wave1.html

Worksheet: wave.mw

Fourier Sine Series

Solution to the Wave Equation

This worksheet examines the sine series that arises in a separation of variables

solution for the wave equation :

= 0 < x < L u(0,t)=u(L,t)=0

with initial condition

u(x,0)=F(x)=L/2-abs(x-L/2) 0 < x< L

(x,0)=G(x)=0 0 < x< L

> restart:with(plots):

> L:=1;c:=1;

(1)

Least Squares Approximation of The Initial Condition

Let's approximate the initial condition F(x) with the sum of sines that came from separation of variables.

> F:=x->L/2-abs(x-L/2);

(1.1)

> plot(F(x), x=0..L);

Plot_2d

> f:=n-> sin (n*Pi*x/L);

(1.2)

> a:=n->(2/L)*int(F(x)*f(n),x=0..L);

proc (n) options operator, arrow; `+`(`/`(`*`(2, `*`(int(`*`(F(x), `*`(f(n))), x = 0 .. L))), `*`(L))) end proc (1.3)

> seq(a[n]=a(n),n=1..10);

(1.4)

> g:= M->sum(a(m)*f(m),m=1..M);

(1.5)

> h:=g(3);

(1.6)

> plot({F(x),h},x=0..L);

Plot_2d

> s:=seq(g(2*M-1),M=1..6);

(1.7)

> plot({s,F(x)},x=0..L);

Plot_2d

Wave Equation Solution via Fourier Sine Series

We can now reconstitute our solution to the heat equation, with the initial condition we derived.

> u:=M->sum(a(m)*sin(m*Pi*x/L)*cos(c*m*Pi*t/L),m=1..M);

(2.1)

> up:=u(10);

(2.2)

> plot3d(up,x=0..L,t=0..2.0,style=patchcontour,shading=zhue,axes=boxed);

Plot_2d

> animate(up,x=0..L,t=0..2*L,frames=51);

Plot_2d

>