Partial Differential EquationsHigher-dimensional PDE: Vibrating rectangular membranes and nodes.Anton DzhamayDepartment of MathematicsThe University of MichiganAnn Arbor, MI 48109wPage: http://www.math.lsa.umich.edu/~adzhamemail: adzham@umich.eduCopyright \251 2004 by Anton DzhamayAll rights reserved

In this worksheet we consider some examples of the vibrating patterns of rectangular membranes. Such membranes are described by the 2-dimensional wave equation NiMvLSUlZGlmZkc2JS0lInVHNiUlInhHJSJ5RyUidEdGLEYsKiYlImNHIiIjLSUmRGVsdGFHNiNGJyIiIg==. we are mainly interested in the product solution obtained by the method of separation of variables, such product solutions of the wave equations are also called standing waves. In particular, we consider intricate patterns of nodal curves appearing when there is more than one eigenfunction corresponding to the same eigenvalue (this happens, for example, for a square membrane.

Some packages that we use in this worksheet:restart: with(plottools): with(plots):

First we define the spatial eigenfunction NiMtJiUkUGhpRzYkJSJuRyUibUc2JCUieEclInlH of the Dirichlet boundary problem:Phi:=unapply(sin(n*Pi*x/L)*sin(m*Pi*y/H),n,m):'Phi[n,m](x,y)'=Phi(n,m);The corresponding eigenvalue NiMmJSdsYW1iZGFHNiQlIm5HJSJtRw== then is lambda:=unapply((n*Pi/L)^2+(m*Pi/H)^2,n,m):'lambda[n,m]'=lambda(n,m);The corresponding time-dependent term is T:=unapply(A[n,m]*cos(c*sqrt(lambda(n,m))*t)+B[n,m]*sin(c*sqrt(lambda(n,m))*t),n,m):T(n,m);The product solution then is u:=unapply(Phi(n,m)*(T(n,m)),n,m):u(n,m);The general solution is the superposition of basic solutions:sum(sum(u(n,m),n=1..infinity),m=1..infinity);In this worksheet we consider only initial displacements:B[n,m]:=0;A[n,m]:=1;And thereforeT:=unapply(A[n,m]*cos(c*sqrt(lambda(n,m))*t)+B[n,m]*sin(c*sqrt(lambda(n,m))*t),n,m):T(n,m);and the product solution is simplyu:=unapply(Phi(n,m)*(T(n,m)),n,m):u(n,m);The period of (NiQlIm5HJSJtRw== )-oscillation isP:=unapply( (2*Pi)/(c*sqrt(lambda(n,m))),n,m):P(n,m);We also introduce the following procedure that will help us to plot solutions together with corresponding nodal curves:wave:=proc(sol) display([ contourplot3d(sol,x=-0.01..1.01*L,y=-0.01..1.01*H,contours=[0],color=red,thickness=3,numpoints=600), plot3d(sol,x=0..L,y=0..H,shading=zhue)],scaling=constrained,axes=boxed); end:

For a generic rectangle eigenvalues are simple.c:=1:L:=2:H:=3:This is the "lowest energy" mode with 'lambda(1,1)'=lambda(1,1);We consider the Dirichlet boundary conditions, so the boundary of the membrane does not move.animate(wave,[u(1,1)],t=0..P(1,1));Note that NiQvJSJuRyIiIi8lIm1HIiIj mode has a fixed line through the middle (at NiMvJSJ5RyomJSJIRyIiIiIiIyEiIg==). Such a line (time-independent zero-level of NiMtJSJ1RzYlJSJ4RyUieUclInRH) is called the nodal curve.animate(wave,[u(1,2)],t=0..P(1,2));There are two nodal curves for NiQvJSJuRyIiJC8lIm1HIiIi mode:animate(wave,[u(3,1)],t=0..P(3,1));And this is how the nodal curves for NiQvJSJuRyIiJC8lIm1HIiIj mode lookslike:animate(wave,[u(3,2)],t=0..P(3,2));Same oscillations viewed from the top:display(animate(wave,[u(3,2)],t=0..P(3,2)),orientation=[-90,0],insequence=true);Note that for general NiMlIkxH and NiMlIkhH different modes have different eigenvalues and therefore have different frequences of oscillation:'lambda(2,3)'=lambda(2,3);'lambda(3,2)'=lambda(3,2);Such a combination will not be a standing wave and will not have nodal curves (the zero-levels of NiMtJSJ1RzYlJSJ4RyUieUclInRH are time-dependent:)display(animate(wave,[u(2,3)+u(3,2)],t=0..2*P(2,3)*P(3,2)),orientation=[-90,0],insequence=true,frames=1000);Clean-up:c:='c':L:='L':H:='H':

For a square membrane different modes can have the same eigenvalue, which leads to non-trivial nodal curves and interesting vibrating patterns.c:=1:L:=2:H:=2:animate(wave,[u(1,3)],t=0..P(1,3),title="u[1,3]");animate(wave,[u(3,1)],t=0..P(3,1),title="u[3,1]");Note that the eigenvalues NiMmJSdsYW1iZGFHNiQiIiIiIiQ= and NiMmJSdsYW1iZGFHNiQiIiQiIiI= are the same:'lambda[1,3]'=lambda(1,3); 'lambda[3,1]'=lambda(3,1);The sum of two eigenfunctions with the same eigenvalue is still an eigenfunction with the same eigenvalue (and so adding a time-dependent term again produces a standing wave), but the nodal curve and the vibration pattern is very different from the "product" behavoir of individual eigenfunctions:animate(wave,[u(1,3)+u(3,1)],t=0..P(1,3),title="u[1,3]+u[3,1]");We collect a few more interesting examples below:animate(wave,[u(1,3)-u(3,1)],t=0..P(1,3),title="u[1,3]-u[3,1]");animate(wave,[u(1,3)+u(3,1)/3],t=0..P(1,3),title="u[1,3]+u[3,1]/3");animate(wave,[u(1,3)-2/3*u(3,1)],t=0..P(1,3),title="u[1,3]-2/3*u[3,1]");animate(wave,[u(2,3)+u(3,2)],t=0..P(2,3),title="u[2,3]+u[3,2]");animate(wave,[u(2,3)+u(3,2)/3],t=0..P(2,3),title="u[2,3]+u[3,2]/3");animate(wave,[u(1,4)+u(4,1)],t=0..P(1,4),title="u[1,4]+u[4,1]");animate(wave,[u(1,4)+sqrt(2/3)*u(4,1)],t=0..P(1,4),title="u[1,4]+sqrt(2/3)*u[4,1]");animate(wave,[u(1,4)+sqrt(2/3)*u(4,1)/3],t=0..P(1,4),title="u[1,4]+sqrt(2/3)*u[4,1]/3");display(animate(wave,[u(6,1)+u(1,6)],t=0..P(1,6)),orientation=[-90,0],insequence=true,frames=1000, title="u[6,1]+u[1,6]");display(animate(wave,[u(6,1)+0.5*u(1,6)],t=0..P(1,6)),orientation=[-90,0],insequence=true,frames=1000, title="u[6,1]+u[1,6]/2");

Walter A. Strauss, Partial Differential Equations: An Introduction, Wiley, 1992Richard Haberman, Elementary Applied Partial Differential Equations, 3rd edition, Prentice HallStanley J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover 1982

"While every effort has been made to validate the solutions in this worksheet, Waterloo Maple Inc. and the contributors are not responsible for any errors contained and are not liable for any damages resulting from the use of this material."