fdtd/fd_seismometer_solution.ipynb

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    "<div style='background-image: url(\"../../share/images/header.svg\") ; padding: 0px ; background-size: cover ; border-radius: 5px ; height: 250px'>\n",
    "    <div style=\"float: right ; margin: 50px ; padding: 20px ; background: rgba(255 , 255 , 255 , 0.7) ; width: 50% ; height: 150px\">\n",
    "        <div style=\"position: relative ; top: 50% ; transform: translatey(-50%)\">\n",
    "            <div style=\"font-size: xx-large ; font-weight: 900 ; color: rgba(0 , 0 , 0 , 0.8) ; line-height: 100%\">Computational Seismology</div>\n",
    "            <div style=\"font-size: large ; padding-top: 20px ; color: rgba(0 , 0 , 0 , 0.5)\">Finite Differences - Seismometer Equation</div>\n",
    "        </div>\n",
    "    </div>\n",
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    "<p style=\"width:20%;float:right;padding-left:50px\">\n",
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    "\n",
    "\n",
    "---\n",
    "\n",
    "This notebook is part of the supplementary material\n",
    "to [Computational Seismology: A Practical Introduction](https://global.oup.com/academic/product/computational-seismology-9780198717416?cc=de&lang=en&#),\n",
    "Oxford University Press, 2016.\n",
    "\n",
    "\n",
    "##### Authors:\n",
    "* Ashim Rijal ([@ashimrijal](https://github.com/ashimrijal))\n",
    "* Heiner Igel ([@heinerigel](https://github.com/heinerigel))\n"
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    "This exercise covers the following aspects:\n",
    "\n",
    "* Solving seismometer equation with finite difference method\n",
    "* Getting familiar with seismometer response function\n",
    "---"
   ]
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    "## Basic Equations\n",
    "\n",
    "** Please refer to the Exercise 4.19 from the book.**\n",
    "\n",
    "The seismometer equation is given by\n",
    "$$\n",
    "\\ddot{x} + 2\\epsilon \\dot{x} + \\omega^2_0 x = - \\ddot{u}\n",
    "$$\n",
    "\n",
    "Where,\n",
    "\n",
    "$ \\epsilon $ is the damping parameter,\n",
    "\n",
    "$ \\omega_0 $ is the eigenfrequency,\n",
    "\n",
    "$ \\ddot{u}(t) $ is the ground motion by which the seismometer is excited, and\n",
    "\n",
    "$ x(t) $ is the motion of the seismometer mass.\n",
    "\n",
    "We replace the time derivative by centered finite-differentiation\n",
    "$$\n",
    "\\dot{x} \\ \\approx \\ \\frac{x (t + \\mathrm{d}t) - x ( t- \\mathrm{d}t)} {2\\mathrm{d}t}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\ddot{x} \\ \\approx \\ \\frac{x( t+ \\mathrm{d}t)-2x(t) + x( t- \\mathrm{d}t)} {\\mathrm{d}t^2}\n",
    "$$\n",
    "\n",
    "Now, solving for $ x(t+\\mathrm{d}t) $ the extrapolation scheme is\n",
    "\n",
    "$$\n",
    "x(t+\\mathrm{d}t) = \\frac{ - \\ddot{u}(t) \\mathrm{d}t^2 + (2-\\omega^2_0 \\mathrm{d}t^2) x(t) + (\\epsilon \\mathrm{d}t-1) x(t-\\mathrm{d}t)} {(1+\\epsilon \\mathrm{d}t)}\n",
    "$$\n"
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    "### Exercise\n",
    "\n",
    "** Part 1**\n",
    "\n",
    "While running the following cells frequency of forcing (the frequency of ground motion) and the damping parameter will be asked to enter. First try using undamped seismometer (i.e. h = 0) for some forcing frequency (eg. 0.1 Hz, 1 Hz, 2Hz, 3Hz, 4Hz, 5Hz, etc.) and interpret the results.\n",
    "\n",
    "** Part 2**\n",
    "\n",
    "Now try frequency of forcing of your choice (eg. 1 HZ) and try to search for suitable damping parameter (h).\n",
    "\n",
    "**Message: Once you become familiar with all the codes below you can go to the Cell tab on the toolbar and click Run All.**"
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    "#Configuration step (Please run it before the simulation code!)\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib\n",
    "# Show Plot in The Notebook\n",
    "matplotlib.use(\"nbagg\")\n",
    "import matplotlib.pyplot as plt\n",
    "matplotlib.rcParams['figure.facecolor'] = 'w'                 # remove grey background"
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    "#Initialization of parameters\n",
    "\n",
    "f0 = 1.                                                       # eigenfrequency of seismometer (hertz)\n",
    "w = 2. * np.pi * f0                                           # in radians per second\n",
    "dt = .01                                                      # time increment for numerical scheme\n",
    "isnap = 2                                                     # snapshot frequency for visualization\n",
    "\n",
    "# Central frequency of forcing or ground motion (will be asked)\n",
    "fu0 = 1.0\n",
    "# Uncomment for interactivity.\n",
