Mathematics 173 -- Applied Math 1

A Boundary Value Problem for the 2D Wave Equation

A rectangular drumhead in shape of [0,L] x [0,H]

has a vertical displacement function u( x,y,t ) satisfying

= 0 all x,t

= 0 all y,t

if L/3 <= x <= L/2, H/3 <= y <= H/2 and 0 otherwise

The boundary condition on might represent a sharp blow

to the drumhead with a stick or mallet, for example(!)

> assume(m,integer): assume(n,integer):

The normal modes of vibration for this problem are

The eigenvalues for the 2D boundary value problem are:

> lambda:=(m,n)->(n*Pi/L)^2 + (m*Pi/H)^2;

The coeffficients are all zero by the initial condition on u. The

coefficients are given by the following formula, derived from the orthogonality

relation for the 2D eigenfunctions

> B := (m,n) -> evalf( -4/(c*sqrt(lambda(m,n))*L*H)*Int(Int((.1)*sin(n*Pi*x/L)*sin(m*Pi*y/H),y=H/3..H/2),x=L/3..L/2));

The partial sum of the Fourier series solution -- the superposition of normal

modes that satisfies the given initial conditions on u and

> uPS:=(x,y,t,M,N)->add(add(B(m,n)*sin(c*sqrt(lambda(m,n))*t)*sin(n*Pi*x/L)*sin(m*Pi*y/H),m=1..M),n=1..N);

Warning, `n` in call to `add` is not local

Warning, `m` in call to `add` is not local

We use animation to show the motion of the vibrating drumhead:

> L:=10; H:=5; c:=.1;

> PSu:=uPS(x,y,t,10,10):

The initial vertical velocity: (NOTE: this series shows a strong 2D Gibbs phenomenon

effect(!))

> plot3d(subs(t=0,diff(PSu,t)),x=0..L,y=0..H,grid=[30,30],axes=BOXED);

> with(plots):

> animate3d(PSu,x=0..L,y=0..H,t=0..100,frames=100,axes=BOXED);

>