Trinh @ Bath

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vpde_lecture25 [2020/03/31 08:25]
trinh 		vpde_lecture25 [2020/04/04 21:05]
trinh
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 This lecture will do apply separation of variables and Fourier series in order to solve for the wave equation on a finite interval.  This lecture will do apply separation of variables and Fourier series in order to solve for the wave equation on a finite interval. 
 +
 +<html>
 +<iframe width="560" height="315" src="https://www.youtube.com/embed/VMz3R6d2ZZ8" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
 +</html>
  
 ===== Definition 16.1 (1D wave equation with homogeneous Dirichlet BCs ===== ===== Definition 16.1 (1D wave equation with homogeneous Dirichlet BCs =====
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 ===== Example 16.2 (Imagining the modes) ===== ===== Example 16.2 (Imagining the modes) =====
  
-I want to help you imagine what the modes, $u_n$, look like. For simplicity, take the length $L = \pi$. Also, we may take $c = 1$. +I want to help you imagine what the modes, $u_n$, look like. For simplicity, take the length $L = \pi$. Also, we may take $c = 1$. The individual modes we want to examine are:  
 +$$ 
 +u_n(x, t) = \sin\left(nx\right) \left[ A_n \cos(nct) + B_n \sin(nct)\right] 
 +$$ 
 + 
 +<Code:Matlab linenums:1 |Demonstration of the modes> 
 +% Code for MA20223 30 Mar 2020 
 +clear 
 +close all 
 + 
 +% Length and time 
 +L = pi; T = 2*pi/(c*n); 
 + 
 +% Function 
 +n = 2; c = 1; 
 +un = @(x,t) sin(n*x/L).*cos(n*c*t/L); 
 + 
 +% Make vectors for space and time 
 +x = linspace(0, pi, 50); t = linspace(0, 2*T, 50); 
 + 
 +% Create a mesh of x vs. t 
 +[X,T] = meshgrid(x,t); 
 + 
 +% Matrix of U values to imagine the surface 
 +U = sin(n*X).*cos(n*T); 
 + 
 +figure(1); subplot(1,2,1); xlabel('x'); ylabel('u'); 
 + 
 +subplot(1,2,2); 
 +% Plot the surface and make it pretty 
 +s = surf(X,T,U); set(s, 'EdgeColor', 'none', 'FaceColor', 'interp'); 
 +view([-48, 17]); xlabel('x'); ylabel('t'); zlabel('u'); 
 +hold on 
 + 
 +% Plot an animation in time 
 +for j = 1:length(t)     
 +    tt = t(j); uu = un(x,tt); 
 +     
 +    subplot(1,2,1); 
 +    plot(x, uu);   title(['t = ', num2str(tt)]); 
 +    ylim([-1,1]); xlim([0, pi]); 
 +     
 +    subplot(1,2,2); 
 +    if j == 1 
 +        p = plot3(x, tt*ones(size(x)), un(x, t(j)), 'LineWidth', 3); 
 +        pause; 
 +    else 
 +        set(p, 'XData', x, 'YData', tt*ones(size(x)), 'ZData', un(x,t(j))); 
 +    end 
 +     
 +    drawnow 
 +    shg 
 +end 
 +</Code> 
 + 
 +==== Implementation of the Fourier series ==== 
 + 
 +Now we need to look to solve for the coefficients of our series by applying the boundary conditions. We have that  
 +$$ 
 +u(x, t) = \sum_{n=0}^\infty \sin\left(\frac{n\pi x}{L}\right) \left[ A_n \cos\left(\frac{n\pi ct}{L}\right) + B_n \sin\left(\frac{n\pi ct}{L}\right)\right] 
 +$$ 
 + 
 +Imposing the initial displacement, we have  
 +$$ 
 +u_0(x) = \sum_{n=0}^\infty A_n \sin\left(\frac{n\pi x}{L}\right),  
 +$$ 
 + 
 +which we recognise as the sine series for the odd $2L$-periodic extension of the function $u_0(x)$ originally defined on $[0, L]$. So the coefficients are (see theorem 12.7) 
 +$$ 
 +A_n = \frac{2}{L} \int_0^L u_0(x) \sin\left(\frac{n\pi x}{L}\right).  
 +$$ 
 + 
 +Imposing the initial velocity, we have 
 +$$ 
 +v_0(x) = \sum_{n=1}^\infty \left(\frac{n\pi c}{L}\right)B_n \sin\left(\frac{n\pi x}{L}\right) 
 +$$ 
 + 
 +Again we recognise this as the sine series, so we now need to equate 
 +$$ 
 +\left(\frac{n\pi c}{L}\right)B_n = \frac{2}{L} \int_0^L v_0(x) \sin\left(\frac{n\pi x}{L}\right).  
 +$$
Last modified: 2020/04/04 21:33