    "# fu0 = float(input('Give frequency of forcing (e.g. f=1 Hz) : '))\n",
    "\n",
    "# Damping parameter of seismometer (will be asked)\n",
    "h = 0.5\n",
    "# Uncomment for interactivity.\n",
    "# h = float(input('Give damping parameter (e.g. h=0.5) : '))\n",
    "\n",
    "# Initialize ground motion\n",
    "\n",
    "# Initialize parameters for ground motion\n",
    "p = 1. / fu0                                                  # period\n",
    "nts = int(2. * p / dt)                                        # time steps\n",
    "uii = np.zeros(nts)                                           # ground motion\n",
    "t0 = p / dt                                                   # time (S)\n",
    "a = 4. / p                                                    # half-width (so called sigma)\n",
    "\n",
    "# we use derivative of a Gaussian as our ground motion\n",
    "for it in range(nts):\n",
    "    t = (it - t0) * dt\n",
    "    uii[it] = -2 * a * t * np.exp(-(a * t) ** 2)\n",
    "\n",
    "nt = int(round(5. * 1. / fu0 / dt))                           # total number of time steps\n",
    "src = np.zeros(nt)                                            # initialize an source array of zeros\n",
    "src[0:len(uii)] = uii                                         # make derivative of gaussian as source\n",
    "\n",
    "# End initialization of ground motion\n",
    "\n",
    "\n",
    "# Initial conditions\n",
    "eps = h * w                                                   # damping factor\n",
    "x = np.zeros(nt)\n",
    "xnow = 0\n",
    "xold = 0\n",
    "x_vector = np.zeros(nt)"
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       "\n",
       "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch(cursor)\n",
       "    {\n",
       "    case 0:\n",
       "        cursor = 'pointer';\n",
       "        break;\n",
       "    case 1:\n",
       "        cursor = 'default';\n",
       "        break;\n",
       "    case 2:\n",
       "        cursor = 'crosshair';\n",
       "        break;\n",
       "    case 3:\n",
       "        cursor = 'move';\n",
       "        break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message(\"ack\", {});\n",
       "}\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = \"image/png\";\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src);\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data);\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig[\"handle_\" + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "}\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function(e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e)\n",
       "        e = window.event;\n",
       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\n",
       "\n",
       "    return {\"x\": x, \"y\": y};\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys (original) {\n",
       "  return Object.keys(original).reduce(function (obj, key) {\n",
       "    if (typeof original[key] !== 'object')\n",
       "        obj[key] = original[key]\n",
       "    return obj;\n",
       "  }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function(event, name) {\n",
       "    var canvas_pos = mpl.findpos(event)\n",
       "\n",
       "    if (name === 'button_press')\n",
       "    {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * mpl.ratio;\n",
       "    var y = canvas_pos.y * mpl.ratio;\n",
       "\n",
       "    this.send_message(name, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.key_event = function(event, name) {\n",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, {key: value,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
       "    if (name == 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message(\"toolbar_button\", {name: name});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n",
       "\n",
       "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function() {\n",
       "        comm.close()\n",
       "    };\n",
       "    ws.send = function(m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function(msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data'])\n",
       "    });\n",
       "    return ws;\n",
       "}\n",
       "\n",
       "mpl.mpl_figure_comm = function(comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element.get(0);\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error(\"Failed to find cell for figure\", id, fig);\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    var width = fig.canvas.width/mpl.ratio\n",
       "    fig.root.unbind('remove')\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable()\n",
       "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
