Section 11.3-Motion of a stringby Alain Goriely, goriely@math.arizona.edu, (http://www.math.arizona.edu/~goriely) Abstract: This section illustrates Section 11.3 in Kreyszig 's book (8th ed.)Application Areas/Subjects: Engineering, Applied MathematicsKeywords: String Motion, 1D-Wave equation See Also: Other Worksheets in the same package.Prerequisites: plotsNote: Send me an e-mail (comments-criticisms) if you use this worksheet.restart;assume(n,integer):with(plots): setoptions(thickness=2): #set the tickness of the lines in the plots

Let NiMvJSJ1Ry1GJDYkJSJ4RyUidEc= be the vertical displacement of the string. Under suitable assumptions (see 11.2), the displacement obeys the following PDEs (the so-called 1D wave equation):NiMvLSUlZGlmZkc2JCUidUctJSIkRzYkJSJ0RyIiIyomJSJjR0YsLUYlNiRGJy1GKTYkJSJ4R0YsIiIiwhere NiMlImNH is the speed of the wave (NiMvKiQlImNHIiIjKiYlJHJob0ciIiIlIlRHISIi : density of the string/ tension in the string) The string is attached at NiMvJSJ4RyUiTEc=, that is we have the boundary conditions NiMvLSUidUc2JCIiISUidEdGJw== and NiMvLSUidUc2JCUiTEclInRHIiIh Here we explore different solutions for the string, starting with initial data. Look at the animations of the string!

The following functions are the solutions for boundary conditions NiMvLSUidUc2JCIiISUidEdGJw== and NiMvLSUidUc2JCUiTEclInRHIiIhSu:=u=(A[n]*cos(lambda[n]*t)+B[n]*sin(lambda[n]*t))*sin(n*Pi*x/L);Slambda:=lambda[n]=c*n*Pi/L;Let us verify that this is indeed a solution of the equation:Diff(u,`$`(t,2)) - c^2*Diff(u,`$`(x,2))=eval(subs(Su,diff(u,`$`(t,2)) - c^2*diff(u,`$`(x,2))));We want to see what these modes look like: Start with n=1, the FUNDAMENTAL MODE:

M1:=subs(n=1,L=2*Pi,c=2,subs(Su,Slambda,B[n]=1,A[n]=0,u));plot(subs(t=Pi/2,M1),x=0..2*Pi,thickness=3);

If we start with the fundamental mode at time t=0, it will evolve in the following way: animate( M1,x=0..2*Pi,t=Pi/2..2*Pi+Pi/2,frames=30,color=red,thickness=3);

See the effect of higher modes (also called harmonics)M2:=subs(n=2,L=2*Pi,c=2,subs(Su,Slambda,B[n]=0,A[n]=1,u)); M3:=subs(n=3,L=2*Pi,c=2,subs(Su,Slambda,B[n]=0,A[n]=1,u)); M4:=subs(n=4,L=2*Pi,c=2,subs(Su,Slambda,B[n]=0,A[n]=1,u)); M5:=subs(n=5,L=2*Pi,c=2,subs(Su,Slambda,B[n]=0,A[n]=1,u));plot(subs(t=0,[M2,M3,M4,M5]),x=0..2*Pi,title="Higher modes, n=2,3,4",thickness=3);

animate( {M2,M4},x=0..2*Pi,t=0..Pi,frames=30,color=red,thickness=3);Observe the presence of nodes: The Nth mode has (N-1) nodes (places at which the string does not move).Observe also that the mode N=4 oscillates twice as fast as the mode N=2.

Take NiMvJSJMRyomIiIjIiIiJSNQaUdGJw== and NiMlImZH like:f:=(1-Heaviside(x-Pi/2))*x+Heaviside(x-Pi/2)*(Pi-x);plot(f,x=0..Pi,thickness=3);Compute the coefficients of the solutionNiMvLSUidUc2JCUieEclInRHLSUkc3VtRzYkKigmJSJBRzYjJSJuRyIiIi0lJGNvc0c2IyomJiUnbGFtYmRhR0YvRjFGKEYxRjEtJSRzaW5HNiMqJkYwRjFGJ0YxRjEvRjA7RjElKWluZmluaXR5Rw==with NiQvJSJjRyIiIi8mJSdsYW1iZGFHNiMlIm5HRio=The Coefficients NiMmJSJBRzYjJSJuRw== are given by:NiMvJiUiQUc2IyUibkcqKCIiIyIiIiUjUGlHISIiLSUkaW50RzYkKiYtJSJmRzYjJSJ4R0YqLSUkc2luRzYjKiZGJ0YqRjRGKkYqL0Y0OyIiIUYrRio=A:= 2/Pi*(int(x*sin(m*x),x=0.. Pi/2)+int((Pi-x)*sin(m*x),x=Pi/2..Pi));S:=(N,x)->sum(subs(m=k,A)*sin(k*x)*cos(k*t),k=1..N);Consider increasing approximations for t=0, N=4, 10, 20plot(subs(t=0,[S(4,x),S(10,x), S(20,x)]),x=0..Pi,color=[blue,black,red],thickness=3);Take N=100 for a good approximationplot(subs(t=0,[S(100,x)]),x=0..Pi,color=[red],thickness=3);St:=S(100,x):Now, we take a few snapshots at regular intervals t=0,plot([subs(t=0,St),subs(t=Pi/6,St),subs(t=2*Pi/6,St), subs(t=3*Pi/6,St),subs(t=4*Pi/6,St),subs(t=5*Pi/6,St),subs(t=6*Pi/6,St)],x=0..Pi,color=red,thickness=3);

animate( St,x=0..Pi,t=0..2*Pi,frames=31,color=blue,thickness=3);

f:=Pi*x-x^2;A:= 2/Pi*int(f*sin(m*x),x=0.. Pi); S:=(N,x)->sum(subs(m=k,A)*sin(k*x)*cos(k*t),k=1..N): St:=S(100,x):animate( St,x=0..Pi,t=0..2*Pi,frames=31,color=green,thickness=3);

f:=x*(Pi*x-x^2);A:= 2/Pi*int(f*sin(m*x),x=0.. Pi); S:=(N,x)->sum(subs(m=k,A)*sin(k*x)*cos(k*t),k=1..N): St:=S(20,x):animate( St,x=0..Pi,t=0..2*Pi,frames=31,color=magenta,thickness=3);

f:=x^3*(Pi*x-x^2);A:= 2/Pi*int(f*sin(m*x),x=0.. Pi); S:=(N,x)->sum(subs(m=k,A)*sin(k*x)*cos(k*t),k=1..N): St:=S(20,x):animate( St,x=0..Pi,t=0..2*Pi,frames=31,color=yellow,thickness=3);

A:= 2/Pi*(int(x/2*sin(m*x),x=0.. 1/2)+int((1/2-x/2)*sin(m*x),x=1/2..1)); S:=(N,x)->sum(subs(m=k,A)*sin(k*x)*cos(k*t),k=1..N): St:=S(150,x):animate( St,x=0..Pi,t=0..2*Pi,frames=31,color=gold,thickness=3);

A:= 2/Pi*(int(sin(Pi*x*20)*sin(m*x),x=0..1/20)); S:=(N,x)->sum(subs(m=k,A)*sin(k*x)*cos(k*t),k=1..N): St:=S(200,x):animate( St,x=0..Pi,t=0..2*Pi,frames=51,color=blue,thickness=2,numpoints=200);

E. Kreyszig : Advanced Engineering Mathematics (8th Edition) John Wiley New York (1999)Disclaimer: 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.