2-D WAVE EQUATION, Rectangular Coordinates (Drum)

Utt = Uxx + Uyy t>0, (x,y) in (0,2)x(0,1)

U = 0 t>0, (x,y) in Bndy of (0,2)x(0,1)

U=f & Ut = g t=0, (x,y) in (0,2)x(0,1)

f:=(x,y)->0;

g:=(x,y)->(Heaviside(x-.9)-Heaviside(x-1.1))*(Heaviside(y-.4)-Heaviside(y-.6))/.04;

plot3d({f(x,y),g(x,y)},x=0..2,y=0..1);

lam:=(n,m)-> -(n*Pi/2)^2 - (m*Pi)^2;

omega:=(n,m)->evalf(sqrt(-lam(n,m)));

u:=(k,l,t,x,y)->sum(sum((c(n,m)*cos(omega((n,m))*t) + d(n,m)*sin(omega((n,m))*t))*sin(n*Pi*x/2)*sin(m*Pi*y),n=1..k),m=1..l);

c:=(n,m)->int(int(f(x,y)*sin(n*Pi*x/2)*sin(m*Pi*y),x=0..2),y=0..1)/int(int((sin(n*Pi*x/2)^2)*(sin(m*Pi*y))^2,x=0..2),y=0..1);

d:=(n,m)->(omega((n,m))^(-1))*int(int(g(x,y)*sin(n*Pi*x/2)*sin(m*Pi*y),x=0..2),y=0..1)/int(int((sin(n*Pi*x/2)^2)*(sin(m*Pi*y))^2,x=0..2),y=0..1);

u(2,5,t,x,y);

with(plots):animate3d(u(5,5,t,x,y),x=0..2,y=0..1,t=0..3,frames=200);

2-D WAVE EQUATION, Rectangular Coordinates (Flag)

Utt = Uxx + Uyy t>0, (x,y) in (0,2)x(0,1)

U(0,y) = Ux(2,y) = 0 t>0, 0<y<1

Uy(x,0) = Uy(x,1) = 0 t>0, 0<x<2

U=f & Ut = g t=0, (x,y) in (0,2)x(0,1)

f:=(x,y)->0;

g:=(x,y)->(Heaviside(x-.9)-Heaviside(x-1.1))*(Heaviside(y-.4)-Heaviside(y-.6))/.04;

plot3d({f(x,y),g(x,y)},x=0..2,y=0..1);

lam:=(n,m)-> -((2*n-1)*Pi/4)^2 - (m*Pi)^2;

omega:=(n,m)->evalf(sqrt(-lam(n,m)));

u:=(k,l,t,x,y)->sum((c(n,0)*cos(omega(n,0)*t) + d(n,0)*sin(omega(n,0)*t))*sin((2*n-1)*Pi*x/4),n=1..k) + sum(sum((c(n,m)*cos(omega(n,m)*t) + d(n,m)*sin(omega(n,m)*t))*sin((2*n-1)*Pi*x/4)*cos(m*Pi*y),n=1..k),m=1..l);

c:=(n,m)->int(int(f(x,y)*sin((2*n-1)*Pi*x/4)*cos(m*Pi*y),x=0..2),y=0..1)/int(int(sin((2*n-1)*Pi*x/4)^2*(cos(m*Pi*y))^2,x=0..2),y=0..1);

d:=(n,m)->(omega((n,m))^(-1))*int(int(g(x,y)*sin((2*n-1)*Pi*x/4)*cos(m*Pi*y),x=0..2),y=0..1)/int(int((sin((2*n-1)*Pi*x/4))^2*(cos(m*Pi*y))^2,x=0..2),y=0..1);

u(2,5,t,x,y);

with(plots):animate3d(u(5,5,t,x,y),x=0..2,y=0..1,t=0..12,frames=200);