       "    fig.close_ws(fig, msg);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.close_ws = function(fig, msg){\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width/mpl.ratio\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message(\"ack\", {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items){\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
       "    titlebar.prepend(buttongrp);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(el){\n",
       "    var fig = this\n",
       "    el.on(\"remove\", function(){\n",
       "\tfig.close_ws(fig, {});\n",
       "    });\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(el){\n",
       "    // this is important to make the div 'focusable\n",
       "    el.attr('tabindex', 0)\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    }\n",
       "    else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager)\n",
       "        manager = IPython.keyboard_manager;\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which == 13) {\n",
       "        this.canvas_div.blur();\n",
       "        event.shiftKey = false;\n",
       "        // Send a \"J\" for go to next cell\n",
       "        event.which = 74;\n",
       "        event.keyCode = 74;\n",
       "        manager.command_mode();\n",
       "        manager.handle_keydown(event);\n",
       "    }\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.find_output_cell = function(html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i=0; i<ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code'){\n",
       "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] == html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel != null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<div id='6bcc3da2-2514-4985-a70a-3bd05e171e23'></div>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Extrapolation scheme and the plots\n",
    "\n",
    "# Initialization of plots\n",
    "\n",
    "# lines1 will plot the seismometer response and lines2 for source function\n",
    "\n",
    "lines1 = plt.plot(np.dot((np.arange(1, nt+1)), dt), x_vector[0:nt] /\n",
    "                  np.max(np.abs(x[0:nt])), color = \"red\", lw = 1.5, label=\"Seismometer response\")\n",
    "lines2 = plt.plot(np.zeros(nt), color = 'blue',lw = 1.5,  label=\"Ground Acceleration\")\n",
    "\n",
    "plt.title(\"At rest\")\n",
    "plt.axis([0, nt*dt, -1, 1])\n",
    "plt.xlabel(\"Time (s)\")\n",
    "plt.ylabel(\"Displacement\")\n",
    "plt.legend(loc=\"upper right\")\n",
    "plt.ion()\n",
    "plt.show()\n",
    "\n",
    "# Begin extrapolation and update the plot\n",
    "\n",
    "# Extrapolation scheme (Centered finite-difference)\n",
    "\n",
    "for i in range(nt):\n",
    "    if i == 0:\n",
    "        xold = xnow\n",
    "\n",
    "    xnew = (-src[i] * dt ** 2 + (2 - w ** 2 * dt ** 2) * xnow + (eps * dt - 1) * xold) / (1 + eps * dt)\n",
    "\n",
    "    xold = xnow                                               # for next loop present will be past\n",
    "    xnow = xnew                                               # for next step future will be present\n",
    "    x[i] = xnow\n",
    "    x_vector[i] = x[i]\n",
    "\n",
    "# Updating the plots\n",
    "    if not i % isnap:\n",
    "        for l in lines1:\n",
    "            l.remove()\n",
    "            del l\n",
    "        for k in lines2:\n",
    "            k.remove()\n",
    "            del k\n",
    "\n",
    "        lines1  = plt.plot(np.dot((np.arange (1, i+1)), dt), x_vector[0:i] /\n",
    "                          np.max(np.abs (x[0:nt])),color = \"red\",lw = 1.5, label=\"Seismometer response\")\n",
    "        lines2 = plt.plot(np.dot((np.arange (1, i+1)), dt), src[0:i] /\n",
    "                          np.max(src[0:nt]), color = 'blue',lw = 1.5, label=\"Ground Acceleration\")\n",
    "    plt.title(\"F0 = 1Hz, SRC = %.2f Hz, h = %.2f \" % (fu0, h))\n",
    "    plt.gcf().canvas.draw()\n",
    "\n",
    "plt.ioff()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "solution"
    ]
   },
   "source": [
    "## Solutions\n",
    "**Part 1**\n",
    "\n",
    "Let us try frequency of forcing 1 Hz and damping parameter h = 0. Then we see that response of seismometer doesn't come to rest even after the ground motion comes to rest.\n",
    "\n",
    "** Part 2**\n",
    "\n",
    "For low damping (h< 1) there exists a peak in the response function, underdamped case. If h=1, the seismometer comes to rest in the least possible time without overshooting, a case called critically damped where the response curve has no peak. The most common values used in seismometres are close to critical (eg. 0.707) in which seismometers perform optimally. For values greater than 1 (h> 1) the sensitivity of seismometer decreases, a case of overdamping."
   ]
  }
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