%% The vibrating string example

%% 
% Solve the wave equation with initial value f(s), initial velocity 0, 
% where f(s)=s if 0<s <L/2 and L-s if L/2 < s < L.  Here, we take L=10, c=1.

syms s k n t
evalin(symengine,'assume(k,Type::Integer)');

%%
% First we calculate the Fourier sine coefficients of f. We know the
% coefficients with even index are 0 so we only have to find the ones with
% odd index.

b = @(k) 4*int(s*sin((2*k-1)*pi*s/10),s,0,5)/10
          

 

%%
% The nth partial sum of the solution is

u = @(s,t,n) symsum(b(k)*sin((2*k-1)*pi*s/10)*cos((2*k-1)*pi*t/10),k,1,n);


%%
% Here are some plots of the solution u(s,t) versus s for some fixed values
% of t (taking 100 terms in the series for u).



ezplot(u(s,0,100),[0,10])
title('t=0')

%%

ezplot(u(s,10/8,100),[0,10])
hold on 
fplot('s*0',[0,10],'k')
hold off
title('t=10/8')
axis equal
%%

ezplot(u(s,10,100),[0,10])
title('t=10')

%% 
% Here are some plots of u(s,t) versus t for some fixed values of s.

T = 0:0.01:20;
U = inline(vectorize(u(5,t,100)));
plot(T,U(T))
title('s=5')

%%

ezplot(u(10,t,100),[0,20])
title('s=10')

%%
% Here is a movie of the motion of the string.  

S = 0:0.01:10;
for n = 0:100
    U = inline(vectorize(u(s,n,100)));
    plot(S, U(S)), axis([0,10,-5.5,5.5]);
    hold on
    plot(S,0*S,'k'), axis([0,10,-5.5,5.5]);
    hold off
    M(n+1) = getframe;
end


mplay(M,6)