.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde/wave/plot_02_wave_animation.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_pde_wave_plot_02_wave_animation.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_pde_wave_plot_02_wave_animation.py:


Animated: a plucked string splitting, reflecting, and recombining
=================================================================

D'Alembert's solution :math:`u = F(x - ct) + G(x + ct)` says a plucked
string's bump splits into two half-height waves running in opposite
directions. At the fixed ends each reflects upside down; after one full
period :math:`2L/c` the string is back where it started. The leapfrog
solve (solid) runs at Courant number 1, where it reproduces d'Alembert's
formula (dashed) exactly at the grid points.

.. GENERATED FROM PYTHON SOURCE LINES 14-32

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.pde import WaveEquation1D, dalembert_solution
    from mathematicskit.pde.visualizers import animate_solution

    # In Jupyter or JupyterLite, show animations as an HTML/JavaScript player
    # (no ffmpeg needed).
    plt.rcParams["animation.html"] = "jshtml"


    def pluck(x):
        return np.maximum(0.0, 1.0 - np.abs(x - 0.3) / 0.1)


    string = WaveEquation1D(pluck, n=201, c=1.0)
    sol = string.solve(2.0, dt=string.dx, method="leapfrog")








.. GENERATED FROM PYTHON SOURCE LINES 33-35

One full period of the string
-----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 35-39

.. code-block:: Python


    anim = animate_solution(sol, exact=lambda x, t: dalembert_solution(pluck, x, t, length=1.0), max_frames=81)
    anim




.. container:: sphx-glr-animation

    .. raw:: html

        
     <link rel="stylesheet"
     href="https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/css/font-awesome.min.css">
     <script language="javascript">
       function isInternetExplorer() {
         ua = navigator.userAgent;
         /* MSIE used to detect old browsers and Trident used to newer ones*/
         return ua.indexOf("MSIE ") > -1 || ua.indexOf("Trident/") > -1;
       }

       /* Define the Animation class */
       function Animation(frames, img_id, slider_id, interval, loop_select_id){
         this.img_id = img_id;
         this.slider_id = slider_id;
         this.loop_select_id = loop_select_id;
         this.interval = interval;
         this.current_frame = 0;
         this.direction = 0;
         this.timer = null;
         this.frames = new Array(frames.length);

         for (var i=0; i<frames.length; i++)
         {
          this.frames[i] = new Image();
          this.frames[i].src = frames[i];
         }
         var slider = document.getElementById(this.slider_id);
         slider.max = this.frames.length - 1;
         if (isInternetExplorer()) {
             // switch from oninput to onchange because IE <= 11 does not conform
             // with W3C specification. It ignores oninput and onchange behaves
             // like oninput. In contrast, Microsoft Edge behaves correctly.
             slider.setAttribute('onchange', slider.getAttribute('oninput'));
             slider.setAttribute('oninput', null);
         }
         this.set_frame(this.current_frame);
       }

       Animation.prototype.get_loop_state = function(){
         var button_group = document[this.loop_select_id].state;
         for (var i = 0; i < button_group.length; i++) {
             var button = button_group[i];
             if (button.checked) {
                 return button.value;
             }
         }
         return undefined;
       }

       Animation.prototype.set_frame = function(frame){
         this.current_frame = frame;
         document.getElementById(this.img_id).src =
                 this.frames[this.current_frame].src;
         document.getElementById(this.slider_id).value = this.current_frame;
       }

       Animation.prototype.next_frame = function()
       {
         this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));
       }

       Animation.prototype.previous_frame = function()
       {
         this.set_frame(Math.max(0, this.current_frame - 1));
       }

       Animation.prototype.first_frame = function()
       {
         this.set_frame(0);
       }

       Animation.prototype.last_frame = function()
       {
         this.set_frame(this.frames.length - 1);
       }

       Animation.prototype.slower = function()
       {
         this.interval /= 0.7;
         if(this.direction > 0){this.play_animation();}
         else if(this.direction < 0){this.reverse_animation();}
       }

       Animation.prototype.faster = function()
       {
         this.interval *= 0.7;
         if(this.direction > 0){this.play_animation();}
         else if(this.direction < 0){this.reverse_animation();}
       }

       Animation.prototype.anim_step_forward = function()
       {
         this.current_frame += 1;
         if(this.current_frame < this.frames.length){
           this.set_frame(this.current_frame);
         }else{
           var loop_state = this.get_loop_state();
           if(loop_state == "loop"){
             this.first_frame();
           }else if(loop_state == "reflect"){
             this.last_frame();
             this.reverse_animation();
           }else{
             this.pause_animation();
             this.last_frame();
           }
         }
       }

       Animation.prototype.anim_step_reverse = function()
       {
         this.current_frame -= 1;
         if(this.current_frame >= 0){
           this.set_frame(this.current_frame);
         }else{
           var loop_state = this.get_loop_state();
           if(loop_state == "loop"){
             this.last_frame();
           }else if(loop_state == "reflect"){
             this.first_frame();
             this.play_animation();
           }else{
             this.pause_animation();
             this.first_frame();
           }
         }
       }

       Animation.prototype.pause_animation = function()
       {
         this.direction = 0;
         if (this.timer){
           clearInterval(this.timer);
           this.timer = null;
         }
       }

