mem1-2.mws   Vibrating of circular membrane (example2 for wave2-1.doc) - Standing Waves

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restart;

>	with(plots):

Warning, the name changecoords has been redefined

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a:=0.05;

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r1:=1;

Eigenvalues:

>	w(x):=BesselJ(0,x*r1);

>	plot(w(x),x=0..30);

>	lambda:=array(1..50);

>	n:=1: for m from 1 to 300 do z:=fsolve(w(x)=0,x=m/10..(m+1)/10); if type (z,float) then lambda[n]:=z: n:=n+1 fi od:

>	for i to 5 do lambda[i] od;

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N:=n-1;

>	n:='n':i:='i':

Initial conditions:

>	f(r):=BesselJ(0,lambda[4]*r);

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g(r):=0;

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n:=4;

Fourier coefficients:

>	b[n]:=int(r*f(r)*BesselJ(0,lambda[n]*r),r=0..r1)/int(r*BesselJ(0,lambda[n]*r)^2,r=0..r1);

>	d[n]:=int(r*g(r)*BesselJ(0,lambda[n]*r),r=0..r1)/int(r*BesselJ(0,lambda[n]*r)^2,r=0..r1)/a/lambda[n];

>	u[n](r,t):=BesselJ(0,lambda[n]*r)*(b[n]*cos(lambda[n]*a*t)+d[n]*sin(lambda[n]*a*t));

>	u(r,t):=u[n](r,t):

>	u(r,0):=subs(t=0,u(r,t)):

>	u(r,1):=subs(t=1,u(r,t)):

>	u(r,2):=subs(t=2,u(r,t)):

>	u(r,3):=subs(t=3,u(r,t)):

>	u(r,4):=subs(t=4,u(r,t)):

>	u(r,5):=subs(t=5,u(r,t)):

>	plot({f(r),u(r,0),u(r,1),u(r,2),u(r,3),u(r,4),u(r,5)},r=0..r1,color=black,axes=boxed);

>	un(-r,t)=u(r,t):

>	T:=2*Pi/a/lambda[1];

>	animate({un(-r,t),u(r,t)},r=-1..1,t=0..T,frames=100,color=blue,axes=boxed);

>	cylinderplot([r,theta,u(r,5)],r=0..r1,theta=0..2*Pi,grid=[50,50]);

>	animate3d([r,theta,u(r,t)],r=0..r1,theta=0..2*Pi,t=0..2*T,coords=cylindrical,style=patchnogrid,frames=100);

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