Wave_equation_1D_fixed_endpoints.gif(274 × 121 pixels, file size: 129 KB, MIME type: image/gif, looped, 99 frames, 4.9 s)

Description
English: Illustration of solution of one-dimensional wave equation: a gaussian wave on a string fixed at both ends. The wave reflects from each end with a 180° phase shift.
Date 24 August 2007, 01:27 (UTC)
Source self-made, with en:Matlab
Author Oleg Alexandrov

Matlab Logo.png This diagram was created with MATLAB.

Public domain I, the copyright holder of this work, release this work into the public domain. This applies worldwide.
In some countries this may not be legally possible; if so:
I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

MATLAB source code

% A wave travelling on a string with
% fixed endpoints

function main()

   % KSmrq's colors
   red    = [0.867 0.06 0.14];
   blue   = [0, 129, 205]/256;
   green  = [0, 200,  70]/256;
   yellow = [254, 194,   0]/256;
   white = 0.99*[1, 1, 1];
   
   % length of the string and the grid
   L = 5;
   N = 151;
   X=linspace(0, L, N);

   h = X(2)-X(1); % space grid size
   c = 0.5; % speed of the wave
   tau = 0.25*h/c; % time grid size
   
   K = 5; % steepness of the bump
   S = 0; % shift the wave
   f=inline('exp(-K*(x-S).^2)', 'x', 'S', 'K'); % a gaussian as an initial wave
   df=inline('-2*K*(x-S).*exp(-K*(x-S).^2)', 'x', 'S', 'K'); % derivative of f

   % wave at time 0 and tau
   U0 = 0*f(X, S, K);
   U1 = U0 - 2*tau*c*df(X, S, K);
   
   U = 0*U0; % current U

   Big=10000;
   Ut = zeros(Big, N);
   Ut(1, :) = U0;
   Ut(2, :) = U1;
   
   % hack to capture the first period of the wave
   min_k = 2*N; k_old = min_k; turn_on = 0;

   for j=3:Big

      last_j = j;
      
      %  fixed end points
      U(1)=0; U(N)=0;
      
      % finite difference discretization in time
      for i=2:(N-1)
         U(i) = (c*tau/h)^2*(U1(i+1)-2*U1(i)+U1(i-1)) + 2*U1(i) - U0(i);
      end

      Ut(j, :) = U;
      
      % update info, for the next iteration
      U0 = U1; U1 = U;

      % hack to capture the first period of the wave
      k = find ( abs(U) == max(abs(U)) );
      k = k(1);

      if k > N/2
         turn_on = 1;
      end

      min_k = min(min_k, k_old);
      if k > min_k & min_k == k_old & turn_on == 1
         break;
      end
      k_old = k; 
      
   end

   % truncate to the first period
   last_j = last_j - 1;
   Ut = Ut(1:last_j, :);

  % shift the wave by a certain amount
   shift = floor(last_j/4);
   Vt=Ut;
   Ut((last_j-shift+1):last_j, :) = Vt(1:shift, :);
   Ut(1:(last_j-shift), :)        = Vt((shift+1):last_j, :);

   num_frames = 100;
   spacing=floor(last_j/num_frames)
   
   % plot the wave
   for j=1:(last_j-spacing+1)

      U = Ut(j, :);

      if rem(j, spacing) == 1

         figure(1); clf; hold on;
         axis equal; axis off; 
         lw = 3; % linewidth
         plot(X, U, 'color', red, 'linewidth', lw);
	 
         % plot the ends of the string
         small_rad = 0.06;
         ball(0, 0, small_rad, red);
         ball(L, 0, small_rad, red);
	 
         % size of the window
         ys = 1.1;
         axis([-small_rad, L+small_rad, -ys, ys]);
      
         % small markers to keep the bounding box fixed when saving to eps
         plot(-small_rad, ys, '*', 'color', white);
         plot(L+small_rad, -ys, '*', 'color', white);

         frame_no = floor(j/spacing)+1;
         frame=sprintf('Frame%d.eps', 1000+frame_no);
         disp(frame)
         saveas(gcf, frame, 'psc2');
         
      end
   end
   
function ball(x, y, radius, color) % draw a ball of given uniform color 
   Theta=0:0.1:2*pi;
   X=radius*cos(Theta)+x;
   Y=radius*sin(Theta)+y;
   H=fill(X, Y, color);
   set(H, 'EdgeColor', color);

% The gif image was creating with the command 
% convert -antialias -loop 10000  -delay 15 -compress LZW Frame10* Movie.gif

Captions

One-dimensional wave equation

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depicts

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Wikimedia username: Oleg Alexandrov
author name string: Oleg Alexandrov

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copyright license

released into the public domain by the copyright holder

inception

24 August 2007

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current01:27, 24 August 2007Thumbnail for version as of 01:27, 24 August 2007274 × 121 (129 KB)Oleg Alexandrov{{Information |Description=Illustration of en:Wave equation |Source=self-made, with en:Matlab |Date=~~~~~ |Author= Oleg Alexandrov }} {{PD-self}} Category:Waves Category:Partial differential equations [[Catego
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