45044, 45036 membrane 2 boundaries Hi, Yama Masu, Below I give you a schema for the solution of your membrane consisting
- To: mathgroup at smc.vnet.net
- Subject: [mg45143] 45044, 45036 membrane 2 boundaries Hi, Yama Masu, Below I give you a schema for the solution of your membrane consisting
- From: CAP F <Ferdinand.Cap at eunet.at>
- Date: Thu, 18 Dec 2003 06:55:17 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
of two squares. When you do the calculations as proposed you may
possibly extend the collocation points all over the full 4 quadrants
because the symmetry properties in polar coordinates are other than in
cartesian coordinates. Observe if the matrix of the determinant is well
conditioned, see then the method on page 276.
AS for your problem of plucking impulse like a string, you find the
solution in equations (4.3.37) and (4.3.38) on page174.Greetings F
Cap
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Cell["\<\
(*Masu membrane with 2 homogeneous boundary conditions \
along an outer square of length 2 and an inner square of length \
1.Show situation and determine n collocation points.First on outer \
square : *)
xs[1] = -1.; ys[1] = -1.; xs[2] = -1.; ys[2] = 1.; xs[3] = 1.; ys[3] \
= 1.;
xs[4] = 1.; ys[4] = -1.; xs[5] = xs[1]; ys[5] = ys[1];
pl1 = ListPlot[Table[{xs[i], ys[i]}, {i, 1, 5}], AspectRatio -> 1,
PlotStyle -> PointSize[1/50], PlotJoined -> True, \
DisplayFunction->Identity]
xi[1]=-0.5;yi[1]=-0.5;xi[2]=-0.5;yi[2]=0.5;xi[3]=0.5;yi[3]=0.5; \
xi[4]=0.5;yi[4]=-0.5;xi[5]=xi[1];yi[5]=yi[1];
pl2=ListPlot[Table[{xi[i], yi[i]}, {i,1,5}], AspectRatio\[Rule]1, \
PlotStyle\[Rule]PointSize[1/50], PlotJoined->True, DisplayFunction\
\[Rule]Identity]
pl3=Show[pl1, pl2, DisplayFunction\[Rule]Identity]
Clear[x,y,n,f,r];n=20;dxa=0.5/(n/10);
Table[x[i+1]=-1.0+i*dxa,{i,0,n/5}];
Table[y[i+1]=1.0,{i,0,n/5}];
Table[x[i+n/4]=-1.0,{i,1,n/5+1}];
Table[y[i+n/4+1]=0+i*dxa,{i,0,n/5}];\
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(* on inner square of length 1. *) \
dxi=0.25/(n/10);Table[x[i+1+n/2]=-0.5+i*dxi,{i,0,n/5}];
Table[y[i+1+n/2]=0.5, {i,0,n/5}];
Table[x[i+1+n/5+n/2]=-0.5, {i,1,n/5+1}];
Table[y[i+2+n/5+n/2]=0.+i*dxi, {i,0,n/5}];
pl4=ListPlot[Table[{x[l], y[l]}, {l,1,n}], \
PlotStyle\[Rule]PointSize[1/40], AspectRatio\[Rule]1., PlotRange\
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(* Now switch to polar coordinates *)Clear [r,\[CurlyPhi]]; \
Table[\[CurlyPhi][l] = N[ArcTan[x[l], y[l]]], {l, 1, n}];
Table[r[l] = N[Sqrt[x[l]^2 + y[l]^2]], {l, 1, n}];
Clear[u, U];
(* Investigate the solution of the membrane equation in polar \
coordinates r,\[CurlyPhi]
D[u[r, \[CurlyPhi]], {r, 2}] + D[u[r, \[CurlyPhi]], r]/r + D[u[r, \
\[CurlyPhi]], {\[CurlyPhi], 2}]/r^2 + k^2*u[r, \[CurlyPhi]] == 0 *)
Clear[u, U]; u[r, \[CurlyPhi]] = U[r]*Cos[m*\[CurlyPhi]];
\
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Clear[u]; u[r, \[CurlyPhi]] = (A[m]*BesselJ[m, k*r] + \
B[m]*BesselY[m, k*r])*Cos[m*\[CurlyPhi]]\
\>", "Input",
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(* Thus this solution can describe a circular ring \
membrane, which is topological identical with the Masu membrane. The \
FABER theorem on the lowest eigenvalue of a membrane of arbitrary \
form but same surface area, delivers the lower bound of the \
eigenvalue of the Masu membrane. For the circular ring membrane of \
outer radius ra=1.09769 and inner radius ri = 0.5, which has the same \
area as the Masu membrane, the eigenvalue k is given by :
BesselJ[p,k*ra]*BesselY[p,k*ri]-BesselY[p,k*ra]*BesselJ[p,k*ri]==0.
The homogeneous boundary conditions on both boundaries in the \
collcation points in polar coordinates now read :
u[r[l],\[CurlyPhi][l]]==0 for {l,1,n}
This gives a homogeneous system of linear equations for the unknown \
A[m] and B[m]. In order that this system has a nontrivial solution, \
its determinant must be zeor. This condition gives the eigenvalue k \
which may be compared with the result of the circular ring membrane. \
The solution u[r,\[CurlyPhi]] should be plotted and checked if it \
satisfeis the boundary conditions in the collocation points *)
\
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