Circular Membrane

This worksheet allow one to plot modes of vibration for a circular  membrane with fixed homogeneous boundary conditions and to visualize the vibrations in time. The radius is a and the wave speed is given by c.
 

>	restart: a:=1: c:=1:  with(plots):

Warning, the name changecoords has been redefined

The frequency of oscillation of the nm-th mode  is given by

>	nu:=(m,n)->c*evalf(BesselJZeros(m,n));

The product solutions take the following form for a zero initial velocity, though one can easily add a phase shift in time to simulate other motions.

>	u:=(m,n,r,theta,t)->cos(c*nu(m,n)*t)*BesselJ(m,nu(m,n)/c*r)*cos(m*theta);

The initial profile of the membrane:

>	m0:=1: n0:=1: plot3d([r*cos(theta),r*sin(theta),u(m0,n0,r,theta,0)],r=0..a,theta=0..2*Pi,grid=[40,40]);

The evolution of this profile

>	animate3d([r*cos(theta),r*sin(theta),u(m0,n0,r,theta,t)],r=0..a,theta=0..2*Pi,t=0..6/nu(m0,n0),frames=20);

The following code shows how to save animations as animated gifs. The first line sets up the plot to be gif and the third line reverts the plotting back oto the screen.

>	plotsetup(gif,plotoutput=`cmesh23.gif`,plotoptions=`height=200,width=200`):

>	m0:=2: n0:=3: animate3d([r*cos(theta),r*sin(theta),u(m0,n0,r,theta,t)],r=0..a,theta=0..2*Pi,t=0..6/nu(m0,n0),frames=20);

>	plotsetup(inline);

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