       Animation.prototype.play_animation = function()
       {
         this.pause_animation();
         this.direction = 1;
         var t = this;
         if (!this.timer) this.timer = setInterval(function() {
             t.anim_step_forward();
         }, this.interval);
       }

       Animation.prototype.reverse_animation = function()
       {
         this.pause_animation();
         this.direction = -1;
         var t = this;
         if (!this.timer) this.timer = setInterval(function() {
             t.anim_step_reverse();
         }, this.interval);
       }
     </script>

     <style>
     .animation {
         display: inline-block;
         text-align: center;
     }
     input[type=range].anim-slider {
         width: 374px;
         margin-left: auto;
         margin-right: auto;
     }
     .anim-buttons {
         margin: 8px 0px;
     }
     .anim-buttons button {
         padding: 0;
         width: 36px;
     }
     .anim-state label {
         margin-right: 8px;
     }
     .anim-state input {
         margin: 0;
         vertical-align: middle;
     }
     </style>

     <div class="animation">
       <img id="_anim_img98314137524e4e00b229275a2f3a6a55">
       <div class="anim-controls">
         <input id="_anim_slider98314137524e4e00b229275a2f3a6a55" type="range" class="anim-slider"
                name="points" min="0" max="1" step="1" value="0"
                oninput="anim98314137524e4e00b229275a2f3a6a55.set_frame(parseInt(this.value));">
         <div class="anim-buttons">
           <button title="Decrease speed" aria-label="Decrease speed" onclick="anim98314137524e4e00b229275a2f3a6a55.slower()">
               <i class="fa fa-minus"></i></button>
           <button title="First frame" aria-label="First frame" onclick="anim98314137524e4e00b229275a2f3a6a55.first_frame()">
             <i class="fa fa-fast-backward"></i></button>
           <button title="Previous frame" aria-label="Previous frame" onclick="anim98314137524e4e00b229275a2f3a6a55.previous_frame()">
               <i class="fa fa-step-backward"></i></button>
           <button title="Play backwards" aria-label="Play backwards" onclick="anim98314137524e4e00b229275a2f3a6a55.reverse_animation()">
               <i class="fa fa-play fa-flip-horizontal"></i></button>
           <button title="Pause" aria-label="Pause" onclick="anim98314137524e4e00b229275a2f3a6a55.pause_animation()">
               <i class="fa fa-pause"></i></button>
           <button title="Play" aria-label="Play" onclick="anim98314137524e4e00b229275a2f3a6a55.play_animation()">
               <i class="fa fa-play"></i></button>
           <button title="Next frame" aria-label="Next frame" onclick="anim98314137524e4e00b229275a2f3a6a55.next_frame()">
               <i class="fa fa-step-forward"></i></button>
           <button title="Last frame" aria-label="Last frame" onclick="anim98314137524e4e00b229275a2f3a6a55.last_frame()">
               <i class="fa fa-fast-forward"></i></button>
           <button title="Increase speed" aria-label="Increase speed" onclick="anim98314137524e4e00b229275a2f3a6a55.faster()">
               <i class="fa fa-plus"></i></button>
         </div>
         <form title="Repetition mode" aria-label="Repetition mode" action="#n" name="_anim_loop_select98314137524e4e00b229275a2f3a6a55"
               class="anim-state">
           <input type="radio" name="state" value="once" id="_anim_radio1_98314137524e4e00b229275a2f3a6a55"
                  >
           <label for="_anim_radio1_98314137524e4e00b229275a2f3a6a55">Once</label>
           <input type="radio" name="state" value="loop" id="_anim_radio2_98314137524e4e00b229275a2f3a6a55"
                  checked>
           <label for="_anim_radio2_98314137524e4e00b229275a2f3a6a55">Loop</label>
           <input type="radio" name="state" value="reflect" id="_anim_radio3_98314137524e4e00b229275a2f3a6a55"
                  >
           <label for="_anim_radio3_98314137524e4e00b229275a2f3a6a55">Reflect</label>
         </form>
       </div>
     </div>


     <script language="javascript">
       /* Instantiate the Animation class. */
       /* The IDs given should match those used in the template above. */
       (function() {
         var img_id = "_anim_img98314137524e4e00b229275a2f3a6a55";
         var slider_id = "_anim_slider98314137524e4e00b229275a2f3a6a55";
         var loop_select_id = "_anim_loop_select98314137524e4e00b229275a2f3a6a55";
         var frames = new Array(81);
    
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         /* set a timeout to make sure all the above elements are created before
            the object is initialized. */
         setTimeout(function() {
             anim98314137524e4e00b229275a2f3a6a55 = new Animation(frames, img_id, slider_id, 59.0,
                                      loop_select_id);
         }, 0);
       })()
     </script>






.. GENERATED FROM PYTHON SOURCE LINES 40-41

.. code-block:: Python

    plt.show()








.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 6.671 seconds)


.. _sphx_glr_download_api_gallery_pde_wave_plot_02_wave_animation.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/pde/wave/plot_02_wave_animation.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_wave_animation.ipynb <plot_02_wave_animation.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_wave_animation.py <plot_02_wave_animation.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_wave_animation.zip <plot_02_wave_animation.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_