r-');\n",
    "plt.show();\n",
    "\n",
    "fig1 = plt.figure(figsize=(12,5));\n",
    "ax1 = plt.subplot(1, 1, 1);\n",
    "plt.axis( [0, 20, 0, 1 ] );\n",
    "plt.xlabel('frequency');\n",
    "for n in range(1,20):\n",
    "    for m in range(1,20):\n",
    "        w = sqrt( n**2 + m**2 )\n",
    "        plt.plot(gda_cvec(w,w),gda_cvec(0,1),'k-');\n",
    "plt.show();\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
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       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "[encoded data, 78688 characters]",
      "text/plain": [
       "<Figure size 640x480 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# animation of result\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from matplotlib import animation, rc\n",
    "from IPython.display import HTML\n",
    "\n",
    "fig, ax = plt.subplots();\n",
    "\n",
    "ax.set_xlim(( ymin, ymax));\n",
    "ax.set_ylim(( xmin, xmax));\n",
    "\n",
    "i=0;\n",
    "PPi = np.squeeze( PP[0:Nx,0:Ny,i] );\n",
    "dmax = np.max(np.abs(PP));\n",
    "dmin = -dmax;\n",
    "mycmap=matplotlib.colormaps['jet'];\n",
    "im=ax.imshow(PPi,cmap=mycm,vmin=dmin,vmax=dmax,aspect='auto',extent=(ymin,ymax,xmax,xmin));\n",
    "plt.xlabel('y');\n",
    "plt.ylabel('x');\n",
    "plt.colorbar(mappable=im);\n",
    "\n",
    "def init():\n",
    "    im.set_data([[]]);\n",
    "    return (im,)\n",
    "\n",
    "def animate(i):\n",
    "    PPi = np.squeeze( PP[0:Nx,0:Ny,i] );\n",
    "    im.set_data(PPi);\n",
    "    return (im,)\n",
    "\n",
    "anim = animation.FuncAnimation(fig, animate, init_func=init,\n",
    "                               frames=int(Nt), interval=int(20), \n",
    "                               blit=True)\n",
    "HTML(anim.to_html5_video())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "animation written\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 78688 characters]",
      "text/plain": [
       "<Figure size 640x480 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# write animation as file\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from matplotlib import animation, rc\n",
    "from IPython.display import HTML\n",
    "\n",
    "fig, ax = plt.subplots();\n",
    "\n",
    "ax.set_xlim(( xmin, xmax));\n",
    "ax.set_ylim(( ymin, ymax));\n",
    "\n",
    "i=0;\n",
    "PPi = np.squeeze( PP[0:Nx,0:Ny,i] );\n",
    "dmax = np.max(np.abs(PP));\n",
    "dmin = -dmax;\n",
    "mycmap=matplotlib.colormaps['jet'];\n",
    "im=ax.imshow(PPi,cmap=mycm,vmin=dmin,vmax=dmax,aspect='auto',extent=(ymin,ymax,xmax,xmin));\n",
    "plt.xlabel('y');\n",
    "plt.ylabel('x');\n",
    "plt.colorbar(mappable=im);\n",
    "\n",
    "def init():\n",
    "    im.set_data([[]]);\n",
    "    return (im,)\n",
    "\n",
    "def animate(i):\n",
    "    PPi = np.squeeze( PP[0:Nx,0:Ny,i] );\n",
    "    im.set_data(PPi);\n",
    "    return (im,)\n",
    "\n",
    "anim = animation.FuncAnimation(fig, animate, init_func=init,\n",
    "                               frames=int(Nt), interval=int(20), \n",
    "                               blit=True)\n",
    "HTML(anim.to_html5_video())\n",
    "\n",
    "writer = animation.PillowWriter(fps=15);\n",
    "anim.save('Lec19_2D_acoustic_wave.gif', writer=writer);\n",
    "print(\"animation written\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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